# On statistical Calderón problems

Kweku Abraham and Richard Nickl

(University of Cambridge)

###### Abstract

For $D$ a bounded domain in $\mathbb{R}^{d},d\geq 2,$ with smooth boundary
$\partial D$, the non-linear inverse problem of recovering the unknown
conductivity $\gamma$ determining solutions $u=u_{\gamma,f}$ of the partial
differential equation

$\begin{split}\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}u)&=0\quad\text{
in }D,\\\ u&=f\quad\text{ on }\partial D,\end{split}$

from noisy observations $Y$ of the Dirichlet-to-Neumann map

$f\mapsto\Lambda_{\gamma}(f)={\gamma\frac{\partial
u_{\gamma,f}}{\partial\nu}}\Big{|}_{\partial D},$

with $\partial/\partial\nu$ denoting the outward normal derivative, is
considered. The data $Y$ consists of $\Lambda_{\gamma}$ corrupted by additive
Gaussian noise at noise level $\varepsilon>0$, and a statistical algorithm
$\hat{\gamma}(Y)$ is constructed which is shown to recover $\gamma$ in
supremum-norm loss at a statistical convergence rate of the order
$\log(1/\varepsilon)^{-\delta}$ as $\varepsilon\to 0$. It is further shown
that this convergence rate is optimal, up to the precise value of the exponent
$\delta>0$, in an information theoretic sense. The estimator $\hat{\gamma}(Y)$
has a Bayesian interpretation in terms of the posterior mean of a suitable
Gaussian process prior and can be computed by MCMC methods.

Keywords: non-linear inverse problems, elliptic partial differential
equations, electrical impedance tomography, asymptotics of nonparametric Bayes
procedures

###### Contents

  1. 1 Introduction
  2. 2 Noise model and electrical impedance tomography
  3. 3 The Bayesian approach to the noisy Calderón problem
    1. 3.1 Prior construction
    2. 3.2 Posterior contraction result
  4. 4 Concluding remarks
  5. 5 Proofs
    1. 5.1 Low rank approximation of $\tilde{\Lambda}_{\gamma}$
    2. 5.2 Forward and inverse continuity results
    3. 5.3 Tests and prior support properties
    4. 5.4 Posterior asymptotics
      1. 5.4.1 Posterior regularity and contraction about $\Lambda_{\gamma_{0}}$
      2. 5.4.2 Proof of Theorem 3
    5. 5.5 Proof of the lower bound Theorem 2
  6. A Laplace–Beltrami eigenfunctions and Sobolev spaces
  7. B Comparison results for Hilbert–Schmidt operators
  8. C Mapping properties of $\Lambda_{\gamma}$ and $\tilde{\Lambda}_{\gamma}$
  9. D Asymptotic comparison of noise models
    1. D.1 A brief overview of the Le Cam distance
    2. D.2 Le Cam deficiency bounds for noisy Calderón problems
  10. E Notation index

##  1 Introduction

Let $D\subset\mathbb{R}^{d},d\geq 2,$ be a bounded domain, which we understand
here to be a connected open set with smooth boundary $\partial D$. For
$\gamma:D\to(0,\infty)$ a conductivity coefficient, consider solutions $u$ to
the _Dirichlet problem_

$\begin{split}\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}u)&=0\quad\text{
in }D,\\\ u&=f\quad\text{ on }\partial D,\end{split}$ (1)

where $\operatorname{\nabla}$ denotes the usual gradient operator and where
$f:\partial D\to\mathbb{C}$ prescribes some boundary values. The parameter
spaces considered in the sequel are of the form

$\displaystyle\Gamma_{m,D^{\prime}}$ $\displaystyle=\Big{\\{}\gamma\in
C(D):\inf_{x\in D}\gamma(x)\geq m,~{}\gamma=1\text{ on }D\setminus
D^{\prime}\Big{\\}},$ (2) $\displaystyle\Gamma^{\alpha}_{m,D^{\prime}}(M)$
$\displaystyle=\\{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{H^{\alpha}(D)}\leq
M\\},\quad M>0,$ (3)

where $m\in(0,1)$ is a fixed constant, $D^{\prime}$ is a domain compactly
supported in $D$ (that is, its closure $\bar{D^{\prime}}$ is contained in
$D$), and $\alpha\geq 0$ measures the regularity of $\gamma$ in the Sobolev
scale. The Sobolev spaces $H^{\alpha}(D),H^{\alpha}(\partial D)$ of complex-
valued functions (and variations thereof) are defined in detail in Appendix A;
the standard $L^{2}(D),L^{2}(\partial D)$ Lebesgue spaces arise as the case
$\alpha=0$, with inner products
$\langle\cdot,\cdot\rangle_{L^{2}(D)},\langle\cdot,\cdot\rangle_{L^{2}(\partial
D)}$, respectively, and $C(D)$ denotes the space of bounded continuous real-
valued functions on $D$, equipped with the sup-norm
$\lVert\cdot\rVert_{\infty}$. Except where otherwise stated, all integrals are
taken with respect to Lebesgue and surface measures on $D$ and $\partial D$
respectively.

The elliptic partial differential equation (PDE) in (1) has, for
$\gamma\in\Gamma_{m,D^{\prime}}$ and $f\in H^{s+1}(\partial D)/\mathbb{C},$
$s\in\mathbb{R}$, a unique weak solution $u_{\gamma,f}$ in the space

$\mathcal{H}_{s}:=\big{(}H^{\min\\{1,s+3/2\\}}(D)\cap
H^{1}_{\textnormal{loc}}(D)\big{)}/\mathbb{C};$

that is (for $\overline{v}$ denoting the complex conjugate of $v$), the
equations

$\begin{split}\int_{D}\gamma\operatorname{\nabla}u\cdot\operatorname{\nabla}\overline{v}&=0\quad\forall
v\in H_{0}^{1}(D),\\\ u&=f\quad\text{ on }\partial D,\end{split}$ (4)

hold simultaneously (for $u\in\mathcal{H}_{s}$) if and only if
$u=u_{\gamma,f}$, where the boundary values of $u$ are defined in a trace
sense. Here and below $/\mathbb{C}$ means that we identify functions $f,f+c$
which are equal up to a scalar $c\in\mathbb{C}$. See Lemma 19 in Appendix C
and its proof for details.

Given a solution $u_{\gamma,f}$ to the Dirichlet problem, one can measure the
_Neumann (boundary) data_

${\gamma\frac{\partial u_{\gamma,f}}{\partial\nu}}\Big{|}_{\partial
D}\equiv{\frac{\partial u_{\gamma,f}}{\partial\nu}}\Big{|}_{\partial
D},~{}~{}\gamma\in\Gamma_{m,D^{\prime}},$

where $\frac{\partial}{\partial\nu}$ denotes the outward normal derivative on
$\partial D$ (again to be understood in a trace sense). It can be shown (see
Lemma 20) that for any $s\in\mathbb{R}$ and any $f\in H^{s+1}(\partial
D)/\mathbb{C}$, the Neumann data lies in the space

$H^{s}_{\diamond}(\partial D):=\\{g\in H^{s}(\partial D):\langle
g,1\rangle_{L^{2}(\partial D)}=0\\}.$ (5)

Thus, we may define the so-called _Dirichlet-to-Neumann map_ ,

$\begin{split}\Lambda_{\gamma}:\quad H^{s+1}(\partial D)/\mathbb{C}&\to
H^{s}_{\diamond}(\partial D),\\\ f&\mapsto{\gamma\frac{\partial
u_{\gamma,f}}{\partial\nu}}\Big{|}_{\partial D},\end{split}$ (6)

which associates to each prescribed boundary value $f$ the Neumann data of the
solution of the PDE 1. The choice to quotient the domain of $\Lambda_{\gamma}$
by $\mathbb{C}$ is natural as the Neumann data is invariant with respect to
addition of scalars.

The _Calderón problem_ [5] is a well studied inverse problem that addresses
the task of recovering interior conductivities $\gamma$ from knowledge of the
boundary data $\Lambda_{\gamma}$. Note that while $\Lambda_{\gamma}$ itself is
a linear operator between Hilbert spaces, the ‘forward map’
$\gamma\to\Lambda_{\gamma}$ is non-linear. Landmark injectivity results by
Sylvester and Uhlmann ($d\geq 3$) and by Nachman ($d=2$) show, however, that
recovery is in principle possible.

######  Theorem (Sylvester & Uhlmann, [45]; Nachman [31]).

If $\Lambda_{\gamma_{1}}=\Lambda_{\gamma_{2}}$ then $\gamma_{1}=\gamma_{2}$.

Nachman [32] and Novikov [37] studied elaborate inversion algorithms that
allow recovery of $\gamma$ if exact knowledge of the entire operator
$\Lambda_{\gamma}$ is available. Moreover Alessandrini [2] (and later Novikov
and Santacesaria [39] for $d=2$) gave ‘stability estimates’ providing
quantitative continuity bounds for the inverse map, and Mandache [28] gave an
‘instability estimate’ showing that these bounds are nearly sharp.

The Calderón problem has since been vigorously studied and an excellent survey
can be found in Uhlmann [47] and also in the lecture notes by Salo [43]. Its
importance partly stems from its applications to electrical impedance
tomography (EIT) – described in more detail in the next section – where
_discrete_ boundary measurements of the operator $\Lambda_{\gamma}$ are
performed to infer the interior conductivity $\gamma$. Any such data comes
with error, and arguably the most natural mathematical description of such
approximate measurements is by a statistical noise model. As the superposition
of many independent errors is well described by a normal distribution (via the
central limit theorem), it is further natural to postulate that this noise
follows a Gaussian law. In algorithmic practice this has already been widely
acknowledged in the general setting of inverse problems, where statistical,
and in particular Bayesian, inversion approaches have flourished in the last
decade since the influential work of Stuart [44]. In the context of EIT we
refer to the articles [22, 21, 24, 42, 12, 10] and the many references
therein. Currently, little theory giving statistical guarantees for the
performance of such Bayesian de-noising methodology is available, particularly
for non-linear problems. Some recent progress has been made in non-linear
settings (see [49, 36, 33, 35, 34, 30, 17]) but no results are available at
present for the Calderón problem described above, and the purpose of the
present paper is to at least partially fill this gap.

We will introduce three natural noise models for such statistical Caldéron
problems, all asymptotically closely related, in the next section. We prove
our main theorems initially in one of these models, and show in Section D.2
that the results are in fact valid in the other models too. The preferred
model for the theoretical development is 12, wherein one observes
$\Lambda_{\gamma}$ corrupted by a Gaussian white noise in an appropriate space
of Hilbert–Schmidt operators. The noise is described by the scalar quantity
$\varepsilon>0$ governing its magnitude and a parameter $r\in\mathbb{R}$
determining its ‘spectral heteroscedasticity’. If we denote by
$P^{\gamma}_{\varepsilon}=P^{\gamma}_{\varepsilon,r}$ the resulting
probability law of the noisy observations $Y$ of $\Lambda_{\gamma}$, then our
main results can be summarised in the following two theorems.

######  Theorem 1.

Let $\alpha>3+d$ be an integer, let $m_{0}\in(0,1)$, $M>1$ be given, and let
$D_{0}$ be a domain in $\mathbb{R}^{d}$ such that $\bar{D}_{0}$ is contained
in $D$.

There exists a measurable function
$\hat{\gamma}=\hat{\gamma}_{\varepsilon}(Y)$ of the observations $Y\sim
P^{\gamma}_{\varepsilon}$ such that

$\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P^{\gamma}_{\varepsilon}(\lVert\hat{\gamma}-\gamma\rVert_{\infty}>C\log(1/\varepsilon)^{-\delta})\to
0,~{}\text{as }\varepsilon\to 0,$

where $\delta>0$ depends only on $d$ and $\alpha$, and $C$ depends only on
$\alpha,$ $M$, $m_{0}$, $D$, $D_{0}$ and $r$.

The estimator $\hat{\gamma}$ in the previous theorem has a natural Bayesian
interpretation in terms of the posterior mean of a suitable Gaussian process
based prior for $\gamma$. Such priors are most effective for recovery of
sufficiently smooth $\gamma$, and the precise bound on $\alpha$ is chosen here
for convenience (see Remark 3 for further discussion). The derivation and
implementation of $\hat{\gamma}$ are described in Section 3, where we give the
more concrete Theorem 3, which implies Theorem 1. We note that $\hat{\gamma}$
can be calculated without knowledge of the bound $M$ for
$\lVert\gamma\rVert_{H^{\alpha}(D)}$.

The slow (logarithmic) convergence rate is not surprising in view of the
folklore that the Calderón problem is a severely ill-posed inverse problem
(cf. also [28]). The following result makes this folklore information-
theoretically precise – it shows that the convergence rate obtained by the
estimator $\hat{\gamma}$ is optimal in the statistical minimax sense, at least
up to the precise value of the exponent $\delta$, for the prototypical case
where $D_{0},D$ are nested balls in $\mathbb{R}^{d}$. We denote by
$\lVert\cdot\rVert$ the standard Euclidean norm on $\mathbb{R}^{d}$.

######  Theorem 2.

Let $D_{0}=\\{x\in\mathbb{R}^{d}:\lVert x\rVert<1/2\\}\subset
D=\\{x\in\mathbb{R}^{d}:\lVert x\rVert<1\\}$, let $\alpha$ be an integer
greater than $2$, and let $m_{0}\in(0,1)$ be arbitrary.

For any $\delta^{\prime}>\alpha(2d-1)/d$ and all $M$ large enough there exists
$c=c(\delta^{\prime},\alpha,d,m_{0},r,M)$ such that

$\inf_{\tilde{\gamma}}\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P^{\gamma}_{\varepsilon}(\lVert\tilde{\gamma}-\gamma\rVert_{\infty}>c\log(1/\varepsilon)^{-\delta^{\prime}})>1/4$

for all $\varepsilon$ small enough, where the infimum extends over all
measurable functions $\tilde{\gamma}=\tilde{\gamma}(Y)$ of the data $Y\sim
P^{\gamma}_{\varepsilon}$.

The particular value $1/4$ in the lower bound is chosen for convenience.
Determining the exact exponent $\delta$ in the minimax convergence rate is a
delicate PDE question related directly to the stability estimates of [2, 39],
and beyond the scope of the present paper. For $d\geq 3$, one could use an
explicit bound of Novikov [38] to show that a choice of $\delta$ satisfying
$\delta/\alpha\to 1$ as $\alpha\to\infty$ is permitted in Theorem 1. This
scales proportionally (in the regularity index $\alpha$) to the exponent
$\delta^{\prime}=\alpha(2d-1)/d$ from the lower bound Theorem 2.

This paper is structured as follows. In Section 2 we introduce the measurement
model we consider in our theorems, and discuss its relationship to physical
measurement models arising in medical imaging practice. In Section 3 we give
the construction of the Bayesian algorithm $\hat{\gamma}$ that solves our
noisy version of the Calderón problem. Section 4 contains some concluding
remarks concerning the main theorems. All proofs and related background
material are relegated to later sections. For convenience, the notation used
is informally gathered in Appendix E.

##  2 Noise model and electrical impedance tomography

We now introduce three scenarios for noisy observations of the operator
$\Lambda_{\gamma}$ from (6), and discuss their relationship at the end of this
subsection. Define

$\tilde{\Lambda}_{\gamma}=\Lambda_{\gamma}-\Lambda_{1}$

where the fixed (deterministic and known) operator $\Lambda_{1}$ is the
Dirichlet-to-Neumann map for the standard Laplace equation, that is, eq. 1
with $\gamma=1$ identically on $D$. We then equivalently consider measuring a
noisy version of $\tilde{\Lambda}_{\gamma}$.

Real-world data involving the Calderón problem arises for example in medical
imaging, namely in _electrical impedance tomography_ , see [22, 21, 24, 42,
12, 10] and references therein. Electrodes are attached to a patient (or some
other physical medium), and are used both to apply voltages and to record the
resulting currents. If we assume the applied voltages are uniform across the
surface of any given electrode, and the electrodes measure the average current
across their surface, we are led to the observation model

$Y_{p,q}=\langle\tilde{\Lambda}_{\gamma}[\psi_{p}],\psi_{q}\rangle_{L^{2}(\partial
D)}+\varepsilon g_{p,q},\quad p,q\leq P,\quad
g_{p,q}\overset{iid}{\sim}N(0,1),~{}\varepsilon>0,$ (7)

where the $\psi_{p}$ are, up to scaling factors, indicator functions
$\mathbbm{1}_{I_{p}}$ of some disjoint measurable subsets $(I_{p})_{p\leq P}$
of $\partial D$ representing the locations of the electrodes. Throughout
$N(0,1)$ denotes the standard normal distribution. In principle the noise
level $\varepsilon>0$ could vary with $p$ and $q$, but choosing scaling
factors $c_{p}$ so that the $\psi_{p}=c_{p}\mathbbm{1}_{I_{p}}$ are
$L^{2}(\partial D)$-orthonormal we expect to be able to realise the above
homoscedastic noise model. Also note that while the inner product
$\langle\cdot,\cdot\rangle_{L^{2}(\partial D)}$ is defined with respect to
complex scalars, $\tilde{\Lambda}_{\gamma}[\psi_{p}]$ takes real values since
$\psi_{p}$ does (see before Lemma 21), and hence it is natural to consider
real-valued noise $g_{p,q}$ and data $Y_{p,q}$.

An alternative noise model considers spectral measurements. Denote by
$(\phi_{k}=\phi_{k}^{(0)}:k\in\mathbb{N}\cup\\{0\\})$ an orthonormal basis of
$L^{2}(\partial D)$ consisting of real-valued eigenfunctions of the
Laplace–Beltrami operator on the compact manifold $\partial D$, described in
more detail in Appendix A. [If $D$ is a disc in $\mathbb{R}^{2}$ these
comprise the usual trigonometric basis, while for $D$ a ball in
$\mathbb{R}^{3}$, they are the spherical harmonics.] By discarding the
constant function $\phi_{0}$ we obtain a basis of the spaces $L^{2}(\partial
D)/\mathbb{C}$ and $L^{2}_{\diamond}(\partial D)=H^{0}_{\diamond}(\partial
D)$. Moreover, appropriate rescaling of these basis functions also provides
orthonormal bases $(\phi_{k}^{(r)}:{k\in\mathbb{N}})$ of all $H^{r}(\partial
D)/\mathbb{C}$ and $H^{r}_{\diamond}(\partial D)$ spaces, $r\in\mathbb{R}$.
For some $r\in\mathbb{R}$, we then consider the noisy matrix measurement model

$Y_{j,k}=\langle\tilde{\Lambda}_{\gamma}[\phi_{j}^{(r)}],\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}+\varepsilon g_{j,k},\quad j\leq J,k\leq K,\quad
g_{j,k}\overset{iid}{\sim}N(0,1),~{}\varepsilon>0,$ (8)

where again it is natural to consider real-valued noise $g_{j,k}$ only. The
parameter $r$ can in principle be chosen by the experimenter and reflects how
the signal-to-noise ratio varies with frequency: as $r$ increases, the signal
at high frequencies (i.e. at larger values of $j$) decreases compared to the
signal at low frequencies. Likely the most realistic choices are $r=0$ (which
will allow for comparison of the models 7 and 8), and $r=1$, in which case the
signal-to-noise ratio is the same across all frequencies: since
$\Lambda_{\gamma}$ maps $H^{1}(\partial D)/\mathbb{C}$ to $L^{2}(\partial D)$
isomorphically (Lemma 20), the signal magnitude
$\lVert\Lambda_{\gamma}[\phi_{j}^{(1)}]\rVert_{L^{2}(\partial D)}$ is of order
$1$ for all $j$. A similar reasoning
($\lVert\phi_{k}^{(0)}\rVert_{L^{2}(\partial D)}=1$ for all $k$) underpins the
choice of the $L^{2}(\partial D)$-inner product in 8.

