# Existence of Approximately Macroscopically Unique States

Huaxin Lin

###### Abstract

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the
$C^{*}$-algebra of bounded operators on $H.$ Suppose that
$T_{1},T_{2},...,T_{n}$ are self-adjoint operators in $B(H).$ We show that, if
commutators $[T_{i},T_{j}]$ are sufficiently small in norm, then
“Approximately Macroscopically Unique” states always exist for any values in a
synthetic spectrum of the $n$-tuple of self-adjoint operators. This is
achieved under the circumstance for which the $n$-tuple may not be
approximated by commuting ones. This answers a question proposed by David
Mumford for measurements in quantum theory. If commutators are not small in
norm but small modulo compact operators, then “Approximate Macroscopic
Uniqueness” states also exist.

##  1 Introduction

In quantum mechanics, macroscopic observables may be represented by bounded
self-adjoint operators $T_{1},T_{2},...,T_{n}$ in a Hilbert space $H.$
Commutators $[T_{i}T_{j},T_{j}T_{i}]$ are related to the uncertainty principle
in their measurements and small commutators indicate more precise
measurements. Let $v\in H$ with $\|v\|=1$ which gives a vector state. The
expected value when measuring an observable given by $T_{j}$ is ${\rm
exp}_{T_{j}}(v)=\langle T_{j}v,v\rangle,$ $1\leq j\leq n,$ (see p.178 of
[13]). Most interesting vector states are eigenstates, the vector states given
by eigenvectors. The joint expected value associated with $v$ is $({\rm
exp}_{T_{1}}(v),{\rm exp}_{T_{2}}(v),...,{\rm
exp}_{T_{n}}(v))\in\mathbb{R}^{n}.$ However, it is difficult to get states
which are joint eigenstates. Strictly speaking, one should not even expect to
have any eigenvector states at all much less the joint eigenvectors. In fact,
a self-adjoint operator may not have any eigenvalues, in general. But every
point in the spectrum of a self-adjoint operator is an approximate eigenvalue.

In his recent book [13], David Mumford proposed to study “near eigenvectors”
for some set of human observables which are called “Approximately
Macroscopically Unique” states. Mumford explained that these states describe a
world recognizable to us with no maybe-dead/maybe-alive cats. Given a vector
state $v\in H,$ variance and standard deviation of the measurement made by
$T_{j}$ are defined by ($1\leq j\leq n$)

$\displaystyle{\rm
var}_{T_{j}}(v)=\langle(T_{j}-\exp_{T_{j}}(v)I)^{2}v,v\rangle\,\,\,{\rm
and}\,\,\,$ (e 1.1) $\displaystyle{\rm sd}_{T_{j}}(v)=\sqrt{{\rm
var}_{T_{j}}(v)}=\|(T_{j}-{\rm exp}_{T_{j}}I)v\|.$ (e 1.2)

The following definition was given by Mumford (p. 179, II. AMU states, Chapter
14 of [13]):

######  Definition 1.1.

Let $T_{1},T_{2},...,T_{n}$ be a given $n$-tuple of self-adjoint bounded
operators. Define

$\displaystyle{\rm AMU}(\\{T_{j}:1\leq j\leq n;\sigma\\})=\\{v\in
H:\|v\|=1,\,\,\,{\rm sd}_{T_{j}}(v)<\sigma,\,\,1\leq j\leq n\\},$ (e 1.3)

where $\sigma>0$ is a given tolerance.

Following Mumford, immediately one has the question:

######  Question 1.2.

Is the set in (e 1.3) non-empty when the commutators $[T_{i}T_{j}-T_{j}T_{i}]$
are sufficiently small?