If we formally take the limit $J,K\to\infty$ in 8, we obtain a model of
Gaussian white noise on a space of Hilbert–Schmidt operators as follows. For
$j,k\in\mathbb{N}$, let $b_{jk}^{(r)}:H^{r}(\partial D)\to L^{2}(\partial D)$
denote the tensor product operator

$b_{jk}^{(r)}(f)=\phi_{j}^{(r)}\otimes\phi_{k}^{(0)}(f):=\langle
f,\phi_{j}^{(r)}\rangle_{H^{r}(\partial D)}\phi_{k}^{(0)},\quad f\in
H^{r}(\partial D),$ (9)

and define the space of linear operators

$\mathbb{H}_{r}:=\Big{\\{}T:H^{r}(\partial D)\to L^{2}(\partial
D),~{}T=\sum_{j,k=1}^{\infty}t_{jk}b_{jk}^{(r)}:t_{jk}\in\mathbb{R},\sum_{j,k=1}^{\infty}t_{jk}^{2}<\infty\Big{\\}}.$
(10)

The elements of $\mathbb{H}_{r}$ are the ‘Hilbert–Schmidt’ operators between
(the real-valued subsets of) the Hilbert spaces $H^{r}(\partial D)$ and
$L^{2}(\partial D)$, see [4] Chapter 12. Moreover, $\mathbb{H}_{r}$ is itself
a (real) Hilbert space for the inner product

$\langle
S,T\rangle_{\mathbb{H}_{r}}=\sum_{j,k=1}^{\infty}s_{jk}t_{jk}\equiv\sum_{j,k=1}^{\infty}\langle
S\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}\langle
T\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}.$

We then consider observing a realisation of the Gaussian process

$\begin{split}\big{(}Y(T)&=\langle\tilde{\Lambda}_{\gamma},T\rangle_{\mathbb{H}_{r}}+\varepsilon\mathbb{W}(T):T\in\mathbb{H}_{r}\big{)},~{}\varepsilon>0,\text{
where }\\\
\mathbb{W}(T)&\equiv\langle\mathbb{W},T\rangle_{\mathbb{H}_{r}}:=\sum_{j,k=1}^{\infty}g_{jk}\langle
T\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)},\text{ for
}g_{jk}\overset{iid}{\sim}N(0,1).\end{split}$ (11)

This makes sense rigorously only if
$\tilde{\Lambda}_{\gamma}\in\mathbb{H}_{r}$, and it is proved in Appendix C,
Lemma 21, that this is indeed the case for any
$\gamma\in\Gamma_{m,D^{\prime}}$ and any $r\in\mathbb{R}$.

The process $\mathbb{W}$ so defined is a Gaussian white noise (isonormal
process; see, e.g., p.19 in [15]) indexed by the (real) Hilbert space
$\mathbb{H}_{r}$. A closed form description of the data in 11 is therefore

$Y=\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W},\quad\varepsilon>0.$ (12)

We write $P^{\gamma}_{\varepsilon,r}$ for the law of $Y$ in this last model.
We often suppress the parameter $r$, and write $P^{\gamma}_{\varepsilon}$ for
the probability law and $E^{\gamma}_{\varepsilon}$ for the corresponding
expectation operator.

The continuous model 12 is more convenient for the application of PDE
techniques and facilitates a clearer exposition in the proofs to follow. We
prove our main results Theorems 1 and 2 in that model initially. We will show
that the models 12 and 7 are asymptotically closely related to each other in a
rigorous ‘Le Cam’ sense, and that as a consequence, Theorems 1 and 2 also hold
in the ‘electrode model’ 7: see Theorem 27 for a precise statement. The
intuitive idea can be summarised as follows: Model 12 with $r=0$ contains all
the information available in model 7 simply by evaluating $Y(T)$ with
$T=\psi_{p}\otimes\psi_{q}$ for each $p,q\leq P$ in 11; conversely, as
$P\to\infty$, one can approximate Laplace–Beltrami eigenfunctions via linear
combinations of indicator functions (under appropriate conditions on the sets
$I_{p},p\leq P$), and in doing so, given data from model 7 we approximately
recover data from model 8 and ultimately then also from 12. We refer to
Appendix D for rigorous details, where also the (simpler) equivalence of the
models 8 and 12 is established.

##  3 The Bayesian approach to the noisy Calderón problem

We now construct the estimator $\hat{\gamma}$ featuring in Theorem 1.
Following the Bayesian approach to inverse problems advocated by A. Stuart
[44], we will construct $\hat{\gamma}$ in terms of the posterior mean arising
from a certain Gaussian process prior. In the context of the EIT inverse
problem a Bayesian approach was proposed already in [22], and conceptually
related work appears in fact much earlier in Diaconis [9] who further traces
some of the key ideas back to H. Poincaré – see Chapter XV, §216, in [40] for
what is possibly the first proposal of an infinite-dimensional Gaussian series
prior in a numerical analysis context.

To this end we need to first establish the existence of a posterior
distribution in our measurement setting. In the Gaussian white noise model 12,
the log-likelihood function can be derived from the Cameron–Martin theorem in
a suitable Hilbert space: precisely, the law $P^{\gamma}_{\varepsilon}$ of $Y$
is dominated by the law $P^{1}_{\varepsilon}$ of $\varepsilon\mathbb{W}$, with
log-likelihood function

$\ell(\gamma)\equiv\log
p^{\gamma}_{\varepsilon}(Y):=\log\frac{dP^{\gamma}_{\varepsilon}}{dP^{1}_{\varepsilon}}(Y)=\frac{1}{\varepsilon^{2}}\langle
Y,\tilde{\Lambda}_{\gamma}\rangle_{\mathbb{H}_{r}}-\frac{1}{2\varepsilon^{2}}\lVert\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}^{2},~{}~{}\gamma\in\Gamma_{m,D^{\prime}}.$
(13)

See Section 7.4 in [33] for a detailed derivation, which requires Borel-
measurability (ensured by Lemma 6 below) of the map
$\gamma\mapsto\tilde{\Lambda}_{\gamma}$ from the (Polish) space
$\Gamma_{m,D^{\prime}}$ equipped with the
$\lVert\cdot\rVert_{\infty}$-topology into the Hilbert space $\mathbb{H}_{r}$.

Then for any prior (Borel) probability measure $\Pi$ on
$\Gamma_{m,D^{\prime}}$, the posterior distribution given observations $Y$ is
given by

$\Pi(B\mid
Y)=\frac{\int_{B}p^{\gamma}_{\varepsilon}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)}{\int_{\Gamma_{m,D^{\prime}}}p^{\gamma}_{\varepsilon}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)},\quad
B\subset\Gamma_{m,D^{\prime}}\text{ Borel-measurable},$ (14)

see again Section 7.4 in [33] (and also [14], eq (1.1)). We denote by
$E^{\Pi}[\cdot]$ the expectation operator according to the prior, and by
$E^{\Pi}[\mathrel{\;\cdot\;}\mid Y]$ the expectation according to the
posterior.

What precedes is stated for a prior defined on the conductivities $\gamma$.
The priors introduced below will be of the form $\gamma=\Phi\circ\theta$ for a
suitable link function $\Phi$, where $\theta$ takes values in a _linear_
space, so that a Gaussian process prior can be assigned to $\theta$.
Composition with the map $\Phi$ described in Section 3.1 gives rise to a
measurable bijection between the parameter spaces for $\theta$ and for
$\gamma$, and the comments above therefore equivalently yield the existence of
the posterior distribution for $\theta$.

###  3.1 Prior construction

We define the prior on $\theta$ in terms of a base prior $\Pi^{\prime}$. For
the base prior we assume the following – we refer, e.g., to [15, Sections 2.1
and 2.6] for the basic definitions of Gaussian measures and processes and
their reproducing kernel Hilbert spaces (RKHS).

######  Assumption 1.

Let $\Pi^{\prime}$ be a centred Gaussian Borel probability measure on the
Banach space $C_{u}(D)$ of uniformly continuous real-valued functions on $D$,
and let $\alpha,\beta$ be integers satisfying $\alpha>\beta>2+d/2$. Assume
$\Pi^{\prime}(H^{\beta}(D))=1$ and that the RKHS
$(\mathcal{H},\lVert\cdot\rVert_{\mathcal{H}})$ of $\Pi^{\prime}$ is
continuously embedded into the Sobolev space $H^{\alpha}(D)$.

Natural candidates for such priors are restrictions to $D$ of Gaussian
processes whose covariances are given by Whittle–Matérn kernels, see [14],
p.313 and p.575 – in these cases one can satisfy the assumption for any
$2+d/2<\beta<\alpha-d/2$ by taking $\mathcal{H}$ to coincide with the Sobolev
space $H^{\alpha}(D)$. The restriction to integer-valued $\alpha,\beta$ is
convenient to simplify some proofs.

In the proofs that follow we will require that the true $\gamma_{0}$ is in the
‘interior’ of the support of the induced prior on $\gamma$, so recalling that
Theorem 1 is stated uniformly over $\Gamma^{\alpha}_{m_{0},D_{0}}(M)$, we
choose $0<m_{1}<m_{0}<1$ and a domain $D_{1}$ such that $\bar{D}_{0}\subset
D_{1},\bar{D}_{1}\subset D$. Then let $\zeta:D\to[0,1]$ be a smooth cutoff
function, identically one on $D_{0}$ and compactly supported in $D_{1}$. For a
link function $\Phi:\mathbb{R}\to(m_{1},\infty)$ to be specified, we define
the prior $\Pi=\Pi_{\varepsilon}$ as the (Borel) law in $C_{u}(D)$ of the
random function

$\gamma=\Phi\circ\theta,~{}~{}\theta(x)=\theta_{\varepsilon}(x)=\varepsilon^{d/(\alpha+d)}\zeta(x)\theta^{\prime}(x),~{}x\in
D,~{}\theta^{\prime}\sim\Pi^{\prime},$ (15)

where, in a slight abuse of notation, we use the notation $\Pi$ for the prior
laws both of $\theta$ and of the induced conductivity
$\gamma=\Phi\circ\theta$. The link function $\Phi$ will be required to be
regular in the sense of [34], that is to say, $\Phi$ is a smooth bijective
function satisfying $\Phi(0)=1$, $\Phi^{\prime}>0$ on $\mathbb{R}$, and
$\lVert\Phi^{(j)}\rVert_{\infty}<\infty$ for all integers $j\geq 1$, with
inverse function denoted by $\Phi^{-1}:(m_{1},\infty)\to\mathbb{R}$. We refer
to [34] Example 8 where a regular link function is exhibited, and to [34]
Lemma 29 for basic properties of such functions. In particular we note that
there are constants $C=C(\Phi)$, $c=c(\Phi,m_{0},m)$,
$C^{\prime}=C^{\prime}(\Phi,\alpha)$ and
$c^{\prime}=c^{\prime}(\Phi,\alpha,m_{0},m)$ such that for any bounded
functions $\theta,\theta_{0}$, any integer $\alpha\geq d/2$ and any
$\gamma_{0},\gamma\in\Gamma_{m,D_{1}},~{}m\in(m_{1},m_{0})$,

$\displaystyle\lVert\Phi\circ\theta-\Phi\circ\theta_{0}\rVert_{\infty}$
$\displaystyle\leq C\lVert\theta-\theta_{0}\rVert_{\infty},$ (16)
$\displaystyle\lVert\Phi^{-1}\circ\gamma-\Phi^{-1}\circ\gamma_{0}\rVert_{\infty}$
$\displaystyle\leq c\lVert\gamma-\gamma_{0}\rVert_{\infty}$ (17)
$\displaystyle\lVert\Phi\circ\theta\rVert_{H^{\alpha}(D)}$ $\displaystyle\leq
C^{\prime}(1+\lVert\theta\rVert_{H^{\alpha}(D)}^{\alpha}),$ (18)
$\displaystyle\lVert\Phi^{-1}\circ\gamma_{0}\rVert_{H^{\alpha}(D)}$
$\displaystyle\leq
c^{\prime}(1+\lVert\gamma_{0}\rVert_{H^{\alpha}(D)}^{\alpha}).$ (19)

The first inequality is an immediate consequence of the mean value theorem and
the third is given in [34] Lemma 29. The second and fourth inequalites follow
from the same arguments, applied to the function $\Phi^{-1}$ (this can be seen
to be regular on the domain $[m,\infty)$ for $m>m_{1}$ by considering explicit
formulas for its derivatives).

###  3.2 Posterior contraction result

For the following result we define

$\xi_{\varepsilon,\delta}=\log(1/\varepsilon)^{-\delta},\quad\varepsilon,\delta>0.$
(20)

######  Theorem 3.

For some $m_{0}\in(0,1)$, $D_{0}$ compactly contained in $D$, and $M>0$,
suppose that the true conductivity $\gamma_{0}$ belongs to the set

$\Gamma_{m_{0},D_{0}}\cap\\{\Phi\circ\theta:\theta\in\mathcal{H},\lVert\theta\rVert_{\mathcal{H}}\leq
M\\},$ (21)

and define $\theta_{0}=\Phi^{-1}\circ\gamma_{0}$. For $\Pi^{\prime}$
satisfying Assumption 1 let $\Pi$ be the prior arising from 15, and denote by
$\Pi(\mathrel{\;\cdot\;}\mid Y)$ the posterior distribution for $\theta$
arising from observations $Y$ in the model 12. Then there exist constants
$C=C(M,m_{0},m_{1},D,D_{0},D_{1},\Phi,\zeta,r,\alpha,\beta)>0$ and
$\delta=\delta(d,\beta)>0$ such that

$\Pi\big{(}\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y\big{)}\to^{P_{\varepsilon}^{\gamma_{0}}}0\text{ as }\varepsilon\to 0.$ (22)

Moreover, if $E^{\Pi}[\theta\mid Y]$ denotes the (Bochner) mean of
$\Pi(\mathrel{\;\cdot\;}\mid Y)$, then for any $K>C$,

$\sup_{\gamma_{0}}P^{\gamma_{0}}_{\varepsilon}\big{(}\lVert E^{\Pi}[\theta\mid
Y]-\theta_{0}\rVert_{\infty}>K\xi_{\varepsilon,\delta}\big{)}\to 0~{}\text{as
}\varepsilon\to 0,$ (23)

where the supremum extends over all $\gamma_{0}$ in the set (21).

Theorem 3, whose proof is given in Section 5.4, immediately implies Theorem 1:
Indeed, given an integer $\alpha>3+d$, let $\Pi$ be a prior from 15 whose base
prior $\Pi^{\prime}$ satisfies Assumption 1 with RKHS
$\mathcal{H}=H^{\alpha}(D)$. [For instance, take a Whittle-Matérn prior,
noting that a choice of integer $\beta>2+d/2$ is then admissible.] Let
$\hat{\theta}=E^{\Pi}[\theta\mid Y]$ be the associated posterior mean and
define $\hat{\gamma}=\Phi\circ\hat{\theta}$. Then 16 and 23 will imply Theorem
1, so it suffices to show that the conditions of Theorem 1 imply those of
Theorem 3, in particular that for any
$\gamma_{0}\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)$, there exists an
$M^{\prime}=M^{\prime}(\alpha,M,m_{0},D,D_{0})$ such that
$\lVert\theta_{0}\rVert_{H^{\alpha}(D)}\leq M^{\prime}.$ But this is immediate
from 19.

##  4 Concluding remarks

######  Remark 1 (Computational aspects).

The posterior mean $E^{\Pi}[\theta\mid Y]$ can be calculated via MCMC or
expectation-propagation methods (naturally in the discretisations 7 or 8 of
our continuous model 12). This allows one to bypass potential difficulties
encountered by optimisation based methods in non-linear settings such as the
present one. For instance, the MAP estimates studied in [21, 34] may not
provably recover global optima in the EIT setting, since the non-linearity of
the map $\theta\mapsto\Lambda_{\Phi(\theta)}\equiv\Lambda_{\gamma}$ implies
that the associated least squares criterion is not necessarily convex.
Variational methods such as those proposed in [20] for EIT also typically
require a convex relaxation to be efficiently computable, see [23].

The pCN algorithm [8] allows one to sample from posterior distributions in
general inverse problems as long as the forward map
$\theta\mapsto\Lambda_{\Phi(\theta)}$ can be evaluated, which in our setting
has the basic cost of (numerically) solving the standard elliptic PDE 1. Even
in the absence of log-concavity of the posterior measure one can give sampling
guarantees for this algorithm, see [18], so that the approximate computation
of $E^{\Pi}[\theta\mid Y]$ by the sample average $(1/M)\sum_{m}\theta_{m}$ of
the pCN Markov chain is provably possible at any given noise level
$\varepsilon$. Related work on MCMC-based approaches in the setting of
electrical impedance tomography can be found in [22, 42, 10], wherein also
many further references can be found. Instead of MCMC methods one can also
resort to variational Bayes methods – see for example [12], where computation
of the posterior mean is addressed specifically for the EIT problem relevant
in the present paper.

######  Remark 2 (Estimation of $\gamma$ at the boundary).

In Theorem 1 we assume that the true conductivity $\gamma_{0}$ equals $1$ on
the complement of some known set $D_{0}\subset D$. In the proofs we work
mostly with the _differences_ $\Lambda_{\gamma}-\Lambda_{\gamma_{0}}$ and the
results can be expected to extend to $\gamma_{0}$ being known (and positive)
on $D\setminus D_{0}$. Estimation of the value of $\gamma_{0}$ at $\partial D$
is an ‘easier’ (less ill-posed) problem and is addressed, in a statistical
setting, in [6].

######  Remark 3 (Smoothness of $\gamma$).

In Theorem 1 we assume $\alpha>3+d$, and hence that the true conductivity is
sufficiently smooth in a Sobolev sense. In terms of the proofs, this
requirement primarily arises from the stability estimates employed ([2]
requires a minimal Sobolev smoothness of $\gamma$ of degree $\beta>2+d/2$),
and further from analytical support properties of Gaussian process priors (for
a Whittle–Matérn process with RKHS $H^{\alpha}(D)$ to be supported in
$H^{\beta}(D)$, one needs $\alpha>\beta+d/2$), necessitating $\alpha>2+d$. To
avoid technicalities with non-integer Sobolev spaces, we strengthen our
hypothesis to $\alpha>3+d$.

An interesting direction for further research would seek to replace the
smoothness assumption on $\gamma_{0}$ with different structural assumptions.
For example, if we consider a class of functions that are piecewise constant
on a known finite collection of subsets of $D$, better than logarithmic
stability results are available in [3], and faster convergence rates may then
be obtained. Note that this would require different methods, e.g., prior
measures that can model discontinuous functions, which is beyond the scope of
the present paper.

##  5 Proofs

The proof of Theorem 3 follows the template devised in [30] for a very
different inverse problem. We first show that the posterior distribution
induced on the operators $\Lambda_{\gamma}$ contracts around
$\Lambda_{\gamma_{0}}$ (see Theorem 13), by combining tools from Bayesian
nonparametric statistics [48, 14] with analytical properties of the
‘regression’ operators $\Lambda_{\gamma}$ (specifically of their low-rank
approximations). Dealing with the unboundedness of Gaussian priors for
$\theta$ and with the non-linearity of the composite forward map
$\theta\mapsto\Lambda_{\Phi(\theta)}$ poses a main challenge in the proof. As
in [30] this challenge is overcome using the rescaling in 15 of the base prior
$\Pi^{\prime}$. For such priors the posterior distribution concentrates on
sufficiently regular conductivities $\gamma$ that the stability estimates of
[2, 39] – which we show to hold also for the information-theoretically
relevant norms here – can be applied, resulting in contraction of the
posterior distributions of $\gamma$ and $\theta$ about $\gamma_{0}$ and
$\theta_{0}$ respectively. Finally, to deduce consistency of the posterior
mean, we adapt a quantitative uniform integrability argument from [30] to the
present situation. The proof of Theorem 2 follows from the ‘instability’
estimate in [28] and common information-theoretic lower bound techniques for
statistical estimators as in [46, 15].

######  Remark.

The model 12 naturally considers $\Lambda_{\gamma}$ as an element of the real
Hilbert space $\mathbb{H}_{r}$ consisting of Hilbert-Schmidt operators between
the sets of real-valued functions of $H^{r}(\partial D)/\mathbb{C}$ and
$L^{2}_{\diamond}(\partial D)$, respectively. Various results in the
literature – such as the stability and instability results of [2] and [28] to
be used below – regard $\Lambda_{\gamma}$ as an element of the space
$\mathbb{H}_{r,\mathbb{C}}$ of Hilbert-Schmidt operators between complex-
valued function spaces $H^{r}(\partial D)/\mathbb{C},L^{2}_{\diamond}(\partial
D)$. This difference is inconsequential since $\mathbb{H}_{r}$ embeds into
$\mathbb{H}_{r,\mathbb{C}}$ via the unique extension
$T_{\mathbb{C}}(f+ig)=T(f)+iT(g)$ for $T\in\mathbb{H}_{r}$, and the
Hilbert–Schmidt norm of this larger space, when restricted to
$\mathbb{H}_{r}$, coincides with the $\mathbb{H}_{r}$ norm. In fact, since we
regard $b_{jk}^{(r)}\in\mathbb{H}_{r}$ in 9 as a map from $H^{r}(\partial D)$
to $L^{2}_{\diamond}(\partial D)$, strictly speaking we have already embedded
$\mathbb{H}_{r}$ into $\mathbb{H}_{r,\mathbb{C}}$ in this way.