As pointed out by Mumford that it is natural to assume that the commutators
have small norm as operators in the $C^{*}$-algebra $B(H)$ of all bounded
linear operators on $H,$ a constraint on macroscopic variables (see p. 118,
Chapter 14 of [13]). In the First International Congress of Basic Science held
in 2023, David Mumford gave the opening plenary lecture on Consciousness,
robots and DNA ([12]). At the very end of this magnificent lecture, he asked
whether there are, nearby, within a small tolerance, an $n$-tuple of commuting
self-adjoint operators so one may project them to a common eigenvector
subspace. A version of the question may be reformulated as follows:

######  Question 1.3.

Let $\epsilon>0$ and $n\in\mathbb{N}$ be a positive integer. When is there a
constant $\delta>0$ such that the following statement holds? If $H$ is any
separable Hilbert space and $T_{1},T_{2},...,T_{n}\in B(H)$ are self-adjoint
with $\|T_{j}\|\leq 1$ ($1\leq j\leq n$) such that

$\displaystyle\|T_{j}T_{i}-T_{i}T_{j}\|<\delta,\,\,\,1\leq i,j\leq n,$ (e 1.4)

there exist self-adjoint operators $S_{1},S_{2},...,S_{n}$ on $H$ such that

$\displaystyle S_{j}S_{i}=S_{i}S_{j}\,\,\,{\rm
and}\,\,\,\|S_{i}-T_{i}\|<\epsilon,\,\,\,1\leq i,j\leq n.$ (e 1.5)

When $n=2$ and $H$ is any finite dimensional Hilbert space (no bound on the
dimension) the same problem is known as von-Neumann-Kadison-Halmos problem for
almost commuting self-adjoint matrices (see [5] and [6]). In that case, the
answer is that such $\delta$ always exists (independent of the dimension) [9]
(see also [2] and [7]).

In quantum mechanics, one often heard “ when commutators tend to zero, or the
quantity $\hbar\to 0,$ we recover the classical system”. Indeed, if
observables in the system are compact self-adjoint operators, by the
affirmative solution to von-Neumann-Kadison-Halmos problem above ([9]), a pair
of observables can be approximated by commuting observables when the norm of
commutators $[T_{1},T_{2}]$ is small ($n=2$). In general, when these
observables are not compact, as pointed out by Mumford (see also Example 4.6
of [2]) there is a serious problem with this statement. If in classical
system, observables are commuting, and in quantum systems, observables have
non-zero (but small) commutators, then “$\hbar\to 0,$ or commutators tend to
zero” do not necessarily recover the classical system. In fact, in general, no
matter how small $\hbar$ or commutators are, observables could be far away
from any commuting ones. There are topological obstacles (the obvious one is
Fredholm index and there are hidden ones—see Proposition 5.5 and Proposition
6.4 in [11]). It is shown in [11] that when an approximate synthetic spectrum
and essential synthetic spectrum of the $n$-tuple are close, the answer to 1.3
is affirmative. This result may be interpreted as when one assumes that any
“local” measurement is not too far off from some “outside” measurements, when
commutators are sufficiently small, observables can be approximate by
commuting ones (see Remark 6.6 of [11]).

But, of course, Mumford is right about the existence of AMU states. Instead of
trying to find a condition on a quantum system with small commutators which
allows one to find near-by commuting observables, we may directly answer
Question 1.2. Theorem 1.4 below states that, when commutators are sufficiently
small, there are indeed always some approximately macroscopically unique
(pure) states for multiple observables simultaneously.

######  Theorem 1.4.

Let $n\in\mathbb{N}$ and $\epsilon>0.$ There exists $\delta(n,\epsilon)>0$
satisfying the following: Suppose that $H$ is an infinite dimensional
separable Hilbert space and $T_{1},T_{2},...,T_{n}\in B(H)$ are self-adjoint
operators with $\|T_{j}\|\leq 1$ ($1\leq j\leq n$) such that

$\displaystyle\|T_{i}T_{j}-T_{j}T_{i}\|<\delta,\,\,\,\,1\leq i,j\leq n.$ (e
1.6)

Then, for any $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n})),$ there exists $v\in H$ with
$\|v\|=1$ such that

$\displaystyle\max_{1\leq i\leq n}|{\rm
exp}_{T_{i}}(v)-\lambda_{i}|=\max_{1\leq i\leq n}|\langle
T_{i}v,v\rangle-\lambda_{i}|<\epsilon\,\,\,{\rm and}\,\,\,$ (e 1.7)
$\displaystyle\max_{1\leq i\leq n}\\{{\rm sd}_{T_{j}}(v)\\}=\max_{1\leq i\leq
n}\|(T_{i}-{\rm exp}_{T_{i}}(v)\cdot I)v\|<\epsilon.$ (e 1.8)

Here $s{\rm Sp}^{\eta}((T_{1},T_{2},...,T_{n}))$ is the set of
$\eta$-synthetic-spectrum (see Definition 2.7). This shows that in a
macroscopic system, if one assumes a suitable commutator bound, AMU states
always exist. Moreover, every point
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})$ in $s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n}))$ can be approximated by a joint
expected value for which AMU states exist.