###  5.1 Low rank approximation of $\tilde{\Lambda}_{\gamma}$

A key idea used in various proofs that follow is that we can project the
operator $\tilde{\Lambda}_{\gamma}$ onto a finite-dimensional subspace and
incur only a small error. We recall from 9 the orthonormal basis
$(b_{jk}^{(r)})_{j,k\in\mathbb{N}}$ of $\mathbb{H}_{r}$ consisting of tensor
product operators

$b_{jk}^{(r)}(f)=\phi_{j}^{(r)}\otimes\phi_{k}^{(0)}(f):=\langle
f,\phi_{j}^{(r)}\rangle_{H^{r}(\partial D)}\phi_{k}^{(0)},\quad f\in
H^{r}(\partial D)/\mathbb{C},~{}j,k\in\mathbb{N},$

where the Laplace–Beltrami eigenfunctions $(\phi_{j}^{(r)})_{j\in\mathbb{N}}$
were introduced before 8. An operator $U\in\mathbb{H}_{r}$ has coefficients
$\langle U,b_{jk}^{(r)}\rangle_{\mathbb{H}_{r}}=\langle
U\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}$ with respect to
this basis, and we define the projection map $\pi_{JK}$ by

$\pi_{JK}U=\sum_{j\leq J}\sum_{k\leq K}\langle
U\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial D)}b_{jk}^{(r)}.$ (24)

######  Lemma 4.

For constants $m\in(0,1),M>1$ and some domain $D^{\prime}$ compactly contained
in $D$, let $\gamma\in\Gamma_{m,D^{\prime}}$ be bounded by $M$ on $D$. For any
$\nu>0$ there is a constant $C=C(\nu,D,D^{\prime},r)>0$ such that

$\lVert\tilde{\Lambda}_{\gamma}-\pi_{JK}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}\leq
C\tfrac{M}{m}\min(J,K)^{-\nu}.$

###### Proof.

Apply Lemma 18 from Appendix B with $s=0$, $p=r-\nu(d-1)$ and $q=\nu(d-1)$,
and note $\lVert\tilde{\Lambda}_{\gamma}\rVert_{H^{p-(d-1)}(D)\to
H^{q}(D)}\leq C\tfrac{M}{m}$ for such a constant $C$ by Lemma 20. ∎

The proofs of the stability results for the Calderón problem in the next
section involve the $H^{1/2}(\partial D)/\mathbb{C}\to H^{-1/2}(\partial D)$
operator norm, which we denote by $\lVert\cdot\rVert_{*}$. To connect this
norm to the information-theoretically relevant $\mathbb{H}_{r}$-norm, the
following consequence of Lemma 4 will be useful.

######  Lemma 5.

For $m\in(0,1)$, $M_{0},M_{1}>1$ and $D^{\prime}$ a domain compactly contained
in $D$, let $\gamma_{0},\gamma\in\Gamma_{m,D^{\prime}}$ be bounded on $D$ by
$M_{0}$ and $M_{1}$ respectively. Then there are constants $C_{1}$ and $C_{2}$
depending only on $r$, $D$ and $D_{0}$ such that if
$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq 1$ then

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
C_{1}\big{(}\tfrac{M_{1}+M_{0}}{m}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\big{)}^{1/2},$
(25)

and if $\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
1$ then

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq
C_{2}\big{(}\tfrac{M_{1}+M_{0}}{m}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\big{)}^{1/2}.$
(26)

###### Proof.

For $J>0$ and $\nu>0$ to be chosen, by Lemma 4 we have

$\lVert\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}\leq
C\tfrac{M_{1}}{m}J^{-\nu},$

for a constant $C=C(\nu,D,D^{\prime},r)$, and a corresponding bound holds for
$\lVert\tilde{\Lambda}_{\gamma_{0}}-\pi_{JJ}\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}$.
An application of Lemma 17 with $s=0$, $p=d-1/2,$ and $q=-1/2$, also yields
(with $x_{+}=\max(x,0)$, and $\mathcal{L}_{2}(H^{d-1/2},H^{-1/2})$ a space of
Hilbert–Schmidt operators as defined in Appendix B)

$\displaystyle\lVert\pi_{JJ}\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}$
$\displaystyle\leq
C^{\prime}(1+J^{1/(d-1)})^{1/2+(d-1/2-r)_{+}}\lVert\pi_{JJ}(\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}})\rVert_{\mathcal{L}_{2}(H^{d-1/2},H^{-1/2})}$
$\displaystyle\leq C^{\prime}(1+J^{1/(d-1)})^{(d+\lvert
r\rvert)}\lVert\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathcal{L}_{2}(H^{d-1/2},H^{-1/2})}$
$\displaystyle\leq c^{\prime}(1+J^{1/(d-1)})^{(d+\lvert
r\rvert)}\lVert\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{*},$

for constants $C^{\prime},c^{\prime}$ depending on $D,r$, where we use Lemma
18 to obtain the final inequality. Since
$\Lambda_{\gamma}-\Lambda_{\gamma_{0}}=\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}$,
we deduce, for a constant $C^{\prime\prime}$ that

$\begin{split}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}&\leq\lVert\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}+\lVert\tilde{\Lambda}_{\gamma_{0}}-\pi_{JJ}\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}+\lVert\pi_{JJ}\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\\\
&\leq
C^{\prime\prime}\big{(}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}J^{-\nu}+J^{(d+\lvert
r\rvert)/(d-1)}\lVert\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{*}\big{)}.\end{split}$

Since
$\lVert\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{*}\leq 1$,
we can choose an integer $J$ to balance the two terms up to a constant (take
$J=\lfloor\big{(}\frac{m}{M_{0}+M_{1}}\lVert\tilde{\Lambda}_{\gamma}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{*}\big{)}^{-(d-1)/(\nu(d-1)+d+\lvert
r\rvert)}\rfloor$). This yields, for a constant $c^{\prime\prime}$,

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
c^{\prime\prime}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}\Big{(}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}^{-1}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\Big{)}^{\nu(d-1)/(\nu(d-1)+d+\lvert
r\rvert)}.$

Choosing $\nu=(d+\lvert r\rvert)/(d-1)$ yields 25.

For (26), given that
$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathcal{L}_{2}(H^{1/2}(\partial
D)/\mathbb{C},H^{-1/2}(\partial D))}$ (using the general fact that the
Hilbert–Schmidt norm upper bounds the operator norm of a linear operator
between separable Hilbert spaces), and the observation that the proof of Lemma
4 can be adapted to apply with the $\mathcal{L}_{2}(H^{1/2}(\partial
D)/\mathbb{C},H^{-1/2}(\partial D))$ norm in place of the $\mathbb{H}_{r}$
norm, an almost identical argument to the above yields

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq
C\big{(}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}J^{-\nu}+J^{(r-1/2)_{+}/(d-1)}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\big{)}.$

Choosing $J$ to balance the terms yields

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq
C^{\prime}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}\Big{(}\big{(}\tfrac{M_{1}+M_{0}}{m}\big{)}^{-1}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\Big{)}^{\nu(d-1)/(\nu(d-1)+(r-1/2)_{+})},$

and the result follows from noting that the exponent is at least 1/2 for
$\nu>(r-1/2)_{+}/(d-1)$. ∎

###  5.2 Forward and inverse continuity results

We now prove the following estimates for the maps
$\gamma\mapsto\Lambda_{\gamma},\Lambda_{\gamma}\to\gamma$.

######  Lemma 6.

For $m\in(0,1),$ $M_{0},M_{1}>1$ and $D^{\prime}$ a domain compactly contained
in $D$, let $\gamma,\gamma_{0}\in\Gamma_{m,D^{\prime}}$ be bounded on $D$ by
$M_{1}$ and $M_{0}$ respectively. Then there exist constants
$C=C(r,D,D^{\prime})$, $\tau=\tau(D)$ such that

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
C\tfrac{M_{0}M_{1}}{m^{2}}\lVert\gamma-\gamma_{0}\rVert_{\infty}^{1/2},$

whenever
$\lVert\gamma-\gamma_{0}\rVert_{\infty}\leq\tau\frac{m^{2}}{M_{0}M_{1}}$.

######  Lemma 7.

For some $\beta>2+d/2$, some $m\in(0,1),$ $M>0$ and some domain $D^{\prime}$
compactly contained in $D$, suppose
$\gamma,\gamma_{0}\in\Gamma^{\beta}_{m,D^{\prime}}(M)=\\{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{H^{\beta}(D)}\leq
M\\}$. Then there exist constants $C$ and $\tau$ depending only on $M$, $D$,
$D^{\prime}$, $m$, $\beta$ and $r$ such that, for a constant
$\delta=\delta(d,\beta)>0$,

$\lVert\gamma-\gamma_{0}\rVert_{\infty}\leq
C\lvert\log\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\rvert^{-\delta},$

whenever
$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq\tau$.

The explicit form of the dependence of the constant in Lemma 6 on $M_{1}$ and
$M_{0}$ will be convenient in the proof of Lemma 11.

###### Proof of Lemma 6.

We initially show, for some $C=C(D)$, that

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}\leq
CM_{0}\tfrac{m+M_{1}}{m^{2}}\lVert\gamma-\gamma_{0}\rVert_{\infty}.$ (27)

The result then follows from Lemma 5, noting that
$(M_{0}+M_{1})M_{0}(m+M_{1})/m^{3}\leq 4M_{0}^{2}M_{1}^{2}/m^{4}$.

For $\gamma,\gamma_{0}$ as given, let $f\in H^{1/2}(\partial D)/\mathbb{C}$,
and recall we write $u_{\gamma,f}$ for the unique solution in
$\mathcal{H}_{-1/2}\equiv H^{1}(D)/\mathbb{C}$ to the Dirichlet problem on $D$
with conductivity $\gamma$ and boundary data $f$, whose existence is
guaranteed by Lemma 19, and similarly for $u_{\gamma_{0},f}$. The equivalence
class of functions $u_{\gamma,f}-u_{\gamma_{0},f}$ has a representative $w\in
H_{0}^{1}(D)$, which is easily seen to (weakly) solve the PDE

$\displaystyle\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}w)$
$\displaystyle=\operatorname{\operatorname{\nabla}\cdot}\big{(}(\gamma_{0}-\gamma)\operatorname{\nabla}u_{\gamma_{0},f}\big{)}\quad$
$\displaystyle\text{ in }D,$ (28) $\displaystyle w$ $\displaystyle=0\quad$
$\displaystyle\text{ on }\partial D.$

We have the dual representation (see remark (ii) in Appendix A)

$\Big{\lVert}\frac{\partial w}{\partial\nu}\Big{\rVert}_{H^{-1/2}(\partial
D)}=\sup\Big{\\{}\big{\lvert}\Big{\langle}\frac{\partial
w}{\partial\nu},v\Big{\rangle}_{L^{2}(\partial D)}\big{\rvert}:v\in
H^{1/2}(\partial D),\lVert v\rVert_{H^{1/2}(\partial D)}=1\Big{\\}}.$

For $v\in H^{1/2}(\partial D)$, by a standard trace theorem (e.g. Chapter I,
Theorem 9.4 of [27]) there exists $V\in H^{1}(D)$ such that $V|_{\partial
D}=v$ and $\lVert V\rVert_{H^{1}(D)}\leq C\lVert v\rVert_{H^{1/2}(\partial
D)}$ for a constant $C=C(D)$. Repeatedly applying the divergence theorem
(recalling that $\gamma=\gamma_{0}=1$ on $\partial D$) and the Cauchy–Schwarz
inequality, and using 28, we deduce

$\displaystyle\Big{\lvert}\int_{\partial D}\overline{v}\frac{\partial
w}{\partial\nu}\Big{\rvert}$
$\displaystyle=\Big{\lvert}\int_{D}\overline{V}\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}w)+\int_{D}\gamma\operatorname{\nabla}\overline{V}\cdot\operatorname{\nabla}w\Big{\rvert}$
$\displaystyle\leq\Big{\lvert}\int_{D}\overline{V}\operatorname{\operatorname{\nabla}\cdot}\big{(}(\gamma_{0}-\gamma)\operatorname{\nabla}u_{\gamma_{0},f}\big{)}\Big{\rvert}+\lVert\gamma\rVert_{\infty}\lVert
V\rVert_{H^{1}(D)}\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)}$
$\displaystyle\leq\Big{\lvert}\int_{D}(\gamma_{0}-\gamma)\operatorname{\nabla}\overline{V}\cdot\operatorname{\nabla}u_{\gamma_{0},f}\Big{\rvert}+C\lVert\gamma\rVert_{\infty}\lVert
v\rVert_{H^{1/2}(\partial D)}\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)}$
$\displaystyle\leq C\lVert v\rVert_{H^{1/2}(\partial
D)}\big{(}\lVert\gamma_{0}-\gamma\rVert_{\infty}\lVert\operatorname{\nabla}u_{\gamma_{0},f}\rVert_{L^{2}(D)}+M_{1}\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)}\big{)},$

hence

$\Big{\lVert}\frac{\partial w}{\partial\nu}\Big{\rVert}_{H^{-1/2}(\partial
D)}\leq
C\big{(}\lVert\gamma_{0}-\gamma\rVert_{\infty}\lVert\operatorname{\nabla}u_{\gamma_{0},f}\rVert_{L^{2}(D)}+M_{1}\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)}\big{)}.$
(29)

A weak solution $w$ to 28 by definition satisfies, for any $v\in
H_{0}^{1}(D)$,

$\int_{D}\gamma\operatorname{\nabla}w\cdot\operatorname{\nabla}\overline{v}=\int_{D}(\gamma_{0}-\gamma)\operatorname{\nabla}u_{\gamma_{0},f}\cdot\operatorname{\nabla}\overline{v}.$

In particular this applies with $v=w$, hence, since $\gamma\geq m$ on $D$, we
apply the Cauchy–Schwarz inequality to deduce

$m\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)}^{2}\leq\lVert\gamma_{0}-\gamma\rVert_{\infty}\lVert\operatorname{\nabla}u_{\gamma_{0},f}\rVert_{L^{2}(D)}\lVert\operatorname{\nabla}w\rVert_{L^{2}(D)},$

which, returning to 29, shows that

$\Big{\lVert}\frac{\partial w}{\partial\nu}\Big{\rVert}_{H^{-1/2}(\partial
D)}\leq
C\lVert\operatorname{\nabla}u_{\gamma_{0},f}\rVert_{L^{2}(D)}\lVert\gamma_{0}-\gamma\rVert_{\infty}\big{(}1+\tfrac{M_{1}}{m}\big{)}.$

Applying the trace result Theorem 9.4 in [27] Chapter I now to each
representative of the equivalence class $f\in H^{1/2}(D)/\mathbb{C}$ and
optimising, there exists $F\in H^{1}(D)/\mathbb{C}$ such that $F|_{\partial
D}=f$ and $\lVert F\rVert_{H^{1}(D)/\mathbb{C}}\leq C\lVert
f\rVert_{H^{1/2}(\partial D)/\mathbb{C}}$ for a constant $C=C(D)$. By
definition of a weak solution to 1 we have

$\int_{D}\gamma_{0}\operatorname{\nabla}u_{\gamma_{0},f}\cdot\operatorname{\nabla}\overline{(u_{\gamma_{0},f}-F)}=0,$

and again applying the Cauchy–Schwarz inequality we deduce

$\lVert\operatorname{\nabla}u_{\gamma_{0},f}\rVert_{L^{2}(D)}\leq
C\tfrac{M_{0}}{m}\lVert f\rVert_{H^{1/2}(\partial D)/\mathbb{C}}.$ (30)

Overall we have shown

$\big{\lVert}(\Lambda_{\gamma}-\Lambda_{\gamma_{0}})f\big{\rVert}_{H^{-1/2}(\partial
D)}\equiv\Big{\lVert}\frac{\partial
w}{\partial\nu}\Big{\rVert}_{H^{-1/2}(\partial D)}\leq
C\big{(}1+\tfrac{M_{1}}{m}\big{)}\tfrac{M_{0}}{m}\lVert\gamma-\gamma_{0}\rVert_{\infty}\lVert
f\rVert_{H^{1/2}(\partial D)/\mathbb{C}}.$

Taking the supremum over all $f$ with $H^{1/2}(\partial D)/\mathbb{C}$ norm
equal to 1, 27 follows. ∎

###### Proof of Lemma 7.

When $d\geq 3$, Theorem 1 in Alessandrini [2] states that there exist
constants $\delta=\delta(d,\beta)$ and $C=C(M,m,D,D^{\prime},\beta)$ such that
there is a (monotone) function $\omega$ satisfying

$\lVert\gamma-\gamma_{0}\rVert_{\infty}\leq
C\omega(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*}),\quad\omega(t)\leq\log(1/t)^{-\delta}\text{
for }t<e^{-1}.$ (31)

Likewise, when $d=2$, we can use Theorem 1.1 in Novikov & Santacesaria [39] in
conjunction with the usual reduction of the Calderón problem to a suitable
Schrödinger equation (e.g., see Lemma 4.8 and the proof of Theorem 4.1 in
[43]) to show that 31 still holds in this case. [In [39], an a priori bound
for $\lVert\gamma\rVert_{C^{2}}$ is assumed, which is derived here from the
bound $M$ on $\lVert\gamma\rVert_{H^{\beta}}$ using a Sobolev embedding.]

To proceed, we appeal to Lemma 5, noting that $M$ upper bounds
$\lVert\gamma\rVert_{\infty}$ and $\lVert\gamma_{0}\rVert_{\infty}$ (up to a
multiplicative constant) by a Sobolev embedding. Thus, for a constant
$C^{\prime}$ depending on $M,$ $m$, $D,$ $D^{\prime}$, $\beta$ and $r$ we have

$\displaystyle\omega(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{*})$
$\displaystyle\leq\omega(C^{\prime}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}^{1/2})$
$\displaystyle\leq(\tfrac{1}{4}\log(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}^{-1}))^{-\delta},$

provided
$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}<\min(e^{-2}(C^{\prime})^{-2},(C^{\prime})^{-4},1).$
The result follows. ∎

###  5.3 Tests and prior support properties

In this section we prove two main auxiliary results. First, we prove the
existence of certain statistical test functions required in the contraction
theorem given in Section 5.4. Instead of using robust ‘Hellinger-
distance’-based testing as in [48, 30], it is more convenient in the present
setting (in part to deal with necessary boundedness restrictions on $\gamma$)
to deduce the existence of tests from the existence of certain estimators with
sufficiently good concentration properties (following ideas in [16]).

Recall $\Gamma_{m_{1},D_{1}}$ is a superset of $\Gamma_{m_{0},D_{0}}$ on which
the prior on $\gamma=\Phi\circ\theta$ concentrates its mass.

######  Lemma 8.

Let $\gamma_{0}\in\Gamma_{m_{0},D_{0}}$ be bounded by $M_{0}$ on $D$. Let
$\eta_{\varepsilon}>0$ satisfy
$\eta_{\varepsilon}\varepsilon^{-(1-a)}\to\infty$ as $\varepsilon\to 0$ for
some $0<a<1$. For any $\kappa>0$ and $M_{1}>1$, there exists a constant
$C=C(\kappa,m_{1},D,D_{1},M_{1},M_{0})$ and tests (i.e. $\\{0,1\\}$-valued
measurable functions of the data) $\psi=\psi_{\varepsilon}(Y)$ with $Y\sim
P_{\varepsilon}^{\gamma}$ from 12 such that for all $\varepsilon$ small enough

$\max(E^{\gamma_{0}}_{\varepsilon}\psi,\sup\\{E^{\gamma}_{\varepsilon}[1-\psi]:\gamma\in\Gamma_{m_{1},D_{1}},\lVert\gamma\rVert_{\infty}\leq
M_{1},\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\geq
C\eta_{\varepsilon}\\})\leq e^{-\kappa(\eta_{\varepsilon}/\varepsilon)^{2}}.$

###### Proof.

We prove the existence of an estimator $\hat{\Lambda}$ satisfying that for any
$\kappa>0,$ there exists a constant
$C^{\prime}=C^{\prime}(\kappa,m_{1},D_{1},M_{0},M_{1},D)$ such that for all
$\varepsilon$ small enough,

$\sup\\{P^{\gamma}_{\varepsilon}(\lVert\hat{\Lambda}-\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}>C^{\prime}\eta_{\varepsilon}):\gamma\in\Gamma_{m_{1},D_{1}},\lVert\gamma\rVert_{\infty}\leq\max(M_{0},M_{1})\\}\leq
e^{-\kappa(\eta_{\varepsilon}/\varepsilon)^{2}}.$ (32)

Then, setting $C=2C^{\prime}$ and
$\psi_{\varepsilon}(Y)=\mathbbm{1}\\{\lVert\hat{\Lambda}-\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>\tfrac{1}{2}C\eta_{\varepsilon}\\},$
the result follows from an application of the triangle inequality (see, e.g.,
the proof of Proposition 6.2.2 in [15]).

Define an estimator $\hat{\Lambda}$ by $\hat{\Lambda}=\sum_{j,k\leq
J}\hat{\Lambda}_{jk}b_{jk}^{(r)}$, where
$J=J_{\varepsilon}=\lfloor\eta_{\varepsilon}/\varepsilon\rfloor$ and

$\hat{\Lambda}_{jk}=\langle
Y,b_{jk}^{(r)}\rangle_{\mathbb{H}_{r}}=\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}+\varepsilon g_{jk},~{}Y\sim P^{\gamma}_{\varepsilon},$ (33)

where we note
$g_{jk}=\langle\mathbb{W},b_{jk}^{(r)}\rangle_{\mathbb{H}_{r}}\overset{iid}{\sim}N(0,1)$.
Then we have the bias-variance decomposition

$P^{\gamma}_{\varepsilon}(\lVert\hat{\Lambda}-\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}>C^{\prime}\eta_{\varepsilon})\leq\mathbbm{1}\\{\lVert\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}>\tfrac{1}{2}C^{\prime}\eta_{\varepsilon}\\}+P^{\gamma}_{\varepsilon}(\lVert\hat{\Lambda}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}>\tfrac{1}{2}C^{\prime}\eta_{\varepsilon}).$
(34)

Recall, by Lemma 4, for any $\nu>0$ there is a constant
$C_{1}=C_{1}(\nu,r,M_{0},M_{1},m_{1},D_{1},D)$ such that

$\lVert\tilde{\Lambda}_{\gamma}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}\leq
C_{1}J^{-\nu},$ (35)

hence the indicator in 34 is bounded by
$\mathbbm{1}\\{C_{1}J^{-\nu}>\tfrac{1}{2}C^{\prime}\eta_{\varepsilon}\\}$.
Choosing $\nu>(1-a)/a$, one finds that the assumption
$\eta_{\varepsilon}\varepsilon^{-(1-a)}\to\infty$ ensures this term vanishes
for $\varepsilon$ small enough.

For the variance term in 34, observe that by Parseval’s identity

$\lVert\hat{\Lambda}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}^{2}=\sum_{j,k\leq
J}(\hat{\Lambda}_{jk}-\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(r)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)})^{2}=\varepsilon^{2}\sum_{j,k\leq J}g_{jk}^{2}.$

One now applies a standard tail inequality (e.g., Theorem 3.1.9 in [15]) to
the effect that

$\Pr\Big{(}\sum_{j,k\leq J}g_{jk}^{2}\geq J^{2}+2J\sqrt{x}+2x\Big{)}\leq
e^{-x}.$ (36)

For a constant $\kappa>0$, taking
$x=\kappa(\eta_{\varepsilon}/\varepsilon)^{2}$, and for our choice
$J=\lfloor\eta_{\varepsilon}/\varepsilon\rfloor$, we see that for $C^{\prime}$
large enough depending only on $\kappa$ we have

$P^{\gamma}_{\varepsilon}(\lVert\hat{\Lambda}-\pi_{JJ}\tilde{\Lambda}_{\gamma}\rVert_{\mathbb{H}_{r}}^{2}>\frac{1}{2}C^{\prime}\eta_{\varepsilon}^{2})\leq
e^{-\kappa(\eta_{\varepsilon}/\varepsilon)^{2}},$

hence the result. ∎

To proceed define $K(p,q)=E_{X\sim p}\log\frac{p}{q}(X)$ to be the
Kullback–Leibler divergence between distributions with densities $p$ and $q$,
and recall the definition of the probability densities
$p_{\varepsilon}^{\gamma}$ from 13. Also denote by
$\operatorname{Var}_{\gamma}$ the variance operator associated to the
probability measure $P^{\gamma}_{\varepsilon}$. The following is then a
standard result for a white noise model on a Hilbert space.

######  Lemma 9.

Let $\gamma_{0},\gamma_{1}\in\Gamma_{m_{1},D_{1}}$. Then
$K(p^{\gamma_{0}}_{\varepsilon},p^{\gamma_{1}}_{\varepsilon})=\frac{1}{2}\varepsilon^{-2}\lVert\Lambda_{\gamma_{0}}-\Lambda_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}$,
and
$\operatorname{Var}_{\gamma_{0}}\big{(}\log\frac{p^{\gamma_{0}}_{\varepsilon}}{p^{\gamma_{1}}_{\varepsilon}}\big{)}=\varepsilon^{-2}\lVert\Lambda_{\gamma_{0}}-\Lambda_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}$.

###### Proof.

Using the explicit formula 13 for the log-likelihoods, we see that under
$\gamma_{0}$,

$\displaystyle\ell(\gamma_{0})-\ell(\gamma_{1})$
$\displaystyle=\varepsilon^{-2}\langle
Y,\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}\rangle_{\mathbb{H}_{r}}-\tfrac{1}{2}\varepsilon^{-2}\lVert\tilde{\Lambda}_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}^{2}+\tfrac{1}{2}\varepsilon^{-2}\lVert\tilde{\Lambda}_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}$
$\displaystyle=\tfrac{1}{2}\varepsilon^{-2}\lVert\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}+\varepsilon^{-1}\langle\mathbb{W},\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}\rangle_{\mathbb{H}_{r}},$

which is normally distributed with mean
$\tfrac{1}{2}\varepsilon^{-2}\lVert\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}$
and variance
$\varepsilon^{-2}\lVert\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}\rVert_{\mathbb{H}_{r}}^{2}$.
Noting that
$\tilde{\Lambda}_{\gamma_{0}}-\tilde{\Lambda}_{\gamma_{1}}=\Lambda_{\gamma_{0}}-\Lambda_{\gamma_{1}}$,
we deduce the result.∎

We define ‘balls’ $B_{KL}^{\varepsilon}(\eta)$ around the true parameter
$\theta_{0}=\Phi^{-1}\circ\gamma_{0}$ by

$B_{KL}^{\varepsilon}(\eta)=\big{\\{}\theta\in
C_{u}(D):K(p^{\gamma_{0}}_{\varepsilon},p^{\Phi\circ\theta}_{\varepsilon})\leq(\eta/\varepsilon)^{2},\operatorname{Var}_{\gamma_{0}}(\log(p^{\gamma_{0}}_{\varepsilon}/p^{\Phi\circ\theta}_{\varepsilon}))\leq(\eta/\varepsilon)^{2}\big{\\}}.$
(37)

Then the following is an immediate consequence of Lemma 9.

######  Corollary 10.

For any $\eta>0$, $\\{\theta\in
C_{u}(D):\lVert\Lambda_{\Phi\circ\theta}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq\eta\\}\subseteq
B_{KL}^{\varepsilon}(\eta)$.

With the preceding preparations, we can now prove the following support result
for the prior $\Pi$ from Theorem 3, using a result of Li & Linde [26].

######  Lemma 11.

Let $\eta_{\varepsilon}=\varepsilon^{\alpha/(\alpha+d)}$. Under the conditions
of Theorem 3, there exists a constant
$\omega=\omega(\alpha,m_{1},M,D,D_{1},\Phi,r)>0$ such that
$\Pi(B_{KL}^{\varepsilon}(\eta_{\varepsilon}))\geq
e^{-\omega(\eta_{\varepsilon}/\varepsilon)^{2}}$ for all $\varepsilon$ small
enough, uniformly in $\gamma_{0}$ in the set 21.

###### Proof.

By 16 and a Sobolev embedding, there is a constant $M_{0}$ depending only on
$\Phi$, $\alpha$ and $M$ such that $\lVert\gamma_{0}\rVert_{\infty}\leq
M_{0}$. By Lemma 6, we deduce, for a constant $C=C(r,D,D_{1})$,

$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
C\frac{M_{0}\lVert\gamma\rVert_{\infty}}{m_{1}^{2}}\lVert\gamma-\gamma_{0}\rVert_{\infty}^{1/2}$

provided $\lVert\gamma-\gamma_{0}\rVert_{\infty}$ is small enough. It follows
from this calculation and Corollary 10 that for $\eta_{\varepsilon}$ small
enough and some constant $C^{\prime}>0$ we have

$\\{\theta:\lVert\Phi\circ\theta-\gamma_{0}\rVert_{\infty}\leq
C^{\prime}\eta_{\varepsilon}^{2}\\}\subseteq\\{\lVert\Lambda_{\Phi\circ\theta}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq\eta_{\varepsilon}\\}\subseteq
B^{\varepsilon}_{KL}(\eta_{\varepsilon}).$

Appealing again to 16, we further deduce that

$\\{\theta\in C_{u}(D):\lVert\theta-\theta_{0}\rVert_{\infty}\leq
A\eta_{\varepsilon}^{2}\\}\subseteq B_{KL}^{\varepsilon}(\eta_{\varepsilon}),$

for a constant $A=A(\alpha,M,m_{1},D,D_{1},r,\Phi)$, so that it remains to
lower bound $\Pi\big{(}\lVert\theta-\theta_{0}\rVert_{\infty}\leq
A\eta_{\varepsilon}^{2}\big{)}$. Note (recalling Assumption 1 and the
definition of the prior 15) that $\Pi$ has RKHS
$\mathcal{H}_{\varepsilon}=\\{\zeta\theta^{\prime}:\theta^{\prime}\in\mathcal{H}\\}$,
with norm $\lVert\cdot\rVert_{\mathcal{H}_{\varepsilon}}$ satisfying the bound
$\lVert\theta\rVert_{\mathcal{H}_{\varepsilon}}\leq\varepsilon^{-d/(\alpha+d)}\lVert\theta^{\prime}\rVert_{\mathcal{H}}=(\eta_{\varepsilon}/\varepsilon)\lVert\theta^{\prime}\rVert_{\mathcal{H}},$
for any $\theta^{\prime}$ such that $\zeta\theta^{\prime}=\theta$. Because
$\theta_{0}=\Phi^{-1}\circ\gamma_{0}$ for some
$\gamma_{0}\in\Gamma_{m_{0},D_{0}}$, we see from the definitions of $\Phi$ and
$\zeta$ that $\theta_{0}=\zeta\theta_{0}$, hence we deduce that
$\lVert\theta_{0}\rVert_{\mathcal{H}_{\varepsilon}}\leq(\eta_{\varepsilon}/\varepsilon)\lVert\theta_{0}\rVert_{\mathcal{H}}\leq
M\eta_{\varepsilon}/\varepsilon$. By Corollary 2.6.18 in [15], we then have

$\displaystyle\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}\leq
A\eta_{\varepsilon}^{2})$ $\displaystyle\geq
e^{-\tfrac{1}{2}\lVert\theta_{0}\rVert_{\mathcal{H}_{\varepsilon}}^{2}}\Pi(\lVert\theta\rVert_{\infty}\leq
A\eta_{\varepsilon}^{2})$ $\displaystyle\geq
e^{-\frac{1}{2}M^{2}(\eta_{\varepsilon}/\varepsilon)^{2}}\Pi^{\prime}\big{(}\lVert\theta^{\prime}\rVert_{\infty}\leq
A\tfrac{\eta_{\varepsilon}^{3}}{\varepsilon}\big{)}.$

Next, since $\mathcal{H}$ embeds continuously into $H^{\alpha}(I_{d})$ for
some large enough cube $I_{d}$ (by a standard extension argument for Sobolev
spaces), the unit ball $B_{\mathcal{H}}$ of $\mathcal{H}$ has covering numbers
with respect to the supremum norm
$N=N(B_{\mathcal{H}},\lVert\cdot\rVert_{\infty},\delta)$ (i.e. the smallest
number of $\lVert\cdot\rVert_{\infty}$ balls of radius $\delta$ needed to
cover $B_{\mathcal{H}}$) satisfying

$\log N(B_{\mathcal{H}},\lVert\cdot\rVert_{\infty},\delta)\leq
K\delta^{-d/\alpha}$ (38)

for some constant $K=K(\alpha,D)$ (cf. after Corollary 4.3.38 in [15]). We can
thus apply [26], Theorem 1.2, to see

$\Pi^{\prime}\big{(}\lVert\theta^{\prime}\rVert_{\infty}\leq
A\tfrac{\eta_{\varepsilon}^{3}}{\varepsilon}\big{)}\geq
e^{-A^{\prime}(\frac{\eta_{\varepsilon}^{3}}{\varepsilon})^{-s}},$

for some constant $A^{\prime}=A^{\prime}(A,K)$, where $s$ is such that
$\frac{d}{\alpha}=\frac{2s}{2+s}$, i.e. $s=\frac{2d}{2\alpha-d}$.

Overall, we have shown $\Pi(B_{KL}^{\varepsilon}(\eta_{\varepsilon}))\geq
e^{-\frac{1}{2}M^{2}(\eta_{\varepsilon}/\varepsilon)^{2}}e^{-A^{\prime}(\eta_{\varepsilon}^{3}/\varepsilon)^{-2d/(2\alpha-d)}}$,
where the constant $A^{\prime}$ depends only on $D$, $\alpha$, $M$, $m_{1}$,
$D_{1}$, $r$ and $\Phi$. For
$\eta_{\varepsilon}=\varepsilon^{\alpha/(\alpha+d)}$ we find
$(\eta_{\varepsilon}^{3}/\varepsilon)^{-2d/(2\alpha-d)}=(\eta_{\varepsilon}/\varepsilon)^{2}$,
and the result follows.∎

###  5.4 Posterior asymptotics

####  5.4.1 Posterior regularity and contraction about $\Lambda_{\gamma_{0}}$

The following two results follow ideas from Bayesian nonparametric statistics
[48, 14] combined with the lemmas from the previous section. Together with the
stability estimate Lemma 7, these two estimates allow us to proceed as in [30]
to prove Theorem 3.

######  Lemma 12.

Let $\eta_{\varepsilon},\omega$ be as in Lemma 11. Under the assumptions of
Theorem 3 there exists $M^{\prime}>0$ such that

$\sup_{\gamma_{0}\in\Gamma_{m_{0},D_{0}}\cap\\{\Phi\circ\theta:\theta\in\mathcal{H},\lVert\theta\rVert_{\mathcal{H}}\leq
M\\}}P^{\gamma_{0}}_{\varepsilon}\big{(}\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}\mid
Y)>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\big{)}\to 0,$ (39)

as $\varepsilon\to 0$. The bound 39 also holds with the supremum norm
$\lVert\cdot\rVert_{\infty}$ in place of the $H^{\beta}(D)$ norm.

######  Theorem 13.

Let $\eta_{\varepsilon},\omega$ be as in Lemma 11. Under the conditions of
Theorem 3, there exists $C>0$ such that

$\sup_{\gamma_{0}\in\Gamma_{m_{0},D_{0}}\cap\\{\Phi\circ\theta:\theta\in\mathcal{H},\lVert\theta\rVert_{\mathcal{H}}\leq
M\\}}P^{\gamma_{0}}_{\varepsilon}\big{(}\Pi(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C\eta_{\varepsilon}\mid
Y)>2e^{-(\omega+4)(\eta/\varepsilon)^{2}}\big{)}\to 0,$ (40)

as $\varepsilon\to 0$.

To prove the preceding results, we note that the posterior from 14 can be
written as

$\Pi(B\mid
Y)=\frac{\int_{B}p^{\gamma}_{\varepsilon}(Y)/p^{\gamma_{0}}_{\varepsilon}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)}{\int_{\Gamma_{m_{1},D_{1}}}p^{\gamma}_{\varepsilon}(Y)/p^{\gamma_{0}}_{\varepsilon}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)},\quad
B\subset\Gamma_{m_{1},D_{1}}\text{ measurable}.$ (41)

The following lemma controls the size of the denominator in the last
expression.

######  Lemma 14.

Let $\eta_{\varepsilon},\omega$ be as in Lemma 11. Introduce the event

$L=L_{\omega,\gamma_{0}}=\Big{\\{}\int_{\Gamma_{m_{1},D_{1}}}\frac{p^{\gamma}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)\geq
e^{-(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{\\}}.$

Then, under the assumptions of Theorem 3, we have

$\sup_{\gamma_{0}\in\Gamma_{m_{0},D_{0}}\cap\\{\Phi\circ\theta:\theta\in\mathcal{H},\lVert\theta\rVert_{\mathcal{H}}\leq
M\\}}P^{\gamma_{0}}_{\varepsilon}(L^{c})\to
0,\quad\text{as}~{}~{}\varepsilon\to 0.$ (42)

###### Proof.

Given Lemma 11, the proof is a standard argument based on Chebyshev’s
inequality, Jensen’s inequality and Fubini’s theorem (for example, see [15]
Lemma 7.3.4 for a proof in the setting of white noise on $L^{2}([0,1])$ which
adapts straightforwardly to the current Hilbert space setting, with
probability measure $\nu$ there equal to the renormalised restriction of
$\mathop{}\\!\mathrm{d}\Pi(\gamma)$ to
$B_{KL}^{\varepsilon}(\eta_{\varepsilon})$). ∎

###### Proof of Lemma 12.

Define $L$ as in Lemma 14. By 41, using Fubini’s theorem and the fact that
$E^{\gamma_{0}}_{\varepsilon}\frac{p^{\gamma}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)=1$,
we see that for every Borel set $B$

$E^{\gamma_{0}}_{\varepsilon}[\mathbbm{1}_{L}\Pi(B\mid Y)]\leq
e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}E^{\gamma_{0}}_{\varepsilon}\int_{B}\frac{p^{\gamma}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)=e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}\Pi(B).$

Then, setting $B=\\{\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}\\}$, an
application of Markov’s inequality yields

$P^{\gamma_{0}}_{\varepsilon}\big{(}\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}\mid
Y)>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\big{)}\leq
P^{\gamma_{0}}_{\varepsilon}(L^{c})+e^{(2\omega+6)(\eta_{\varepsilon}/\varepsilon)^{2}}\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}).$

The first term on the right vanishes asymptotically, uniformly across
$\gamma_{0}$ in the given set, by Lemma 14. For the second, we recall 18 and
the definition 15 of the prior. In conjunction with the facts that
$\varepsilon^{-d/(\alpha+d)}=\eta_{\varepsilon}/\varepsilon$ for our choice of
$\eta_{\varepsilon}$ and that
$\lVert\zeta\theta^{\prime}\rVert_{H^{\beta}(D)}\leq
C\lVert\zeta\rVert_{H^{\beta}(D)}\lVert\theta^{\prime}\rVert_{H^{\beta}(D)}$
for some constant $C$ (since $\beta>d/2$), these allow us to deduce that

$\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime})\leq\Pi^{\prime}\Big{(}\lVert\theta^{\prime}\rVert_{H^{\beta}(D)}>\tfrac{\eta_{\varepsilon}}{\varepsilon}(C\lVert\zeta\rVert_{H^{\beta}(D)})^{-1}(M^{\prime}/C^{\prime}-1)^{1/\beta}\Big{)}.$

Since $\eta_{\varepsilon}/\varepsilon\to\infty$ and since
$\Pi^{\prime}(H^{\beta}(D))=1$ by hypothesis, we can apply a version of
Fernique’s theorem, more specifically Theorem 2.1.20 in [15] (see also Example
2.1.6 therein) to deduce that for any $c>0$ there exists a
$M^{\prime}=M^{\prime}(\beta,c,C^{\prime},\zeta)$ such that the last
probability does not exceed $e^{-c(\eta_{\varepsilon}/\varepsilon)^{2}}$.
Taking $c>2\omega+6$ concludes the proof for the $H^{\beta}(D)$ norm, and the
result for the supremum norm follows by a Sobolev embedding and adjusting the
constant $M^{\prime}$. ∎