The universe is even more interesting. We could have large commutators in norm
but “tend to zero” in a different sense. Let us assume that $H$ is an infinite
dimensional separable Hilbert space with an orthonormal basis
$\\{u_{k}:k\in\mathbb{N}\\}$ of $H.$ Let $p_{m}$ be the projection on ${\rm
span}\\{u_{k}:1\leq k\leq m\\}.$ Suppose that

$\displaystyle\lim_{m\to\infty}\|(T_{i}T_{j}-T_{j}T_{i})(1-p_{m})\|=0,\,\,\,1\leq
i,j\leq n.$ (e 1.9)

In other words, the commutators are compact.

Denote by ${\cal K}$ the $C^{*}$-subalgebra of all compact operators. Then we
have the following:

######  Theorem 1.5.

Let $H$ be an infinite dimensional separable Hilbert space and
$T_{1},T_{2},...,T_{n}\in B(H)$ be self-adjoint operators. Suppose that

$\displaystyle T_{i}T_{j}-T_{j}T_{i}\in{\cal K},\,\,\,1\leq i,j\leq n.$ (e
1.10)

Then, for any $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in{\rm
Sp}(\pi_{c}(T_{1}),\pi_{c}(T_{2}),...,\pi_{c}(T_{n})),$ where $\pi_{c}:B(H)\to
B(H)/{\cal K}$ is the quotient map, there exists a sequence $v^{(m)}\in H$
with $\|v^{(m)}\|=1$ such that

$\displaystyle\lim_{m\to\infty}\max_{1\leq i\leq n}|\lambda-{\rm
exp}_{Ti}(v^{(m)})|=0\,\,\,\text{and}\,\,\,$ (e 1.11)
$\displaystyle\lim_{m\to\infty}\max_{1\leq i\leq n}{\rm
sd}_{T_{i}}(v^{(m)})=\lim_{m\to\infty}\max_{1\leq i\leq n}\|(T_{j}-{\rm
exp}_{T_{i}}\cdot I)(v^{(m)})\|=0.$ (e 1.12)

Moreover, for any $\zeta=(\zeta_{1},\zeta_{2},...,\zeta_{n})\subset{\rm
Conv}({\rm Sp}(T_{1},T_{2},...,T_{n})),$ there are unit vectors $u^{(k)}\in H$
such that

$\displaystyle\lim_{k\to\infty}\max_{1\leq j\leq k}|\zeta_{j}-\langle
T_{j}u^{(k)},u^{(k)}\rangle\|=0.$ (e 1.13)

Here ${\rm Sp}(\pi_{c}(T_{1}),\pi_{c}(T),...,\pi_{C}(T_{n}))$ is the essential
joint spectrum of the $n$-tuple self-adjoint operators (see Definition 2.3).
Note that any $\eta$-essential synthetic spectrum contains (see Definition 2.7
and Proposition 2.11) ${\rm
Sp}(\pi_{c}(T_{1}),\pi_{c}(T),...,\pi_{C}(T_{n})).$ These are expected values
of pure quantum states of $B(H),$ or $C,$ the $C^{*}$-subalgebra generated by
$T_{1},T_{2},...,T_{n},1$ vanishing on ${\cal K}.$ These may be called
essential joint expected values.

The condition (e 1.10) means that the commutators tend to zero along any
orthonormal basis of $H$ in the sense of (e 1.9). It may be interpreted that
any measurement with less local (finite dimensional) interference increases
its accuracy. Another way to interpret this is the commutators (or $\hbar$)
vanishes at the far edge of the (non-commutative) universe, or, in operator
algebra’s term, it vanishes in the corona algebra. With this condition on the
macroscopic observables, Theorem 1.5 states that, any essential joint expected
value can be measured (simultaneously) by an AMU state. It should be noted
(see (4) of Remark 4.2) that the condition (e 1.10) does not imply that the
system is a compact perturbation of classical ones.