###### Proof of Theorem 13.

We decompose in a standard way: writing

$S=\\{\gamma\in\Gamma_{m_{1},D_{1}}:\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C\eta_{\varepsilon},~{}\lVert\gamma\rVert_{\infty}\leq
M^{\prime}\\},$

for $M^{\prime}$ the constant of Lemma 12 and $C$ a large enough constant (to
be chosen below), we have, for $\psi$ the test given by Lemma 8 and $L$ as in
Lemma 14,

$\displaystyle\Pi\big{(}\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C\eta_{\varepsilon}\mid
Y\big{)}$ $\displaystyle\leq\Pi\big{(}S\mid
Y\big{)}+\Pi\big{(}\lVert\gamma\rVert_{\infty}>M^{\prime}\mid Y\big{)},$ (43)
$\displaystyle\Pi(S\mid Y)$
$\displaystyle\leq\mathbbm{1}_{L^{c}}+\psi+\Pi(S\mid
Y)\mathbbm{1}_{L}[1-\psi].$

Hence, denoting by $\mathcal{C}$ the event
$\big{\\{}\Pi(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C\eta_{\varepsilon}\mid
Y)>2e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\big{\\}}$, we have

$\displaystyle P^{\gamma_{0}}_{\varepsilon}(\mathcal{C})\leq
P^{\gamma_{0}}_{\varepsilon}\Big{(}\Pi\big{(}\lVert\gamma\rVert_{\infty}>M^{\prime}\mid
Y\big{)}>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)}+P^{\gamma_{0}}_{\varepsilon}\Big{(}\Pi(S\mid
Y)>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)},$ $\displaystyle
P^{\gamma_{0}}_{\varepsilon}\Big{(}\Pi(S\mid
Y)>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)}\leq
P^{\gamma_{0}}_{\varepsilon}(L^{c})+E^{\gamma_{0}}_{\varepsilon}\psi+P^{\gamma_{0}}_{\varepsilon}\Big{(}\Pi(S\mid
Y)\mathbbm{1}_{L}[1-\psi]>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)}.$

In view of Lemmas 12, 8 and 14, it suffices to show that there exists a
constant $C$ such that $P^{\gamma_{0}}_{\varepsilon}\Big{(}\Pi(S\mid
Y)\mathbbm{1}_{L}[1-\psi]>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)}\to
0$, uniformly across $\gamma_{0}$ in the given set, and we note that
$\lVert\gamma_{0}\rVert_{\infty}$ is uniformly bounded in the set by a Sobolev
embedding. Appealing to 41, Fubini’s theorem and again Lemma 8 we have for
every $\kappa>0$ and for $C$ large enough (depending on $\kappa$),

$\displaystyle E^{\gamma_{0}}_{\varepsilon}\Big{[}\Pi(S\mid
Y)\mathbbm{1}_{L}(1-\psi)\Big{]}$ $\displaystyle\leq
e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}E^{\gamma_{0}}_{\varepsilon}\int_{S}\frac{p^{\gamma}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)(1-\psi)(Y)\mathop{}\\!\mathrm{d}\Pi(\gamma)$
$\displaystyle\leq
e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}\int_{S}E_{\varepsilon}^{\gamma}[(1-\psi)(Y)]\mathop{}\\!\mathrm{d}\Pi(\gamma)$
$\displaystyle\leq e^{(\omega+2-\kappa)(\eta_{\varepsilon}/\varepsilon)^{2}},$

hence by Markov’s inequality, the probability in question is bounded by
$e^{(2\omega+6-\kappa)(\eta_{\varepsilon}/\varepsilon)^{2}}$. This tends to
zero, uniformly in $\gamma_{0}$, if $C$ is large enough to permit
$\kappa>2\omega+6$. ∎

####  5.4.2 Proof of Theorem 3

Recall Lemma 7 to the effect that

$\lVert\gamma-\gamma_{0}\rVert_{\infty}\leq
C^{\prime}\lvert\log\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\rvert^{-\delta},$

at least for
$\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}$ small
enough, for some $\delta=\delta(d,\beta)>0$ and a constant $C^{\prime}$
depending only on $M$, $D$, $D_{1}$, $\alpha$, $\beta$, $r$, and an upper
bound for $\lVert\gamma\rVert_{H^{\beta}(D)}$, where we have also used
$\lVert\gamma_{0}\rVert_{H^{\beta}(D)}\leq\lVert\gamma_{0}\rVert_{H^{\alpha}(D)}$,
eq. 18 and the hypothesis on $\gamma_{0}$. Thus, for $M^{\prime}>0$ the
constant of Lemma 12 and $C_{1}>0$ large enough that Theorem 13 applies with
constant $C_{1}$, we have

$(\lVert\gamma\rVert_{H^{\beta}(D)}\leq
M^{\prime})\wedge(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
C_{1}\eta_{\varepsilon})\implies\lVert\gamma-\gamma_{0}\rVert_{\infty}\leq
C^{\prime}\xi_{\varepsilon,\delta}.$

In view of 17, for $\varepsilon$ small enough (noting that since
$C^{\prime}\xi_{\varepsilon,\delta}\to 0$ it is eventually smaller than
$m_{0}-m_{1}$), we further deduce for a constant $C$ that

$\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y)\leq\Pi(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C_{1}\eta_{\varepsilon}\mid
Y)+\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}\mid Y).$ (44)

Then Theorems 13 and 12 imply the contraction rate 22.

To prove consistency of the posterior mean $E[\theta\mid Y]$, introduce, for
$\omega,L$ as in Lemma 14, the event

$\mathcal{A}=L\cap\Big{\\{}\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y)\leq 3e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{\\}},$

and note that

$P^{\gamma_{0}}_{\varepsilon}\big{(}\lVert E^{\Pi}[\theta\mid
Y]-\theta_{0}\rVert_{\infty}>K\xi_{\varepsilon,\delta}\big{)}\leq
P^{\gamma_{0}}_{\varepsilon}(\mathcal{A}^{c})+P^{\gamma_{0}}_{\varepsilon}\big{(}\lVert
E^{\Pi}[\theta-\theta_{0}\mid
Y]\rVert_{\infty}\mathbbm{1}_{\mathcal{A}}>K\xi_{\varepsilon,\delta}\big{)}.$
(45)

In view of 44, observe that $\mathcal{A}^{c}$ is contained in

$L^{c}\cup\Big{\\{}\Pi(\lVert\Lambda_{\gamma}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}>C_{1}\eta_{\varepsilon}\mid
Y)>2e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{\\}}\cup\Big{\\{}\Pi(\lVert\gamma\rVert_{H^{\beta}(D)}>M^{\prime}\mid
Y)>e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{\\}},$

hence, by Lemmas 12, 13 and 14,
$P^{\gamma_{0}}_{\varepsilon}(\mathcal{A}^{c})\to 0$ as $\varepsilon\to 0$,
uniformly in $\gamma_{0}$. Now for the second term in 45, by Jensen’s
inequality and the Cauchy–Schwarz inequality, we have

$\displaystyle\lVert E^{\Pi}[\theta-\theta_{0}\mid
Y]\rVert_{\infty}\mathbbm{1}_{\mathcal{A}}$ $\displaystyle\leq
C\xi_{\varepsilon,\delta}+E^{\Pi}\Big{[}\lVert\theta-\theta_{0}\rVert_{\infty}\mathbbm{1}\\{\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\\}\mathrel{\big{|}}Y\Big{]}\mathbbm{1}_{\mathcal{A}}$
$\displaystyle\leq
C\xi_{\varepsilon,\delta}+\big{(}E^{\Pi}[\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\mid
Y]\big{)}^{1/2}\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y)^{1/2}\mathbbm{1}_{\mathcal{A}},$

and by Markov’s inequality, it follows for $K>C$ that

$\begin{split}&P^{\gamma_{0}}_{\varepsilon}(\lVert
E^{\Pi}[\theta-\theta_{0}\mid
Y]\rVert_{\infty}\mathbbm{1}_{\mathcal{A}}>K\xi_{\varepsilon,\delta})\leq\\\
&\quad\tfrac{1}{(K-C)\xi_{\varepsilon,\delta}}E^{\gamma_{0}}_{\varepsilon}\Big{[}\big{(}E^{\Pi}[\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\mid
Y]\big{)}^{1/2}(\Pi\\{\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y\\})^{1/2}\mathbbm{1}_{\mathcal{A}}\Big{]}.\end{split}$ (46)

Again applying the Cauchy–Schwarz inequality the last expected value is
bounded by

$\quad
E^{\gamma_{0}}_{\varepsilon}\big{[}E^{\Pi}[\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\mid
Y]\mathbbm{1}_{\mathcal{A}}\big{]}^{1/2}E^{\gamma_{0}}_{\varepsilon}\big{[}\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y)\mathbbm{1}_{\mathcal{A}}\big{]}^{1/2}.$

From the definition of the event $\mathcal{A}$ it is immediate that

$E^{\gamma_{0}}_{\varepsilon}\big{[}\Pi(\lVert\theta-\theta_{0}\rVert_{\infty}>C\xi_{\varepsilon,\delta}\mid
Y)\mathbbm{1}_{\mathcal{A}}\big{]}\leq
3e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}},$

while, applying 41 and Fubini’s theorem and since
$E^{\gamma_{0}}_{\varepsilon}\tfrac{p^{\Phi\circ\theta}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)=1$,
we have

$\displaystyle
E^{\gamma_{0}}_{\varepsilon}\big{[}E^{\Pi}[\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\mid
Y]\mathbbm{1}_{\mathcal{A}}\big{]}$ $\displaystyle\leq
e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}E^{\gamma_{0}}_{\varepsilon}\Big{[}\int_{C_{u}(D)}\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\tfrac{p^{\Phi\circ\theta}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}(Y)\mathop{}\\!\mathrm{d}\Pi(\theta)\Big{]}$
$\displaystyle\leq
e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}E^{\Pi}\lVert\theta-\theta_{0}\rVert_{\infty}^{2}.$

Plugging this back into 46, we see

$P^{\gamma_{0}}_{\varepsilon}(\lVert E^{\Pi}[\theta-\theta_{0}\mid
Y]\rVert_{\infty}\mathbbm{1}_{\mathcal{A}}>K\xi_{\varepsilon,\delta})\leq\tfrac{\sqrt{3}}{(K-C)\xi_{\varepsilon,\delta}}\Big{(}e^{(\omega+2)(\eta_{\varepsilon}/\varepsilon)^{2}}E^{\Pi}\lVert\theta-\theta_{0}\rVert_{\infty}^{2}e^{-(\omega+4)(\eta_{\varepsilon}/\varepsilon)^{2}}\Big{)}^{1/2}.$

Note that, since $E^{\Pi}\lVert\theta-\theta_{0}\rVert_{\infty}^{2}\leq
2(\lVert\theta_{0}\rVert_{\infty}^{2}+E^{\Pi}\lVert\theta\rVert_{\infty}^{2})$
and $E^{\Pi^{\prime}}\lVert\theta^{\prime}\rVert^{2}_{\infty}$ is finite
(Exercise 2.1.2 and Theorem 2.1.20a in [15]), a Sobolev embedding combined
with the prior definition 15 implies that
$E^{\Pi}\lVert\theta-\theta_{0}\rVert_{\infty}^{2}$ is bounded uniformly
across the specified $\theta_{0}$’s. Since
$e^{-(\eta_{\varepsilon}/\varepsilon)^{2}}/\xi_{\varepsilon,\delta}\to 0$, we
see, returning to 45, that the result follows.

###  5.5 Proof of the lower bound Theorem 2

Recall the shorthand 20 and also the definition of $K(p,q)$ from before Lemma
9. It is enough to find
$\gamma_{0},\gamma_{1}\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)$ (both allowed to
depend on $\varepsilon$) such that, for some $\mu$ small enough (to be
chosen),

  1. i.

$\lVert\gamma_{1}-\gamma_{0}\rVert_{\infty}\geq\xi_{\varepsilon,\delta^{\prime}}\equiv\log(1/\varepsilon)^{-\delta^{\prime}}$

  2. ii.

$K(p^{\gamma_{1}}_{\varepsilon},p^{\gamma_{0}}_{\varepsilon})\leq\mu.$

Indeed, the standard proof of this reduction (for example as in [15], Theorem
6.3.2, or Chapter 2 in [46]) is as follows. Under condition (i), noting that
$\psi=\mathbbm{1}\\{\lVert\hat{\gamma}-\gamma_{1}\rVert_{\infty}<\lVert\hat{\gamma}-\gamma_{0}\rVert_{\infty}\\}$
yields a test of $H_{0}:\gamma=\gamma_{0}$ against $H_{1}:\gamma=\gamma_{1}$,
we see

$\inf_{\hat{\gamma}}\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P^{\gamma}_{\varepsilon}\big{(}\lVert\hat{\gamma}-\gamma\rVert_{\infty}\geq\tfrac{1}{2}\xi_{\varepsilon,\delta^{\prime}}\big{)}\geq\inf_{\psi}\max(P^{\gamma_{0}}_{\varepsilon}(\psi\not=0),P^{\gamma_{1}}_{\varepsilon}(\psi\not=1)),$

where the latter infimum is over all tests $\psi$. Introducing the event
$A=\big{\\{}\frac{p^{\gamma_{0}}_{\varepsilon}}{p^{\gamma_{1}}_{\varepsilon}}\geq
1/2\big{\\}}$, we see

$P^{\gamma_{0}}_{\varepsilon}(\psi\not=0)\geq
E^{\gamma_{1}}_{\varepsilon}\big{[}\tfrac{p^{\gamma_{0}}_{\varepsilon}}{p^{\gamma_{1}}_{\varepsilon}}\mathbbm{1}_{A}\psi\big{]}\geq\tfrac{1}{2}[P^{\gamma_{1}}_{\varepsilon}(\psi=1)-P^{\gamma_{1}}_{\varepsilon}(A^{c})]$

Thus, writing $p_{1}=P^{\gamma_{1}}_{\varepsilon}(\psi=1)$, we see

$\displaystyle\max(P^{\gamma_{0}}_{\varepsilon}(\psi\not=0),P^{\gamma_{1}}_{\varepsilon}(\psi\not=1))$
$\displaystyle\geq\max(\tfrac{1}{2}(p_{1}-P^{\gamma_{1}}_{\varepsilon}(A^{c})),1-p_{1})$
$\displaystyle\geq\inf_{p\in[0,1]}\max(\tfrac{1}{2}(p-P^{\gamma_{1}}_{\varepsilon}(A^{c})),1-p).$

The infimum is attained when
$\frac{1}{2}(p-P^{\gamma_{1}}_{\varepsilon}(A^{c}))=1-p$ and takes the value
$\frac{1}{3}P^{\gamma_{1}}_{\varepsilon}(A)$ so that

$\inf_{\hat{\gamma}}\sup_{\gamma\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)}P^{\gamma}_{\varepsilon}\big{(}\lVert\hat{\gamma}-\gamma\rVert_{\infty}\geq\tfrac{1}{2}\xi_{\varepsilon,\delta^{\prime}}\big{)}\geq\tfrac{1}{3}P^{\gamma_{1}}_{\varepsilon}(A).$
(47)

Next observe

$\displaystyle P^{\gamma_{1}}_{\varepsilon}(A)$
$\displaystyle=P^{\gamma_{1}}_{\varepsilon}\big{[}\tfrac{p^{\gamma_{1}}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}\leq
2\big{]}=1-P^{\gamma_{1}}_{\varepsilon}\big{[}\log\big{(}\tfrac{p^{\gamma_{1}}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}\big{)}>\log
2\big{]}\geq
1-P^{\gamma_{1}}_{\varepsilon}\big{[}\lvert\log(\tfrac{p^{\gamma_{1}}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}})\rvert>\log
2\big{]}$ $\displaystyle\geq
1-(\log{2})^{-1}E^{\gamma_{1}}_{\varepsilon}\Big{\lvert}\log\big{(}\tfrac{p^{\gamma_{1}}_{\varepsilon}}{p^{\gamma_{0}}_{\varepsilon}}\big{)}\Big{\rvert},$

where we have used Markov’s inequality to attain the final expression. By the
second Pinsker inequality (Proposition 6.1.7b in [15]), using condition (ii)
we can continue the chain of inequalities to see

$P^{\gamma_{1}}_{\varepsilon}(A)\geq 1-(\log
2)^{-1}\big{[}K(p^{\gamma_{1}}_{\varepsilon},p^{\gamma_{0}}_{\varepsilon})+\sqrt{2K(p^{\gamma_{1}}_{\varepsilon},p^{\gamma_{0}}_{\varepsilon})}\big{]}\geq
1-(\log 2)^{-1}(\mu+\sqrt{2\mu}).$

Choosing $\mu$ small enough, we can thus lower bound 47 by

$\tfrac{1}{3}\Big{(}1-\frac{\mu+\sqrt{2\mu}}{\log 2}\Big{)}>\frac{1}{4},$

so that Theorem 2 will follow.

Now we prove the existence of
$\gamma_{0},\gamma_{1}\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)$ satisfying
conditions (i) and (ii). We appeal to Corollary 1 in [28], which says that for
any integer $k\geq 2$, any $q\geq 0$, some $B>0$ and any $\xi\in(0,1)$ there
exist $\gamma_{0},\gamma_{1}$ such that
$\operatorname{supp}(\gamma_{j}-1)\subseteq D_{0}$, $\gamma_{j}\geq 1$ on $D$
for $j=0,1$, and

  1. a.

$\lVert\gamma_{1}-\gamma_{0}\rVert_{\infty}\geq\xi,$

  2. b.

$\lVert\Lambda_{\gamma_{1}}-\Lambda_{\gamma_{0}}\rVert_{H^{-q}(\partial
D)/\mathbb{C}\to H^{q}_{\diamond}(\partial
D)}\leq\exp\big{(}-\xi^{-\frac{d}{(2d-1)k}}\big{)},$

  3. c.

$\max(\lVert\gamma_{1}\rVert_{C^{k}(D)},\lVert\gamma_{0}\rVert_{C^{k}(D)})\leq
B$,

where the norm in (b) is the operator norm and where
$\lVert\cdot\rVert_{C^{k}(D)}$ denotes the usual norm on the space $C^{k}(D)$
of functions with bounded continuous partial derivatives up to order $k$.
(Note that [28] states this with full space $H^{-q}(\partial D)$ in place of
the quotient space, but since $\Lambda_{\gamma_{j}}$ maps constant functions
to $0$ for $j=0,1$, the two operator norms coincide.) For $M$ sufficiently
large, we may take $k=\alpha$ to deduce (noting that $C^{\alpha}(D)$
continuously embeds into $H^{\alpha}(D)$) that there exist
$\gamma_{0},\gamma_{1}\in\Gamma^{\alpha}_{m_{0},D_{0}}(M)$ satisfying (a) and
(b). Taking $\xi=\xi_{\varepsilon,\delta^{\prime}}$ we note that (i) holds by
definition.

For (ii), applying Lemma 18 with $s=0$, $p=\min(d-1,r)$ and
$q=(d-1-r)_{+}\equiv\max(d-1-r,0)=d-1-p$ we see that, for a constant
$C=C(d,r)$,

$\lVert\Lambda_{\gamma_{1}}-\Lambda_{\gamma_{0}}\rVert_{\mathbb{H}_{r}}\leq
C\lVert\Lambda_{\gamma_{1}}-\Lambda_{\gamma_{0}}\rVert_{H^{-q}(\partial
D)/\mathbb{C}\to H^{q}_{\diamond}(\partial D)}.$

Thus, appealing to Lemma 9, we can bound the KL-divergences
$K(p^{\gamma_{1}}_{\varepsilon},p^{\gamma_{0}}_{\varepsilon})$ by

$\varepsilon^{-2}\lVert\Lambda_{\gamma_{1}}-\Lambda_{\gamma_{0}}\rVert_{{\mathbb{H}_{r}}}^{2}\leq
C^{2}\varepsilon^{-2}\lVert\Lambda_{\gamma_{1}}-\Lambda_{\gamma_{0}}\rVert_{H^{-q}\to
H^{q}}^{2}\leq
C^{2}\exp\big{[}2\log(1/\varepsilon)-(\log(1/\varepsilon))^{-\frac{\delta^{\prime}d}{(2d-1)\alpha}}\big{]}.$

Since $\delta^{\prime}>\alpha(2d-1)/d$ by assumption, the final expression
tends to zero as $\varepsilon\to 0$ and in particular is less than the $\mu$
of condition (ii) for $\varepsilon$ small enough.

##  Appendix A Laplace–Beltrami eigenfunctions and Sobolev spaces

In this Appendix, we define the Sobolev spaces $H^{r}(D),$ $H^{r}(\partial D)$
for a bounded domain $D\subset\mathbb{R}^{d}$, with smooth boundary $\partial
D$.

######  Definition ($H^{r}(D)$).

We follow [27] to define these Sobolev spaces: see Chapter I, Sections 1.1,
9.1 and 12.1 (pages 1, 40 and 70 respectively) for details. For
$r\in\mathbb{N}\cup\\{0\\}$ we set

$H^{r}(D)=\\{f\in L^{2}(D):\lVert
f\rVert_{H^{r}(D)}^{2}=\sum_{\lvert\alpha\rvert\leq r}\lVert
D^{\alpha}f\rVert_{L^{2}(D)}^{2}<\infty\\},$

where for a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{d}),$
$\alpha_{j}\in\mathbb{N}\cup\\{0\\}$ for $j\leq d$, of order
$\lvert\alpha\rvert=\sum_{j\leq d}\alpha_{j}$,

$D^{\alpha}=\frac{\partial^{\lvert\alpha\rvert}}{\partial
x_{1}^{\alpha_{1}}\dots\partial x_{d}^{\alpha_{d}}},$

with partial derivatives defined in a weak sense. $L^{2}(D)$ is defined with
respect to the Lebesgue measure on $D$. Note $H^{r}(D)$ is a Hilbert space,
with inner product

$\langle f,g\rangle_{H^{r}(D)}=\sum_{\lvert\alpha\rvert\leq r}\int
D^{\alpha}f\cdot\overline{D^{\alpha}g}.$

For $r\in\mathbb{R}$, $r\geq 0$ we then define $H^{r}(D)$ via interpolation.
Finally, defining

$H^{r}_{0}(D)=\\{f\in
H^{r}(D):\tfrac{\partial^{j}f}{\partial\nu^{j}}\big{|}_{\partial D}=0,0\leq
j<r-1/2\\},$

with the normal boundary derivatives defined in a trace sense, we define
$H^{r}(D),r<0,$ as the topological dual space $(H_{0}^{\lvert
r\rvert}(D))^{*}$, equipped with the dual norm. [Cf. also [27] Chapter I,
Sections 11.1, 11.4 and 12.1 on pages 55, 62 and 70.]

For $C^{\infty}_{c}(D)$ the space of smooth functions compactly supported in
$D$, $H^{1}_{\textnormal{loc}}(D)$ is defined as

$H^{1}_{\textnormal{loc}}(D)=\\{f\text{ locally integrable }:f\phi\in
H^{1}(D)\text{ for all $\phi\in C_{c}^{\infty}(D)$}\\}$

(see [27], Chapter II, Section 3.2 on p125), or, equivalently,

$H^{1}_{\textnormal{loc}}(D)=\\{f\text{ locally integrable }:f|_{U}\in
H^{1}(U)\text{ for all domains $U$ satisfying $\bar{U}\subset D$}\\}.$

To define the Sobolev space $H^{r}(\partial D)$ for the compact boundary
manifold $\partial D$, let
$(\phi_{k}=\phi_{k}^{(0)}:k\in\mathbb{N}\cup\\{0\\})$ be an orthonormal basis
of $L^{2}(\partial D)$ consisting of real-valued eigenfunctions of the
Laplace–Beltrami operator $\operatorname{\mathop{{}\Delta}\nolimits}_{\partial
D}$. The basic properties of such a basis are gathered, for example, in Chavel
[7], Chapter I. Let $\lambda_{k}\geq 0$ be the corresponding eigenvalues:

$-\operatorname{\mathop{{}\Delta}\nolimits}_{\partial
D}\phi_{k}=\lambda_{k}\phi_{k}.$

(By convention these are assumed to have been sorted in increasing order.)

######  Definition ($H^{r}(\partial D)$).

For $r\geq 0$, we define

$H^{r}(\partial D)=\\{f\in L^{2}(\partial D)\text{ s.t.
}\sum_{k=0}^{\infty}(1+\lambda_{k})^{r}\lvert\langle
f,\phi_{k}\rangle_{L^{2}(\partial D)}\rvert^{2}=:\lVert
f\rVert_{H^{r}(\partial D)}^{2}<\infty\\},$

where the space $L^{2}(\partial D)$ is defined relative to the surface element
on $\partial D$. For $r<0$, we define $H^{r}(\partial D)$ as the completion of
$L^{2}(\partial D)$ with respect to the norm
$\lVert\cdot\rVert_{H^{r}(\partial D)}$ (defined as in the above display).

######  Remarks.

  1. i.

It is immediate that $\\{\phi_{k}:k\in\mathbb{N}\cup\\{0\\})\\}$ is an
orthogonal spanning set of $H^{r}(\partial D)$, and that setting
$\phi_{k}^{(r)}=(1+\lambda_{k})^{-r/2}\phi_{k}$ yields an orthonormal basis of
$H^{r}(\partial D)$.

  2. ii.

This definition of $H^{r}(\partial D)$ coincides with other possible
definitions. For example, for $r=1$ the calculation

$\int_{\partial
D}\operatorname{\nabla}\phi_{k}\cdot\overline{\operatorname{\nabla}\phi_{l}}=-\int_{\partial
D}\phi_{k}\operatorname{\mathop{{}\Delta}\nolimits}_{\partial
D}\phi_{l}=\lambda_{l}\int_{\partial
D}\phi_{k}\phi_{l}=\lambda_{l}\delta_{kl},$

derived via the divergence theorem on a closed manifold (e.g. see [7] eq (35);
note that the manifold $\partial D$ is compact) implies that our definition of
$\lVert\cdot\rVert_{H^{1}(\partial D)}$ is equivalent to the standard
definition $\lVert f\rVert_{H^{1}(\partial D)}^{2}=\lVert
f\rVert_{L^{2}(\partial
D)}^{2}+\lVert\operatorname{\nabla}f\rVert_{L^{2}(\partial D)}^{2},$ and
inductively the same is true for $H^{r}(\partial D)$, $r\in\mathbb{N}$.

For the equivalence of this definition with some other definitions of negative
or non-integer Sobolev spaces, see [27] Chapter I Section 7.3 (p34-37). In
particular note that $H^{-s}(\partial D)$ is the topological dual space of
$H^{s}(\partial D)$ for any $s\in\mathbb{R}$.

  3. iii.

Note that $\phi_{0}$ is a constant function, hence the $H^{r}(\partial
D)/\mathbb{C}$ norm, defined by $\lVert[f]\rVert_{H^{r}(\partial
D)/\mathbb{C}}=\inf_{z\in\mathbb{C}}\lVert f-z\rVert$ for $[f]$ the
equivalence class over $\mathbb{C}$ of a function $f\in H^{r}(\partial D)$,
can also be characterised as

$\lVert[f]\rVert_{H^{r}(\partial
D)/\mathbb{C}}^{2}=\sum_{k=1}^{\infty}(1+\lambda_{k})^{r}\lvert\langle
f,\phi_{k}\rangle_{L^{2}(\partial D)}\rvert^{2}.$ (48)

Recall also $H^{s}_{\diamond}(\partial D)=\\{g\in H^{s}(\partial D):\langle
g,1\rangle_{L^{2}(\partial D)}=0\\}$ (see 5). Note that each $[f]\in
H^{s}(\partial D)/\mathbb{C}$ has a representative $g\in
H^{s}_{\diamond}(\partial D)$, and $\lVert f\rVert_{H^{s}(\partial
D)/\mathbb{C}}=\lVert g\rVert_{H^{s}(\partial D)}$. We thus use the norm 48 on
spaces $H^{s}(\partial D)/\mathbb{C}$ and on $H^{s}_{\diamond}(\partial D)$
without further mention. We also typically write $f$ for the equivalence class
$[f]$ and only comment further on this where necessary.

This ‘spectral’ definition of $H^{r}(\partial D)$ is useful particularly
because Weyl’s law explicitly specifies the scaling of $\lambda_{k}$ as
$k\to\infty$.

######  Lemma 15 (Weyl’s law on a compact closed manifold, e.g. [7] eq.(49)).

Suppose $M$ is a closed compact manifold of dimension $d$. Then

$N(\lambda)=(2\pi)^{-d}\lambda^{d/2}\omega_{d}\operatorname{Vol}(M)+o(\lambda^{d/2}),$

where $N(\lambda)$ is the number of eigenvalues (counted with multiplicity) no
bigger than $\lambda$ and $\omega_{d}$ is the volume of a unit disc in
$\mathbb{R}^{d}$.

######  Corollary 16.

The eigenvalues of the Laplace–Beltrami operator
$\operatorname{\mathop{{}\Delta}\nolimits}_{\partial D}$ satisfy
$C_{1}k^{2/(d-1)}\leq\lambda_{k}\leq C_{2}k^{2/(d-1)}$ for constants
$C_{1},C_{2}$ depending only on $D$. Hence, the eigenfunctions satisfy

$C_{3}(1+k^{\frac{1}{d-1}})^{s-r}\leq\lVert\phi_{k}^{(r)}\rVert_{H^{s}(\partial
D)}\leq C_{4}(1+k^{\frac{1}{d-1}})^{s-r},~{}s,r\in\mathbb{R},$ (49)

for constants $C_{3}$ and $C_{4}$ depending only on $\partial D$ and on the
difference $s-r$. For $k>0$ the same expression holds with the quotient norm
$\lVert\phi_{k}^{(r)}\rVert_{H^{s}(\partial D)/\mathbb{C}}$ in place of
$\lVert\phi_{k}^{(r)}\rVert_{H^{s}(\partial D)}$.

###### Proof.

We apply Weyl’s law on the manifold $\partial D$, which has dimension $d-1$.
Writing $N(\lambda^{-})$ for $\lim_{x\uparrow\lambda}N(x)$ and
$N(\lambda^{+})$ for $\lim_{x\downarrow\lambda}N(x)$, we thus have

$N(\lambda_{k}^{-})\leq k\leq N(\lambda_{k}^{+}).$

It follows that $C\lambda_{k}^{(d-1)/2}+o(\lambda_{k}^{(d-1)/2})\leq k\leq
C\lambda_{k}^{(d-1)/2}+o(\lambda_{k}^{(d-1)/2})$ for the constant
$C=C(D)=(2\pi)^{-(d-1)}\omega_{d-1}\operatorname{Area}(\partial D)$ and hence
we deduce the scaling of the eigenvalues. Then 49 follows from the definition
of $H^{r}(\partial D)$ and the remarks thereafter. ∎

##  Appendix B Comparison results for Hilbert–Schmidt operators

For separable Hilbert spaces $A$ and $B$, we use the notation
$\mathcal{L}(A,B)$ for the space of bounded linear maps $A\to B$ equipped with
the operator norm $\lVert\cdot\rVert_{A\to B}$, and $\mathcal{L}_{2}(A,B)$ for
the space of Hilbert–Schmidt operators $A\to B$ equipped with inner product
$\langle\cdot,\cdot\rangle_{\mathcal{L}_{2}(A,B)}$ (e.g. see [4, Chapter 12]).
Define the orthonormal basis $(b_{jk}^{(p,q)})$ of
$\mathcal{L}_{2}(H^{p},H^{q})$ by

${b_{jk}^{(p,q)}(f)=\phi_{j}^{(p)}\otimes\phi_{k}^{(q)}(f)=\langle
f,\phi_{j}^{(p)}\rangle_{H^{p}}\phi_{k}^{(q)},\quad j,k\in\mathbb{N},}$

(in this Appendix we omit explicit reference to the domain, writing $H^{p}$
for either $H^{p}(\partial D)/\mathbb{C}$ or $H^{p}_{\diamond}(\partial D)$;
as in remark (iii) in Appendix A, both spaces can be identified with
$\overline{\operatorname{span}}\\{\phi_{k}^{(p)}:k\geq 1\\}$, hence the
omission should not cause confusion). The compatibility between our bases of
$H^{\alpha}(\partial D)$ for different $\alpha\in\mathbb{R}$ means that the
subspaces spanned by $(b_{jk}^{(p,q)})_{j\leq J,k\leq K}$ coincide for all $p$
and $q$, and the $\mathcal{L}_{2}(H^{p},H^{q})$-orthogonal projections onto
this subspace coincide with $\pi_{JK}$ (as defined in 24). Corollary 16
implies the following lemmas controlling Hilbert–Schmidt norms for different
domains and codomains in terms of each other, and in terms of operator norms.

######  Lemma 17.

Let $p,q,r,s\in\mathbb{R}$ and let
$T\in\operatorname{span}\\{b_{jk}^{(p,q)}:1\leq j\leq J,~{}1\leq k\leq K\\}$.
Then there is a constant $C$ depending only on $D$ and on the differences
$r-p,$ $s-q$ such that

$\lVert T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}\leq
C(1+J^{1/(d-1)})^{(p-r)_{+}}(1+K^{1/(d-1)})^{(s-q)_{+}}\lVert
T\rVert_{\mathcal{L}_{2}(H^{p},H^{q})},$

where $x_{+}=\max(x,0)$ for $x\in\mathbb{R}$.

###### Proof.

The coefficients $a_{jk}^{(r,s)}$ of $T$ with respect to the basis
$(b_{jk}^{(r,s)})$ are given by

$a_{jk}^{(r,s)}=\langle
T,b_{jk}^{(r,s)}\rangle_{\mathcal{L}_{2}(H^{r},H^{s})}\equiv\sum_{p}\langle
T\phi_{p}^{(r)},b_{jk}^{(r,s)}(\phi_{p}^{(r)})\rangle_{H^{s}}=\langle
T\phi_{j}^{(r)},\phi_{k}^{(s)}\rangle_{H^{s}}$

and we see from Corollary 16 that

$\lvert a_{jk}^{(r,s)}\rvert\leq
C(1+j^{1/(d-1)})^{p-r}(1+k^{1/(d-1)})^{s-q}\lvert a_{jk}^{(p,q)}\rvert$ (50)

for a constant $C$ depending only on $D$ and the differences $r-p,s-q$.

Upper bounding $(1+j^{1/(d-1)})^{(p-r)}\leq(1+J^{1/(d-1)})^{(p-r)_{+}}$ for
$j\leq J$, and similarly for $k$, we find that

$\lVert T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}^{2}=\sum_{j\leq J,k\leq
K}\lvert a_{jk}^{(r,s)}\rvert^{2}\leq
C(1+J^{1/(d-1)})^{2(p-r)_{+}}(1+K^{1/(d-1)})^{2(s-q)_{+}}\lVert
T\rVert_{\mathcal{L}_{2}(H^{p},H^{q})}^{2},$

hence the result. ∎

######  Lemma 18.

For $p,q,r,s\in\mathbb{R}$ satisfying $p\leq r$ and $q\geq s$, let
$T\in\mathcal{L}(H^{p-(d-1)},H^{q})$. Then we have
$T\in\mathcal{L}_{2}(H^{r},H^{s})$ with, for constant $C$ depending only on
$D$ and the differences $r-p$, $q-s$,

$\lVert T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}\leq C\lVert
T\rVert_{H^{p-(d-1)}\to H^{q}},$

and moreover

$\lVert T-\pi_{JK}T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}\leq C\lVert
T\rVert_{H^{p-(d-1)}\to
H^{q}}\max\big{(}(1+J^{1/(d-1)})^{p-r},(1+K^{1/(d-1)})^{s-q}\big{)}.$

###### Proof.

Firstly, as a consequence of Corollary 16, we have, for a constant $C=C(D)$,

$\lVert T\phi_{j}^{(p)}\rVert_{H^{q}}^{2}\leq\lVert T\rVert_{H^{p-(d-1)}\to
H^{q}}^{2}\lVert\phi_{j}^{(p)}\rVert_{H^{p-(d-1)}}^{2}\leq C\lVert
T\rVert_{H^{p-(d-1)}\to H^{q}}^{2}(1+j^{1/(d-1)})^{-2(d-1)},$ (51)

which is summable over $j$, hence by definition of the space of
Hilbert–Schmidt operators, $T\in\mathcal{L}_{2}(H^{p},H^{q})$ and $\lVert
T\rVert_{\mathcal{L}_{2}(H^{p},H^{q})}\leq C^{\prime}\lVert
T\rVert_{H^{p-(d-1)}\to H^{q}}$. That $T$ lies in
$\mathcal{L}_{2}(H^{r},H^{s})$ and satisfies the the specified bound follows
by monotonicity of $H^{\alpha}$ norms.

Since the $\mathcal{L}_{2}(H^{r},H^{s})$-orthogonal projection maps coincide
for all $r$ and $s$, next defining $a_{jk}^{(r,s)}$ as in the previous proof,
we have from 50 that for a constant $C$,

$\displaystyle\lVert T-\pi_{JK}T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}^{2}$
$\displaystyle=\sum_{j>J\text{ or }k>K}\lvert a_{jk}^{(r,s)}\rvert^{2}$
$\displaystyle\leq C\sum_{j>J\text{ or
}k>K}(1+j^{1/(d-1)})^{2(p-r)}(1+k^{1/(d-1)})^{2(s-q)}\lvert
a_{jk}^{(p,q)}\rvert^{2}$

Since $p\leq r$ and $q\geq s$, we see that

$\displaystyle\sum_{j>J}\sum_{k}(1+j^{1/(d-1)})^{2(p-r)}(1+k^{1/(d-1)})^{2(s-q)}\lvert
a_{jk}^{(p,q)}\rvert^{2}$
$\displaystyle\leq(1+J^{1/(d-1)})^{2(p-r)}\sum_{j>J}\sum_{k}\lvert
a_{jk}^{(p,q)}\rvert^{2}$ $\displaystyle\leq(1+J^{1/(d-1)})^{2(p-r)}\lVert
T\rVert_{\mathcal{L}_{2}(H^{p},H^{q})}^{2}.$

Arguing similarly for the sum over all $j$ and over $k>K$, we deduce that

$\lVert T-\pi_{JK}T\rVert_{\mathcal{L}_{2}(H^{r},H^{s})}^{2}\leq 2C\lVert
T\rVert_{\mathcal{L}_{2}(H^{p},H^{q})}^{2}\max\big{(}(1+J^{1/(d-1)})^{2(p-r)},(1+K^{1/(d-1)})^{2(s-q)}\big{)}.$

The result follows.∎

##  Appendix C Mapping properties of $\Lambda_{\gamma}$ and
$\tilde{\Lambda}_{\gamma}$

In this Appendix we prove the following mapping properties of
$\Lambda_{\gamma}$ and $\tilde{\Lambda}_{\gamma}$ which were used throughout
the main body of the paper. The proofs rely on basic theory for elliptic PDEs,
which can be found in the many books on the subject (we shall refer mostly to
[27]).

######  Lemma 19.

Let $s\in\mathbb{R}$, let $m\in(0,1)$ and let $D^{\prime}$ be a domain
compactly contained in $D$. For $\gamma\in\Gamma_{m,D^{\prime}}$ and $f\in
H^{s+1}(\partial D)/\mathbb{C}$, there is a unique weak solution
$u_{\gamma,f}\in\mathcal{H}_{s}=\big{(}H^{\min\\{1,s+3/2\\}}(D)\cap
H^{1}_{\textnormal{loc}}(D)\big{)}/\mathbb{C}$ to the Dirichlet problem 1.
Moreover, if $u_{1,f}$ is the unique solution when $\gamma=1$, then for any
other $\gamma\in\Gamma_{m,D^{\prime}}$ bounded by $M$ on $D$,
$u_{\gamma,f}-u_{1,f}$ lies in $H^{1}_{0}(D)/\mathbb{C}$ and satisfies the
estimate

$\lVert u_{\gamma,f}-u_{1,f}\rVert_{H^{1}(D)/\mathbb{C}}\leq
C{\tfrac{M}{m}}\lVert f\rVert_{H^{s+1}(\partial D)/\mathbb{C}},$ (52)

for some constant $C=C(D,D^{\prime},s)$.

Lemma 19 is neither novel nor necessarily sharp, but we require explicit
bounds controlling how the constants depend on $\gamma$ for our proofs, which
is why the result is given here.