In section 2, apart from some notations, we provide some review on approximate
synthetic spectrum for $n$-tuples of self-adjoint operators. Section 3
presents the proof of Theorem 1.4 and some preparation for section 4. In
section 4, we first prove Theorem 4.1 which goes beyond both Theorem 1.4 and
Theorem 1.5. We also include a final remark.

##  2 Synthetic spectra

######  Definition 2.1.

Let $A$ be a $C^{*}$-algebra. Denote by $A_{s.a.}$ the set of self-adjoint
elements in $A.$

If $x,y\in A$ and $\epsilon>0,$ we write

$\displaystyle x\approx_{\epsilon}y,\,\,\,{\rm if}\,\,\,\|x-y\|<\epsilon.$ (e
2.14)

######  Definition 2.2.

Let $\Omega$ be a metric space. Denote by $C(\Omega)$ the $C^{*}$-algebra of
all (complex valued) continuous functions on $\Omega.$

Suppose that $x\in\Omega$ and $r>0.$ Define $B(x,r)=\\{y\in\Omega:{\rm
dist}(y,x)<r\\}.$ For any subset $Y\subset\Omega,$ $\overline{Y}$ is the
closure of $Y.$

If $X\subset\Omega$ is a compact subset and $\eta>0,$ denote by
$X_{\eta}=\\{y\in\Omega:{\rm dist}(y,X)<\eta\\}.$

Recall that the Hausdorff distance of two compact subsets of
$X,Y\subset\Omega$ is defined by

$\displaystyle d_{H}(X,Y)=\max\\{\sup_{x\in X}\\{{\rm dist}(x,Y)\\},\sup_{y\in
Y}\\{{\rm dist}(y,X)\\}\\}.$ (e 2.15)

Let $F(\Omega)$ be the set of all non-empty compact subsets of $M.$ Then
$(F(\Omega),d_{H})$ is a compact metric space with the metric $d_{H}.$

######  Definition 2.3.

Let $e_{0}(\xi)=1$ for all $\xi\in\mathbb{R}^{n}$ be the constant function,
$e_{i}$ be a continuous function defined on $\mathbb{R}^{n}$ by
$e_{i}((r_{1},r_{2},...,r_{n}))=r_{i}$ for
$(r_{1},r_{2},...,r_{n})\in\mathbb{R}^{n},$ $i=1,2,...,n.$ This notation will
be used throughout this paper.

Let ${\mathbb{I}}^{n}=\\{(r_{1},r_{2},...,r_{n})\in\mathbb{R}^{n}:|r_{i}|\leq
1\\}.$ Note that $C({\mathbb{I}}^{n})$ is generated by
$\\{e_{i}|_{{\mathbb{I}}^{n}}:0\leq i\leq 1\\}.$

######  Definition 2.4.

Let $A$ be a unital $C^{*}$-algebra and $x\in A.$ Denote by ${\rm sp}(x)$ the
spectrum of $x.$ Suppose that $T_{1},T_{2},...,T_{n}\in A_{s.a.}$ and
$T_{i}T_{j}=T_{j}T_{i},$ $1\leq i,j\leq n.$ Let $C$ be the unital
$C^{*}$-subalgebra generated by $1,T_{1},T_{2},...,T_{n}.$ Put $T_{0}=1.$ Then
there exists a compact subset $\Omega\subset\mathbb{R}^{n}$ and an isomorphism
$\varphi:C(\Omega)\to C$ such that $\varphi(e_{j})=T_{j},$ where $e_{j}$ is
defined above. In other words, $\\{T_{j}:0\leq j\leq n\\}$ generates a
$C^{*}$-subalgebra $C\cong C(\Omega).$

The set $\Omega$ is called the spectrum of the $n$-tuple self-adjoint
operators $T_{1},T_{2},...,T_{n}.$ We will write ${\rm
Sp}((T_{1},T_{2},...,T_{n}))=\Omega.$

######  Definition 2.5.

Let $H$ be an infinite dimensional separable Hilbert space. For $x,y\in H,$
denote by $\langle x,y\rangle$ the inner product of $x$ and $y.$ Denote by
$B(H)$ the $C^{*}$-algebra of all bounded linear operators on $H$ and by
${\cal K}$ the $C^{*}$-algebra of all compact operators on $H.$ Denote by
$\pi_{c}:B(H)\to B(H)/{\cal K}$ the quotient map.