######  Lemma 20.

For some $m\in(0,1)$ and some domain $D^{\prime}$ compactly contained in $D$,
let $\gamma\in\Gamma_{m,D^{\prime}}$. For each $s\in\mathbb{R}$,
$\Lambda_{\gamma}$ is a continuous linear map from $H^{s+1}(\partial
D)/\mathbb{C}$ to $H^{s}_{\diamond}(\partial D)$, and it is continuously
invertible. For each $s,t\in\mathbb{R}$, the shifted operator
$\tilde{\Lambda}_{\gamma}=\Lambda_{\gamma}-\Lambda_{1}$ is a continuous map
from $H^{s}(\partial D)/\mathbb{C}$ to $H^{t}_{\diamond}(\partial D)$.

Moreover, if $\gamma$ also satisfies the bound
$\lVert\gamma\rVert_{\infty}\leq M$, then we have the explicit bounds

$\displaystyle\lVert\Lambda_{\gamma}\rVert_{H^{s+1}(\partial D)/\mathbb{C}\to
H^{s}(\partial D)}\leq C_{1}\frac{M}{m},$ (53)
$\displaystyle\lVert\tilde{\Lambda}_{\gamma}\rVert_{H^{s}(\partial
D)/\mathbb{C}\to H^{t}(\partial D)}\leq C_{2}{\frac{M}{m}},$ (54)

for constants $C_{1}=C(D,D^{\prime},s)$ and $C_{2}=C_{2}(D,D^{\prime},s,t)$.

Given Lemma 20, the following is an immediate consequence of Lemma 18 (recall
$\mathbb{H}_{r}$ is defined in 10, and note that the proofs of the previous
lemmas imply that $\tilde{\Lambda}_{\gamma}$ maps real-valued functions to
real-valued valued functions).

######  Lemma 21.

For any $r\in\mathbb{R}$ and any $\gamma\in\Gamma_{m,D^{\prime}}$,
$\tilde{\Lambda}_{\gamma}\in\mathbb{H}_{r}$.

A key to proving Lemmas 20 and 19 is the following basic fact about harmonic
functions. For convenience of the reader, we include a proof (following Lemma
A.1 in [19]). Recall that a function $h$ is harmonic on some domain $\Omega$
if $\Delta h=0$ on $\Omega$, where $\Delta$ denotes the Laplacian. Note that
as we have assumed $\gamma=1$ on a neighbourhood $D\setminus D^{\prime}$ of
$\partial D$, our solutions $u_{\gamma,f}$ (and in particular, the
_differences_ $u_{\gamma,f}-u_{\gamma_{0},f}$ between solutions) are harmonic
on this neighbourhood.

######  Lemma 22 (Interior smoothness of harmonic functions).

Let $U_{0},U$ be bounded domains such that $\bar{U}\subseteq U_{0}$. Then for
any $s,t\in\mathbb{R}$, there is a constant $C=C(s,t,U,U_{0})$ such that for
any harmonic function $v\in H^{s}(U)$,

$\lVert v\rVert_{H^{s}(U)/\mathbb{C}}\leq C\lVert
v\rVert_{H^{t}(U_{0})/\mathbb{C}}.$

###### Proof.

By monotonicity of $H^{t}$ norms it suffices to prove the result for $s=t+k$
for $k\in\mathbb{N}$. Let $v\in H^{t}(U_{0})$ represent the equivalence class
and choose a domain $U_{1}$ such that $\bar{U}\subseteq
U_{1}\subseteq\bar{U}_{1}\subseteq U_{0}$. Let $\phi$ be a smooth cutoff
function, identically one on $U_{1}$ and compactly supported in $U_{0}$. For
$z\in\mathbb{C}$ we observe that $\tilde{v}:=(v-z)\phi$ satisfies

$\displaystyle\begin{split}\operatorname{\mathop{{}\Delta}\nolimits}\tilde{v}&=F\quad\text{
in }U_{0}\\\ \tilde{v}&=0\quad\text{ on }\partial U_{0},\end{split}$

where
$F=2\operatorname{\nabla}\phi\cdot\operatorname{\nabla}v+(v-z)\operatorname{\mathop{{}\Delta}\nolimits}\phi$.
Then

$\lVert v\rVert_{H^{t+1}(U_{1})/\mathbb{C}}\leq\lVert
v-z\rVert_{H^{t+1}(U_{1})}\leq\lVert\tilde{v}\rVert_{H^{t+1}(U_{0})}\leq
C\lVert F\rVert_{H^{t-1}(U_{0})},$

by standard elliptic boundary value regularity results (e.g. [27] Chapter II
Remark 7.2 on page 188, with $N=\\{0\\}$ there as we are considering the
standard Laplace equation; in the case $t<1$ Remark 7.2 gives the result with
a different norm in place of the $H^{t-1}(U_{0})$ norm on the right, but the
two norms are equivalent when restricted to functions of compact support in
$U_{0}$). Note

$\lVert F\rVert_{H^{t-1}(U_{0})}\leq C(\phi)(\lVert
v\rVert_{H^{t}(U_{0})/\mathbb{C}}+\lVert v-z\rVert_{H^{t-1}(U_{0})}),$

and optimising across $z\in\mathbb{C}$ yields

$\lVert v\rVert_{H^{t+1}(U_{1})/\mathbb{C}}\leq C\lVert
v\rVert_{H^{t}(U_{0})/\mathbb{C}}.$ (55)

Finally, we choose a finite sequence of domains $(U_{j})_{1\leq j\leq k}$ such
that $U_{k}=U$ and, for $1\leq j\leq k$, $\bar{U}_{j}\subset U_{j-1}$;
applying 55 successively on each pair $(U_{j},U_{j-1})$, we deduce the result.
∎

###### Proof of Lemma 19.

We adapt the proof of Theorem A.2 from Hanke et al. [19] to the Dirichlet
setting here. From standard theory for the Laplacian, for $\gamma=1$ and $f\in
H^{s+1}(\partial D)/\mathbb{C}$ there exists a solution $u_{1,f}\in
H^{s+3/2}(D)/\mathbb{C}$ to the Dirichlet problem 1, and this solution
satisfies

$\lVert u_{1,f}\rVert_{H^{s+3/2}(D)/\mathbb{C}}\leq C\lVert
f\rVert_{H^{s+1}(\partial D)/\mathbb{C}}$ (56)

for a constant $C=C(D,s)$ (e.g. see [27] Chapter II, Remark 7.2 on page 188
with $N=\\{0\\}$). Also note that, as a harmonic function, $u_{1,f}\in
H^{1}_{\textnormal{loc}}(D)/\mathbb{C}$ by Lemma 22.

Define the sesquilinear operator $B_{\gamma}$ and the conjugate linear
operator $A$,

$B_{\gamma}(w,v)=\int_{D}\gamma\operatorname{\nabla}w\cdot\operatorname{\nabla}\overline{v},\quad\text{and}\quad
A(v)=-\int_{D}\gamma\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v},$

and consider the equation

$B_{\gamma}(w,v)=A(v)\quad\forall v\in H_{0}^{1}(D).$ (57)

Observe that if $w\in H_{0}^{1}(D)$ solves 57, then $u_{\gamma,f}=w+u_{1,f}$
solves the weak Dirichlet problem 4; note that $u_{\gamma,f}$ so defined lies
in $\mathcal{H}_{s}$. We will use Lax–Milgram theory to show the existence and
uniqueness of such a $w$.

By definition any $\gamma\in\Gamma_{m,D^{\prime}}$ is bounded; consistenly
with the second half of the Lemma let $M$ be an upper bound for
$\lVert\gamma\rVert_{\infty}$. Let $v,w\in H_{0}^{1}(D)$. By the
Cauchy–Schwarz inequality, $\lvert B_{\gamma}(w,v)\rvert\leq M\lVert
w\rVert_{H^{1}(D)}\lVert v\rVert_{H^{1}(D)}$, and we also note
$B_{\gamma}(v,v)\geq m\lVert\operatorname{\nabla}v\rVert_{L^{2}(D)}^{2}\geq
cm\lVert v\rVert_{H^{1}(D)}^{2}$, where the latter inequality, with constant
$c=c(D)$, is the Poincaré inequality (e.g., Corollary 6.31 in [1] applied to
$v\in H^{1}_{0}(D)$). In other words, $B_{\gamma}$ is bounded and coercive.
Next, since $\gamma=1$ on $D\setminus D^{\prime}$, for $v\in H^{1}_{0}(D)$, an
application of the divergence theorem yields

$\displaystyle-\int_{D\setminus
D^{\prime}}\gamma\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}$
$\displaystyle=-\int_{D}\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}+\int_{D^{\prime}}\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}$
$\displaystyle=\int_{D}\overline{v}\Delta u_{1,f}-\int_{\partial
D}\overline{v}\frac{\partial
u_{1,f}}{\partial\nu}+\int_{D^{\prime}}\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}=\int_{D^{\prime}}\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}$

It follows, since $\lVert
1-\gamma\rVert_{\infty}\leq\lVert\gamma\rVert_{\infty}\leq M$, that

$\displaystyle\lvert A(v)\rvert$
$\displaystyle=\Big{\lvert}\int_{D^{\prime}}-\gamma\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}{\overline{v}}-\int_{D\setminus
D^{\prime}}\gamma\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}\Big{\rvert}=\Big{\lvert}\int_{D^{\prime}}(1-\gamma)\operatorname{\nabla}u_{1,f}\cdot\operatorname{\nabla}\overline{v}\Big{\rvert}$
$\displaystyle\leq M\lVert
v\rVert_{H^{1}(D)}\lVert\operatorname{\nabla}u_{1,f}\rVert_{L^{2}(D^{\prime})}.$

By Lemma 22 and recalling 56, there are constants $C$ and $C^{\prime}$
depending only on $D,D^{\prime}$ and $s$ such that

$\lVert\operatorname{\nabla}u_{1,f}\rVert_{L^{2}(D^{\prime})}\leq\lVert
u_{1,f}\rVert_{H^{1}(D^{\prime})/\mathbb{C}}\leq C^{\prime}\lVert
u_{1,f}\rVert_{H^{s+3/2}(D)/\mathbb{C}}\leq C\lVert f\rVert_{H^{s+1}(\partial
D)/\mathbb{C}}.$

Thus, $\lvert A(v)\rvert\leq CM\lVert f\rVert_{H^{s+1}(\partial D)}\lVert
v\rVert_{H^{1}(D)}$. We deduce from the Lax–Milgram theorem (e.g., Theorem 1
in Section 6.2.1 of [11]; note while the theorem there is stated for real
scalars and bilinear maps, the same proof works for complex scalars and
sesquilinear maps) that 57 has a unique solution $w\in H^{1}_{0}(D)$.
Moreover, the equation $B_{\gamma}(w,w)=A(w)$ shows that $\lVert
w\rVert_{H^{1}(D)}$ is upper bounded the operator norm of $A$ divided by the
coercivity constant of $B_{\gamma}$, yielding 52.

It remains to show that the (equivalence class of) function(s) $u_{\gamma,f}$
so constructed is the unique solution in $\mathcal{H}_{s}$ to 4. Since we have
shown uniqueness of $w$, it is enough to show that the difference $h$ between
two $\mathcal{H}_{s}$ solutions lies in $H_{0}^{1}(D)$, as then it must be the
zero function. (We are considering $h$ as a function, rather than an
equivalence class of functions, which we can do by for example choosing a
representative with average zero.) This is clear for $s\geq-1/2$ as then
$\mathcal{H}_{s}=H^{1}(D)/\mathbb{C}$, and can be shown also for $s<-1/2$ as
in [19], Theorem A.2.∎

###### Proof of Lemma 20.

We first remark, for $\gamma\in\Gamma_{m,D^{\prime}}$, that by the divergence
theorem,

$\big{\langle}\tfrac{\partial u}{\partial\nu},1\big{\rangle}_{L^{2}(\partial
D)}=\int_{\partial D}\gamma\tfrac{\partial
u}{\partial\nu}=\int_{D}\operatorname{\operatorname{\nabla}\cdot}(\gamma\operatorname{\nabla}u)=0$

for a solution $u$ to the Dirichlet problem 1, so that it suffices to prove
53, 54, and the continuity of $\Lambda_{\gamma}^{-1}:H^{s}_{\diamond}(\partial
D)\to H^{s+1}(\partial D)/\mathbb{C}$.

We first prove 54, by adapting the proof of Theorem A.3 from [19] and tracking
the constants. Given $f\in H^{s}(\partial D)$ let
$u_{\gamma,f}\in\mathcal{H}_{s-1}$ be the unique solution to the Dirichlet
problem 1 and let $w\in H^{1}_{0}$ be a representative of the function class
$u_{\gamma,f}-u_{1,f}$. Choose a domain $\Omega$ with smooth boundary
$\partial\Omega$, satisfying
$\bar{D}^{\prime}\subset\Omega\subset\bar{\Omega}\subset D.$ Choose also
domains $U,U_{0}$ with smooth boundaries such that

$\partial\Omega\subseteq U\subset\bar{U}\subset U_{0}\subset\bar{U}_{0}\subset
D\setminus D^{\prime}.$

We can apply an appropriate trace theorem ([27] Chapter I Theorem 9.4 for
$t>0$, Chapter II Theorem 6.5 (and Remark 6.4) for $t\leq-3/2$, or Chapter II
Theorem 7.3 for $-3/2<t<1/2$; note in the latter two cases we use that $w$ is
harmonic on $D\setminus\bar{\Omega}$) to $w-z$ and optimise across
$z\in\mathbb{C}$ to see

$\lVert\partial w/\partial\nu\rVert_{H^{t}(\partial D)}\leq C\lVert
w\rVert_{H^{t+3/2}(D\setminus\bar{\Omega})/\mathbb{C}}.$ (58)

Applying [27] Chapter II Remark 7.2 on p.188 with $N=\\{0\\}$ we see

$\lVert w\rVert_{H^{t+3/2}(D\setminus\bar{\Omega})/\mathbb{C}}\leq
C\big{(}\lVert\operatorname{tr}w\rVert_{H^{t+1}(\partial
D)/\mathbb{C}}+\lVert\operatorname{tr}w\rVert_{H^{t+1}(\partial\Omega)/\mathbb{C}}\big{)}=C\lVert\operatorname{tr}w\rVert_{H^{t+1}(\partial\Omega)/\mathbb{C}}.$

Again applying an appropriate trace theorem, this time on a subset of $U$
bounded on one side by $\partial\Omega$, and applying Lemma 22, we see

$\lVert\operatorname{tr}w\rVert_{H^{t+1}(\partial\Omega)/\mathbb{C}}\leq
C\lVert w\rVert_{H^{t+3/2}(U)/\mathbb{C}}\leq C^{\prime}\lVert
w\rVert_{H^{1}(U_{0})/\mathbb{C}}.$

The constants in the above depend on $D$ and (via $\Omega,$ $U$ and $U_{0}$)
on $D^{\prime}$, but are otherwise independent of $\gamma$. Recalling 52,
which, because the smoothness of $f$ here is $s$, tells us $\lVert
w\rVert_{H^{1}(U_{0})/\mathbb{C}}\leq C\big{(}\frac{M}{m}\big{)}\lVert
f\rVert_{H^{s}(D)/\mathbb{C}}$, we overall have $\lVert\partial
w/\partial\nu\rVert_{H^{t}(\partial D)}\leq C\big{(}\frac{M}{m}\big{)}\lVert
f\rVert_{H^{s}(\partial D)/\mathbb{C}}$, so that we have proved 54.

Now we prove 53; given 54, it suffices to show
$\lVert\Lambda_{1}\rVert_{H^{s+1}(\partial D)/\mathbb{C}\to H^{s}(\partial
D)}\leq C\tfrac{M}{m}$ for an appropriate constant $C$. Since $u_{1,f}$ is
harmonic on $D$, applying the trace theorems from [27] as for 58 yields

$\lVert\partial u_{1,f}/\partial\nu\rVert_{H^{s}(\partial D)}\leq C\lVert
u_{1,f}\rVert_{H^{s+3/2}(D)/\mathbb{C}}$

for a constant $C=C(D,s)$. Then [27] Chapter II Remark 7.2 with $N=\\{0\\}$
yields

$\lVert u_{1,f}\rVert_{H^{s+3/2}(D)/\mathbb{C}}\leq C^{\prime}\lVert
f\rVert_{H^{s+1}(\partial D)/\mathbb{C}}$

for a constant $C^{\prime}=C^{\prime}(D,s)$, and 53 follows.

Finally we remark that the same arguments (see Theorem A.3 in [19]) applied to
the inverse of $\Lambda_{\gamma}$, which is the _Neumann-to-Dirichlet map_ ,
show that this is continuous from $H^{s}_{\diamond}(\partial D)$ to
$H^{s+1}(\partial D)/\mathbb{C}$ as claimed. ∎

##  Appendix D Asymptotic comparison of noise models

In this Appendix we formulate and prove the asymptotic comparison results
discussed informally in Section 2.

###  D.1 A brief overview of the Le Cam distance

We first define the Le Cam deficiency and the Le Cam distance, providing the
sense in which the models will be shown to be asymptotically related to each
other. The concepts throughout this Section are drawn from Le Cam’s 1986
monograph [25]. We refer to the expository paper of Mariucci [29] for a
gentler introduction to the area.

######  Definitions.

Statistical experiment

A _statistical experiment_ , or just _experiment_ , is the triple
$(\mathcal{X},\mathcal{F},\\{P_{\theta}\\}_{\theta\in\Theta})$, where for each
$\theta$ in the parameter space $\Theta$, $P_{\theta}$ is a probability
measure on the measurable space $(\mathcal{X},\mathcal{F})$.

Markov Kernel

For measurable spaces $(\mathcal{X}_{i},\mathcal{F}_{i})$, $i=1,2$, a _Markov
kernel with source $(\mathcal{X}_{1},\mathcal{F}_{1})$ and target
$(\mathcal{X}_{2},\mathcal{F}_{2})$_ is a map
$T:\mathcal{X}_{1}\times\mathcal{F}_{2}\to[0,1]$ such that $T(x,\cdot)$ is a
probability measure for each $x\in\mathcal{X}_{1}$, and $T(\cdot,A)$ is
measurable for each $A\in\mathcal{F}_{2}$.

Given a (deterministic) measurable function
$F:\mathcal{X}_{1}\to\mathcal{X}_{2}$ we will denote by $T_{F}$ the Markov
kernel

$T_{F}(x,A)=\mathbbm{1}\\{F(x)\in A\\}.$ (59)

Le Cam deficiency

The _Le Cam deficiency_ between experiments $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$, where
$\mathcal{E}_{i}=(\mathcal{X}_{i},\mathcal{F}_{i},\\{P_{i,\theta}\\}_{\theta\in\Theta})$
for $i=1,2$, for a common parameter space $\Theta$, is

$\delta(\mathcal{E}_{1},\mathcal{E}_{2})=\inf_{T}\sup_{\theta\in\Theta}\lVert
TP_{1,\theta}-P_{2,\theta}\rVert_{\textnormal{TV}},$

where the infimum is over all Markov kernels with source
$(\mathcal{X}_{1},\mathcal{F}_{1})$ and target
$(\mathcal{X}_{2},\mathcal{F}_{2})$. The measure $TP_{1,\theta}$ is defined as

$TP_{1,\theta}(A)=\int_{\mathcal{X}_{1}}T(x,A)\mathop{}\\!\mathrm{d}P_{1,\theta}(x),$

and $\lVert\cdot\rVert_{\textnormal{TV}}$ denotes the total variation norm on
signed measures,

$\lVert\nu\rVert_{\textnormal{TV}}=\sup_{A}\lvert\nu(A)\rvert.$

The Le Cam deficiency satisfies the triangle inequality, but is not symmetric.

Le Cam distance

The _Le Cam distance_ between experiments $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ on a common parameter space $\Theta$ is

$\Delta(\mathcal{E}_{1},\mathcal{E}_{2})=\max(\delta(\mathcal{E}_{1},\mathcal{E}_{2}),\delta(\mathcal{E}_{2},\mathcal{E}_{1})).$

If we identify experiments whose Le Cam distance is zero, this defines a
proper metric.