Let $T\in B(H).$ Recall that the essential spectrum of $T,$ ${\rm
sp}_{ess}(T)={\rm sp}(\pi_{c}(T)),$ is the spectrum of $\pi_{c}(T).$

######  Definition 2.6.

Fix an integer $k\in\mathbb{N}$ and $M\geq 1.$ Define

$P_{k}^{M}=\\{\xi=(x_{1},x_{2},...,x_{n}):x_{j}=m_{j}/k,|m_{j}|\leq
Mk,m_{j}\in\mathbb{Z},\,\,1\leq j\leq n\\}.$

$P_{k}^{M}$ has only finitely many points.

######  Definition 2.7.

Let $n\in\mathbb{N}$ and $M>0.$ In what follows, for each $0<\eta<1,$ we
choose a fixed integer $k=k(\eta)\in\mathbb{N}$ such that
$k=\inf\\{l\in\mathbb{N}:(M+1)/l<{\eta\over{2\sqrt{n}}}\\}.$ Denote
$D^{\eta}=P_{k}^{M}.$ We write $D^{\eta}=\\{x_{1},x_{2},...,x_{m}\\}.$ Then
$D^{\eta}$ is $\eta/2$-dense in ${\mathbb{I}}^{n}.$ Moreover, $D^{\eta}\subset
D^{\delta}$ if $0<\delta<\eta.$

Denote by $M^{n}=\\{(r_{1},r_{2},...,r_{n}):|r_{i}|\leq M\\}.$

Fix $\xi=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in M^{n}.$ Let
$\theta_{\lambda_{i},\eta}\in C([-M,M])$ be such that
$0\leq\theta_{\lambda_{i},\eta}\leq 1,$ $\theta_{\lambda_{i},\eta}(t)=1,$ if
$|t-\lambda_{i}|\leq 3\eta/4,$ $\theta_{\lambda_{i},\eta}(t)=0$ if
$|t-\lambda_{i}|\geq\eta,$ and $\theta_{\lambda_{i}\eta}$ is linear in
$(\lambda_{i}-\eta,\lambda_{i}-3\eta/4)$ and in
$(\lambda_{i}+3\eta/4,\lambda_{i}+\eta),$ $i=1,2,...,n.$ Define, for each
$t=(t_{1},t_{2},...,t_{n})\in\mathbb{R}^{n},$

$\displaystyle\Theta_{\xi,\eta}(t)$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\theta_{\lambda_{i},\eta}(t_{i}).$ (e 2.16)

Suppose that $A$ is a unital $C^{*}$-algebra and $(a_{1},a_{2},...,a_{n})$ is
an $n$-tuple of self-adjoint elements in $A$ with $\|a_{i}\|\leq M$ ($1\leq
i\leq n$). Put

$\displaystyle\Theta_{\xi,\eta}(a_{1},a_{2},...,a_{n})$ $\displaystyle=$
$\displaystyle\theta_{\lambda_{1},\eta}(a_{1})\theta_{\lambda_{2},\eta}(a_{2})\cdots\theta_{\lambda_{n},\eta}(a_{n}).$
(e 2.17)

Note that we do not assume that $a_{1},a_{2},...,a_{n}$ mutually commute and
the product in (e 2.16) has a fixed order.

For $x_{j}=(x_{j,1},x_{j,2},...,x_{j,n})\in D^{\eta},$ we may write
$\theta_{j,i,\eta}:=\theta_{x_{j,i},\eta}$ and
$\Theta_{j,\eta}:=\Theta_{x_{j},\eta},$ $1\leq i\leq n,$ $j=1,2,...,m.$ Set

$\displaystyle s{\rm
Sp}^{\eta}((a_{1},a_{2},...,a_{n})=\bigcup_{{}_{\|\Theta_{j,\eta}(a_{1},a_{2},...,a_{n})\|\geq
1-\eta}}\overline{B(x_{j},\eta)}.$ (e 2.18)

The set $s{\rm Sp}^{\eta}((a_{1},a_{2},...,a_{n}))$ is called
$\eta$-synthetic-spectrum of the $n$-tuple $(a_{1},a_{2},...,a_{n})$ which, by
the definition, is compact.

If $a_{i}a_{j}=a_{j}a_{i}$ for all $1\leq i,j\leq n,$ then ${\rm
Sp}((a_{1},a_{2},...,a_{n}))\subset s{\rm
Sp}^{\eta}((a_{1},a_{2},...,a_{n})).$

The $\eta$-synthetic-spectrum can be tested. Suppose $a_{1},a_{2},...,a_{n}\in
B(H)_{s.a.}$ with $\|a_{i}\|\leq 1$ ($1\leq i\leq n$). In order to have
$\|\Theta_{\lambda,\eta}\|\geq 1-\eta,$ it suffices to have one unit vector
$x\in H$ such that

$\displaystyle\langle\Theta_{\lambda,\eta}(a_{1},a_{2},...,a_{n})x,x\rangle>1-\eta.$
(e 2.19)

######  Definition 2.8.

Let $A$ and $B$ be $C^{*}$-algebras. A linear map $L:A\to B$ is a c.p.c. map
if it is completely positive and contractive.

######  Definition 2.9.

Let $A$ be a unital $C^{*}$-algebra and $a_{1},a_{2},...,a_{n}\in A_{s.a.}$
for some $n\in\mathbb{N}.$ Let us assume that $\|a_{i}\|\leq M$ (for some
$M>0$), $i=1,2,...,n.$ Fix $0<\eta<1/2.$ Suppose that $X\subset M^{n}$ is a
compact subset and $L:C(X)\to A$ is a unital c.p.c. map such that

$\displaystyle{\rm(i)}$ $\displaystyle\|L(e_{j}|_{X})-a_{j}\|<\eta,\,\,\,1\leq
j\leq n,$ (e 2.20) $\displaystyle{\rm(ii)}$
$\displaystyle\|L((e_{j}e_{i})|_{X})-L(e_{i}|_{X})L(e_{j}|_{X})\|<\eta,\,\,\,1\leq
i,j\leq n,\,\,\,{\rm and}\,\,\,$ (e 2.21) $\displaystyle{\rm(iii)}$
$\displaystyle\|L(f)\|\geq 1-\eta$ (e 2.22)

for any $f\in C(X)_{+}$ which has value $1$ on an open ball with the center
$x\in X$ (for some $x$) and the radius $\eta.$ Then we say that $X$ is an
$\eta$-near-spectrum of the $n$-tuple $(a_{1},a_{2},...,a_{n}).$ We write

$nSp^{\eta}((a_{1},a_{2},...,a_{n})):=X.$

If, moreover, $L$ is a unital homomorphism, then we say $X$ is an $\eta$\-
spectrum of the $n$-tuple $(a_{1},a_{2},...,a_{n}).$

Let $X$ be an $\eta$-near-spectrum and $Y$ be a $\delta$-near -spectrum for
$(a_{1},a_{2},...,a_{n}),$ respectively. Suppose that $\delta<\eta,$ then, by
the definition, $Y$ is also an $\eta$-near-spectrum. In particular, if
$nSp^{\delta}((a_{1},a_{2},...,a_{n}))\not=\emptyset,$ then
$nSp^{\eta}((a_{1},a_{2},...,a_{n}))\not=\emptyset.$

######  Proposition 2.10 (Proposition 2.11 of [11]).

Fix $n\in\mathbb{N}.$ For any $\eta>0,$ there exists $\delta(n,\eta)>0$
satisfying the following: Suppose that $A$ is a unital $C^{*}$-algebra and
$a_{i}\in A_{s.a.}$ with $\|a_{i}\|\leq 1,$ $1\leq i\leq n,$ such that

$\displaystyle\|a_{i}a_{j}-a_{j}a_{i}\|<\delta,\,\,\,1\leq i,j\leq n.$ (e
2.23)