######  Remark.

Given any action set $A$, any loss function $L:\Theta\times A\to[0,1]$, and
any decision rule $\rho_{2}:\mathcal{X}_{2}\to A$, there exists a (possibly
randomised) decision rule $\rho_{1}:\mathcal{X}_{1}\to A$ such that, denoting
the risk functions by $R_{j}(\rho_{j},\theta)=E_{X\sim
P_{j,\theta}}L(\theta,\rho_{j}(X)),$ $j=1,2$, we have (see [29] Theorem 2.7)

$R_{1}(\rho_{1},\theta)\leq
R_{2}(\rho_{2},\theta)+\delta(\mathcal{E}_{1},\mathcal{E}_{2}),\quad\forall\theta\in\Theta.$
(60)

We may further take the supremum over $\theta$ and then the infimum over
$\rho_{2}$ (note that the infimum over decision rules $\rho_{1}$ associated as
above to decision rules $\rho_{2}$ upper bounds the infimum over all decision
rules $\rho_{1}$) to see that the minimax risks
$R_{j,\textnormal{minimax}}=\inf_{\rho_{j}}\sup_{\theta}R_{j}(\rho_{j},\theta)$,
$j=1,2$ satisfy

$R_{1,\textnormal{minimax}}\leq
R_{2,\textnormal{minimax}}+\delta(\mathcal{E}_{1},\mathcal{E}_{2}).$ (61)

These equations capture the intuitive definition that the Le Cam deficiency is
the worst-case error we incur when reconstructing a decision rule in
$\mathcal{E}_{2}$ using data from $\mathcal{E}_{1}$ (see also Theorem 27 where
this intuition is made concrete for the models considered here).

We gather the key tools we will use to control Le Cam deficiencies in the
following Lemma. Recall that $K(p,q)$ denotes the Kullback–Leibler divergence
between distributions with densities $p$ and $q$; in an abuse of notation we
will in this section also write $K(P,Q)$ for the Kullback–Leibler divergence
between distributions $P$ and $Q$.

######  Lemma 23.

Let $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ be experiments with a common
parameter set $\Theta$: write
$\mathcal{E}_{j}=(\mathcal{X}_{j},\mathcal{F}_{j},\\{P_{j,\theta}\\}_{\theta\in\Theta})$.

  1. a.

Suppose further that the experiments are defined on a common probability
space, i.e. that $\mathcal{X}_{1}=\mathcal{X}_{2}$ and
$\mathcal{F}_{1}=\mathcal{F}_{2}$. Then

$\Delta(\mathcal{E}_{1},\mathcal{E}_{2})\leq\sup_{\theta\in\Theta}\lVert
P_{1,\theta}-P_{2,\theta}\rVert_{\textnormal{TV}}\leq\sup_{\theta\in\Theta}\sqrt{K(P_{1,\theta},P_{2,\theta})/2}.$
(62)

  2. b.

Let $F:\mathcal{X}_{1}\to\mathcal{X}_{2}$ be any (deterministic) measurable
map. Then

$\delta(\mathcal{E}_{1},\mathcal{E}_{2})\leq\sup_{\theta\in\Theta}\lVert
P_{1,\theta}\circ F^{-1}-P_{2,\theta}\rVert_{\textnormal{TV}}.$ (63)

  3. c.

Let $F:\mathcal{X}_{1}\to\mathcal{X}_{2}$ be a measurable map. Suppose that
$P_{1,\theta}\circ F^{-1}=P_{2,\theta}$ for each $\theta\in\Theta$ and suppose
that $F(X)$ is a sufficient statistic for $X\sim P_{1,\theta}$. Then
$\Delta(\mathcal{E}_{1},\mathcal{E}_{2})=0.$

###### Proof.

  1. a.

The first inequality is immediate from the definition, since the Markov kernel
$T_{\operatorname{Id}}$ corresponding to the identity map
$\operatorname{Id}:\mathcal{X}_{1}\to\mathcal{X}_{2}=\mathcal{X}_{1}$
satisfies $T_{\operatorname{Id}}P=P$ for all probability measures $P$ on
$(\mathcal{X}_{1},\mathcal{F}_{1})$. The second inequality is Pinsker’s
inequality (e.g. Proposition 6.1.7a in [15]).

  2. b.

Observe that $T_{F}P_{1,\theta}(A)=P_{1,\theta}(F(X)\in A)=P_{1,\theta}\circ
F^{-1}(A).$ The result follows.

  3. c.

See [29], Property 3.12.∎

###  D.2 Le Cam deficiency bounds for noisy Calderón problems

First we recall the noise models under consideration. The following displays
are labelled with the notation we will use for experiments with the specified
data, in each case taking the parameter space to be
$\\{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{\infty}\leq M\\}$ for
some constants $m\in(0,1),M>1$ and some domain $D^{\prime}$ compactly
contained in $D$.

For noise level $\varepsilon>0$ and some $P\in\mathbb{N}$, in the ‘electrode’
model 7 we are given data

$\mathcal{E}_{0}:\quad
Y_{p,q}=\langle\tilde{\Lambda}_{\gamma}[\psi_{p}],\psi_{q}\rangle_{L^{2}(\partial
D)}+\varepsilon g_{p,q},\quad p,q\leq P,\quad
g_{p,q}\overset{iid}{\sim}N(0,1),$

where $\psi_{p}=c_{p}\mathbbm{1}_{I_{p}}$ for some disjoint (measurable) sets
$I_{p}=I_{p,P}\subseteq\partial D$, with constants $c_{p}$ chosen such that
the $\psi_{p}$ are $L^{2}(\partial D)$ orthonormal. For some
$\varepsilon>0,~{}r\in\mathbb{R},$ and $J,K\in\mathbb{N}$, in the ‘discrete
spectral’ model 8 we are given data

$\mathcal{E}_{1}^{(r)}:\quad
Y_{j,k}=\langle\tilde{\Lambda}_{\gamma}[\phi_{j}^{(r)}],\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}+\varepsilon g_{j,k},\quad j\leq J,k\leq K,\quad
g_{j,k}\overset{iid}{\sim}N(0,1),$

for a (real-valued) Laplace–Beltrami basis $(\phi_{k}^{(r)}:{k\in\mathbb{N}})$
of $H^{r}(\partial D)/\mathbb{C}$. For $\varepsilon>0$ and $r\in\mathbb{R}$,
in the ‘continuous’ model 12 (see also the equivalent 11) we are given data

$\mathcal{E}_{2}^{(r)}:\quad
Y=\tilde{\Lambda}_{\gamma}+\varepsilon\mathbb{W},\quad\text{$\mathbb{W}$ a
Gaussian white noise indexed by $\mathbb{H}_{r}$.}$

We start with the following result which shows how to approximate the discrete
spectral model using the electrode model.

######  Theorem 24.

Suppose $\cup_{p\leq P}I_{p}=\partial D$ and
$\operatorname{diam}(I_{p})\leq(A/P)^{1/(d-1)}$ for a constant $A$ independent
of $P$, where $\operatorname{diam}S=\sup_{x,y\in S}\lvert x-y\rvert$ denotes
the Euclidean diameter of a subset $S\subset\mathbb{R}^{d}$. Then the one-way
Le Cam deficiency $\delta(\mathcal{E}_{0},\mathcal{E}_{1}^{(0)})$ satisfies

$\delta(\mathcal{E}_{0},\mathcal{E}_{1}^{(0)})\leq
C\big{(}\max(J,K)^{(5d-2)/(2d-2)}+\varepsilon^{-1}\max(J,K)^{3d/(2d-2)}\big{)}P^{-1/(d-1)},$

for some constant $C=C(A,D^{\prime},D,M,m)$, and hence vanishes asymptotically
if $P$ is large enough compared to $\varepsilon^{-1},$ $J$ and $K$.

######  Remarks.

  1. i.

The conditions on $(I_{p})_{p\leq P}$ are only used to prove that we can
approximate any Laplace–Beltrami eigenfunction using a linear combination of
the $(\psi_{p})_{p\leq P}$ with $L^{2}(\partial D)$-norm approximation error
proportional to $P^{-1/(d-1)}$(Lemma 25). If $(\psi_{p})_{p\leq P}$ are such
that we can approximate Laplace–Beltrami eigenfunctions at a rate $f(P)$ then
we achieve the result with $f(P)$ in place of $P^{-1/(d-1)}$ .

  2. ii.

The given conditions are naturally satisfied by ‘evenly spaced’ sets
$(I_{p})_{p\leq P}$ partitioning the boundary $\partial D$, with a constant
$A$ depending only on the domain $D$. This can be seen by considering the
covering numbers $N(\partial D,d_{\partial D},\delta)$ (the smallest number of
$d_{\partial D}$ balls of radius $\delta$ needed to cover $\partial D$) for
$d_{\partial D}$ the geodesic distance. Theorem 4.5 in Geller & Pensenson [13]
applied to the current setting yields

$N(\partial D,d_{\partial D},\delta)\leq A\delta^{-(d-1)}$

for a constant $A=A(D)$, for any $\delta>0$. Taking $\delta=2(A/P)^{1/(d-1)}$
we deduce that there exist $P$ balls of ($d_{\partial D}$)-radius $\delta/2$
covering $D$. To construct $P$ disjoint subsets of diameter at most $\delta$,
we simply assign each $x\in\partial D$ to exactly one of the balls containing
it (note the Euclidean diameter is upper bounded by the geodesic diameter,
because any geodesic path on the surface $\partial D$ is also a path in
Euclidean space).

###### Proof.

Let $(Y_{p,q})$ be the data from experiment $\mathcal{E}_{0}$. Let
$\phi^{P}_{j}$ denote the $L^{2}$-orthogonal projection of $\phi_{j}^{(0)}$
onto $\operatorname{span}\\{\psi_{p}:p\leq P\\}$, and write
$a_{jp}=\langle\phi_{j}^{(0)},\psi_{p}\rangle_{L^{2}(\partial
D)}\in\mathbb{R}$, so that $\phi^{P}_{j}=\sum_{p=1}^{P}a_{jp}\psi_{p}$. Define
$F:\mathbb{R}^{P\times P}\to\mathbb{R}^{J\times K}$ via

$F((u_{pq})_{p,q\leq P})_{jk}=\sum_{p,q\leq P}a_{jp}a_{kq}u_{pq}.$

For parameter set
$\\{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{\infty}\leq M\\}$, let
$\mathcal{E}_{0}^{\prime}$ denote the experiment with data

$\mathcal{E}_{0}^{\prime}:\quad Y_{j,k}^{\prime}=F((Y_{p,q})_{p,q\leq
P})_{jk}=\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{P},\phi_{k}^{P}\rangle_{L^{2}(\partial
D)}+\varepsilon g^{\prime}_{j,k},$ (64)

where we define $g^{\prime}_{j,k}=\sum_{p,q}a_{jp}a_{kq}g_{p,q},$ and let
$\mathcal{E}_{1}^{\prime}$ denote the experiment with data 64 but for i.i.d.
Gaussian noise. By 23b we see that
$\delta(\mathcal{E}_{0},\mathcal{E}_{0}^{\prime})=0$, so by the triangle
inequality we deduce

$\delta(\mathcal{E}_{0},\mathcal{E}_{1}^{(0)})\leq\delta(\mathcal{E}_{0},\mathcal{E}_{0}^{\prime})+\delta(\mathcal{E}_{0}^{\prime},\mathcal{E}_{1}^{\prime})+\delta(\mathcal{E}_{1}^{\prime},\mathcal{E}_{1}^{(0)})\leq\Delta(\mathcal{E}_{0}^{\prime},\mathcal{E}_{1}^{\prime})+\Delta(\mathcal{E}_{1}^{\prime},\mathcal{E}_{1}^{(0)}).$

We control the terms on the right.

#####  $\Delta(\mathcal{E}_{0}^{\prime},\mathcal{E}_{1}^{\prime})$:

The covariance of $(g_{j,k}^{\prime})$ is given by

$\operatorname{Cov}(g^{\prime}_{j,k},g^{\prime}_{l,m})=\langle\phi^{P}_{j},\phi^{P}_{l}\rangle_{L^{2}(\partial
D)}\langle\phi^{P}_{k},\phi^{P}_{m}\rangle_{L^{2}(\partial D)}.$

Writing

$\langle\phi^{P}_{j},\phi^{P}_{l}\rangle_{L^{2}(\partial
D)}=\langle\phi_{j}^{(0)},\phi_{l}^{(0)}\rangle_{L^{2}(\partial
D)}+\langle\phi_{j}^{(0)},\phi^{P}_{l}-\phi_{l}^{(0)}\rangle_{L^{2}(\partial
D)}+\langle\phi^{P}_{j}-\phi_{j}^{(0)},\phi^{P}_{l}\rangle_{L^{2}(\partial
D)},$

and applying the Cauchy–Schwarz inequality (note also that
$\lVert\phi_{l}^{P}\rVert_{L^{2}(\partial
D)}\leq\lVert\phi_{l}^{(0)}\rVert_{L^{2}(\partial D)}=1$) we see that

$\lvert\langle\phi_{j}^{P},\phi_{l}^{P}\rangle_{L^{2}(\partial
D)}-\delta_{jl}\rvert\leq\lVert\phi_{j}^{P}-\phi_{j}^{(0)}\rVert_{L^{2}(\partial
D)}+\lVert\phi_{l}^{P}-\phi_{l}^{(0)}\rVert_{L^{2}(\partial D)}.$

A similar decomposition further yields

$\lvert\langle\phi_{j}^{P},\phi_{l}^{P}\rangle_{L^{2}(\partial
D)}\langle\phi_{k}^{P},\phi_{m}^{P}\rangle-\delta_{jl}\delta_{km}\rvert\leq
4\max_{i\leq\max(J,K)}\lVert\phi_{i}^{P}-\phi_{i}^{(0)}\rVert_{L^{2}(\partial
D)}.$

Lemma 25 controls this $L^{2}$ approximation error, yielding that for a
constant $C=C(A,D)$,

$\lvert\operatorname{Cov}(g_{j,k}^{\prime},g_{l,m}^{\prime})-\delta_{jl}\delta_{km}\rvert\leq
C\max(J,K)^{(1+d/2)/(d-1)}P^{-1/(d-1)}.$

Thus, controlling the Le Cam distance between Gaussian experiments with equal
means by $\sqrt{2}$ times the Frobenius distance between the covariance
matrices (e.g. as in the proof of Theorem 3.1 in [41]) yields, for constants
$C^{\prime},c^{\prime}$,

$\displaystyle\Delta(\mathcal{E}_{0}^{\prime},\mathcal{E}_{1}^{\prime})$
$\displaystyle\leq\sqrt{2}\Big{(}\sum_{j,l\leq J,}\sum_{k,m\leq
K}(\operatorname{Cov}(g^{\prime}_{j,k},g^{\prime}_{l,m})-\delta_{jl}\delta_{km})^{2}\Big{)}^{1/2}\leq
c^{\prime}JK\max(J,K)^{(1+d/2)/(d-1)}P^{-1/(d-1)}$ $\displaystyle\leq
C^{\prime}\max(J,K)^{(5d-2)/(2d-2)}P^{-1/(d-1)}.$

#####  $\Delta(\mathcal{E}_{1}^{\prime},\mathcal{E}_{1}^{(0)})$:

Explicitly calculating the Kullback–Leibler divergence between multivariate
normals with the same covariance matrix (cf. the similar calculation in Lemma
9) and using 23a yields

$\Delta(\mathcal{E}_{1}^{\prime},\mathcal{E}_{1}^{(0)})\leq\tfrac{1}{2}\varepsilon^{-1}\times\sup_{\gamma\in\Gamma_{m,D^{\prime}}:\lVert\gamma\rVert_{\infty}\leq
M}\big{\lVert}\big{(}\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(0)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}-\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{P},\phi_{k}^{P}\rangle_{L^{2}(\partial
D)}\big{)}_{j\leq J,k\leq K}\big{\rVert}_{\mathbb{R}^{J\times K}},$

where the norm on the right is the usual Frobenius or Hilbert–Schmidt norm on
the space of $J\times K$ matrices. By Lemma 20,
$\lVert\tilde{\Lambda}_{\gamma}\rVert_{L^{2}(\partial D)\to L^{2}(\partial
D)}$ is bounded by a constant $C=C(D,D^{\prime},M,m)$, hence applying also
Lemma 25 (as well as the Cauchy–Schwarz inequality) we have for a different
constant $C^{\prime}=C^{\prime}(A,D,D^{\prime},M,m)$,

$\displaystyle(\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{(0)},\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}-\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{P},\phi_{k}^{P}\rangle_{L^{2}(\partial
D)})^{2}$
$\displaystyle=(\langle\tilde{\Lambda}_{\gamma}(\phi_{j}^{(0)}-\phi_{j}^{P}),\phi_{k}^{(0)}\rangle_{L^{2}(\partial
D)}+\langle\tilde{\Lambda}_{\gamma}\phi_{j}^{P},\phi_{k}^{(0)}-\phi_{k}^{P}\rangle_{L^{2}(\partial
D)})^{2}$ $\displaystyle\leq
C^{2}(\lVert\phi_{j}^{(0)}-\phi_{j}^{P}\rVert_{L^{2}(\partial
D)}+\lVert\phi_{k}^{(0)}-\phi_{k}^{P}\rVert_{L^{2}(\partial D)})^{2}$
$\displaystyle\leq C^{\prime}\max(J,K)^{(2+d)/(d-1)}P^{-2/(d-1)}.$

Summing over $j$ and $k$ we deduce
$\Delta(\mathcal{E}_{1}^{\prime},\mathcal{E}_{1}^{(0)})\leq
C^{\prime}\varepsilon^{-1}\max(J,K)^{3d/(2d-2)}P^{-1/(d-1)},$ concluding the
proof. ∎

######  Lemma 25.

Under the hypotheses of Theorem 24, let $\phi^{P}_{j}$ denote the
$L^{2}$-orthogonal projection of $\phi_{j}^{(0)}$ onto
$\operatorname{span}\\{\psi_{p}:p\leq P\\}$. Then there is a constant $C$
depending only on the constant $A$ of Theorem 24 and on $D$ such that, for
$j\leq\max(J,K)$,

$\lVert\phi_{j}^{(0)}-\phi^{P}_{j}\rVert_{L^{2}(\partial D)}^{2}\leq
C\max(J,K)^{(2+d)/(d-1)}P^{-2/(d-1)}.$ (65)

###### Proof.

Since $\phi^{P}_{j}$ as the $L^{2}$-orthogonal projection minimises the
$L^{2}$ distance to $\phi_{j}^{(0)}$ of any function in
$\operatorname{span}\\{\psi_{p}:p\leq P\\}$, for any points $x_{p}\in I_{p}$
we see

$\displaystyle\lVert\phi_{j}^{(0)}-\phi^{P}_{j}\rVert_{L^{2}(\partial D)}^{2}$
$\displaystyle\leq\lVert\phi_{j}^{(0)}-\sum_{p=1}^{P}\phi_{j}^{(0)}(x_{p})\mathbbm{1}_{I_{p}}\rVert_{L^{2}(\partial
D)}^{2}$ $\displaystyle\leq\max_{p\leq
P}(\operatorname{diam}(I_{p})^{2})\sum_{p=1}^{P}\int_{I_{p}}\frac{\lvert\phi_{j}^{(0)}(x)-\phi_{j}^{(0)}(x_{p})\rvert^{2}}{\lvert
x-x_{p}\rvert^{2}}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq(A/P)^{2/(d-1)}\lVert\phi_{j}^{(0)}\rVert_{\textnormal{Lip}}^{2}\operatorname{Area}(\partial
D).$

Using a Sobolev embedding for the compact manifold $\partial D$, we may
estimating the Lipschitz constant of $\phi_{j}^{(0)}$ by a constant times
$\lVert\phi_{j}^{(0)}\rVert_{H^{\kappa}(\partial D)}$ for any
$\kappa>1+(d-1)/2$. In particular, taking $\kappa=1+d/2$, we see that the
final expression is bounded by $C\max(J,K)^{(2+d)/(d-1)}P^{-2/(d-1)}$ for some
$C=C(A,D)$ by Corollary 16. ∎

The following theorem shows that the ‘discrete spectral’ and the continuous
measurement models are very close to each other.

######  Theorem 26.

For any $r\in\mathbb{R}$ and any $\nu>0$ there is a constant
$C=C(\nu,r,D,D_{0},M,m)$ such that the Le Cam distance
$\Delta(\mathcal{E}_{1}^{(r)},\mathcal{E}_{2}^{(r)})$ satisfies

$\Delta(\mathcal{E}_{1}^{(r)},\mathcal{E}_{2}^{(r)})\leq
C\varepsilon^{-1}\min(J,K)^{-\nu}.$

###### Proof.

We introduce the experiments $\mathcal{E}_{i}^{(r)}$, $i=3,4$ with parameter