Then

(1) $X:=s{\rm Sp}^{\eta}((a_{1},a_{2},...,a_{n}))\not=\emptyset$ and

(2) $Y:=nSp^{\eta}((a_{1},a_{2},...,a_{n}))\not=\emptyset.$

######  Proposition 2.11 (Proposition 2.15 of [11]).

Fix $k\in\mathbb{N}.$ For any $\eta>0,$ there exits $\delta(k,\eta)>0$
satisfying the following:

Suppose that $A$ is a unital $C^{*}$-algebra and $a_{1},a_{2},...,a_{k}\in
A_{s.a.}$ with $\|a_{i}\|\leq 1$ ($1\leq i\leq k$) such that
$(a_{1},a_{2},...,a_{k})$ has a non-empty $\delta$-near-spectrum
$X=nSp^{\delta}((a_{1},a_{2},...,a_{n})).$ If $Y$ is also a non-empty
$\delta$-near-spectrum of $(a_{1},a_{2},...,a_{k}),$ and $Z=s{\rm
Sp}^{\eta}((a_{1},a_{2},...,a_{k}))\not=\emptyset,$ then

$\displaystyle d_{H}(X,Y)<\eta\,\,\,{\rm and}\,\,\,X,Y\subset Z\subset
X_{2\eta}.$ (e 2.24)

##  3 The case of small commutators

######  Lemma 3.1.

Let $\Omega$ be a compact metric space, $X\subset\Omega$ a compact subset and
$X_{i}\subset\Omega$ be a sequence of compact subsets. Suppose that

$\displaystyle\lim_{i\to\infty}{\rm dist}_{H}(X_{i},X)=0.$ (e 3.25)

Suppose that ${\cal G}\subset C(\Omega)$ is a finite subset which generates
$C(\Omega)$ and there are a sequence of decreasing numbers $\eta_{i}\searrow
0,$ and a sequence of c.p.c. maps $L_{i}:C(\Omega)\to A_{i}$ for some unital
$C^{*}$-algebra $A_{i}$ such that

$\displaystyle\|L_{i}(fg)-L_{i}(f)L_{i}(g)\|<\eta_{i}\,\,\,\text{for\,\,\,all}\,\,\,f,g\in{\cal
G}\,\,\,\text{and}\,\,\,$ (e 3.26) $\displaystyle\|L_{i}(h)\|\geq 1-\eta_{i}$
(e 3.27)

for any $h\in C(X)_{+}$ with value 1 in an open ball with center at some point
$x\in X_{i}$ and radius $\eta_{i},$ $i=1,2,....$

Let $\Psi:C(\Omega)\to\prod_{i=1}^{\infty}A_{i}/\bigoplus_{i=1}^{\infty}A_{i}$
be the homomorphism defined by $\Psi(f)=\Pi\circ\\{L_{i}(f)\\}$ for all $f\in
C(\Omega),$ where
$\Pi:\prod_{i=1}^{\infty}A_{i}\to\prod_{i=1}^{\infty}A_{i}/\bigoplus_{i=1}^{\infty}A_{i}$
is the quotient map. Then ${\rm ker}\Psi=\\{f\in C(\Omega):f|_{X}=0\\}.$

###### Proof.

First we note, by (e 3.26), that $\Psi=\Pi\circ\\{\L_{i}\\}$ is indeed a
homomorphism, since ${\cal G}$ is a generating set.

Let $f\in C(\Omega)$ such that $f|_{X}=0.$ To show that $\Psi(f)=0,$ let
$\epsilon>0.$ Since $X$ is compact, there is $\delta>0$ such that

$\displaystyle\|f(x)\|<\epsilon\,\,\,{\rm
for\,\,\,all}\,\,\,x\in\\{x\in\Omega:{\rm dist}(x,X)<\delta\\}.$ (e 3.28)

By (e 3.25), there exists $i_{0}\in\mathbb{N}$ such that

$\displaystyle X_{i}\subset\\{x\in\Omega:{\rm dist}(x,X)<\delta\\}\,\,\,{\rm
for\,\,\,all}\,\,\,i\geq i_{0}.$ (e 3.29)

Hence

$\displaystyle\|L_{i}(f|_{X_{i}})\|<\epsilon\,\,\,{\rm
for\,\,\,all}\,\,\,i\geq i_{0}.$ (e 3.30)

It follows that

$\displaystyle\|\Psi(f)\|<\epsilon.$ (e 3.31)

Since this holds for any $\epsilon>0,$ we have $\Psi(f)=0.$ Hence

$\displaystyle\\{f\in C(\Omega):f|_{X}=0\\}\subset{\rm ker}\Psi.$ (e 3.32)

To complete the proof of the lemma, it suffices to show that if
$f\not\in\\{g\in C(\Omega):g|_{X}=0\\},$ then $\Psi(f)\not=0.$ Since
$I=\\{g\in C(\Omega):g|_{X}=0\\}$ is an ideal and
$\Psi(f)^{*}\Psi(f)=\Psi(f^{*}f),$ it suffices show that if $f\not\in\\{g\in
C(\Omega):g|_{X}=0\\}$ and $f\geq 0,$ then $\Psi(f)\not=0.$

Thus we assume that $f\geq 0.$ Then there is $x\in X$ such that $f(x)>0.$

By considering $(1/f(x))f,$ to simplify the notation, we may assume that
$f(x)=1.$ There exists $\delta_{1}>0$ such that $|f(\zeta)|\geq 1/2$ for all
$\zeta\in B(x,\delta_{1}).$ Choose $h\in C(\Omega)_{+}$ such that $0\leq h\leq
1,$ $h(y)=1$ if $y\in B(x,\delta_{1}/2)$ and $h(y)=0$ if $y\not\in
B(x,\delta_{1}).$ Hence $2f\geq h.$

By (e 3.25), there exists $i_{1}\in\mathbb{N}$ such that, for all $i\geq
i_{0},$ there exists $x_{i}\in X_{i}$ such that

$\displaystyle{\rm dist}(x_{i},x)<\delta_{1}/8\,\,\,{\rm
and}\,\,\,\eta_{i}<\delta_{1}/16.$ (e 3.33)

Hence $B(x_{i},\delta_{1}/8)\subset B(x,\delta_{1}/2).$ It follows from (e
3.27) that

$\displaystyle\|L_{i}(2f)\|\geq\|L_{i}(h)\|\geq 1-\eta_{i}\,\,\,{\rm
for\,\,\,all}\,\,\,i\geq i_{1}.$ (e 3.34)

Hence $\Psi(2f)\geq 1$ and $\Psi(f)\not=0.$ ∎

######  Lemma 3.2.

Let $X$ be a compact metric space and $B$ be a $C^{*}$-algebra which is a
corona algebra of real rank zero. Suppose that $\varphi:C(X)\to B$ is a unital
injective homomorphism. Then, for any finite subset
$S=\\{\zeta_{1},\zeta_{2},...,\zeta_{m}\\}\subset X,$ there are mutually
orthogonal non-zero projections $d_{1},d_{2},...,d_{m}\in B$ with
$d=\sum_{i=1}^{m}d_{i}$ such that

$\displaystyle(1-d)\varphi(f)=\varphi(f)(1-d)\,\,\,\text{and}\,\,\,$ (e 3.35)
$\displaystyle\varphi(f)=\sum_{i=1}^{m}f(\zeta_{i})d_{i}+(1-d)\varphi(f)(1-d)\,\,\,{\rm
for\,\,\,all}\,\,\,f\in C(X).$ (e 3.36)

###### Proof.

For each $\zeta_{k}\in X,$ let

$\displaystyle C_{k}=\\{f\in C(X):f(\zeta_{k})=0\\},$ (e 3.37)

an ideal of $C(X).$ Put $D_{k}=\overline{\varphi(C_{k})B\varphi(C_{k})}.$ Then
$D_{k}$ is a $\sigma$-unital hereditary $C^{*}$-subalgebra of $B.$ Since
$\varphi$ is injective, no elements in $D_{k}$ is invertible. Hence
$D_{k}\not=B.$ Since $B$ is a corona algebra, by Pedersen’s double annihilator
theorem (Theorem 15 of [14]),

$\displaystyle D_{k}=(D_{k}^{\perp})^{\perp}.$ (e 3.38)

It follows that $D_{k}^{\perp}=\\{c\in B:cd=dc=0\,\,\,{\rm
for\,\,\,all}\,\,\,d\in D_{k}\\}\not=\\{0\\}.$ Since $B$ has real rank zero,
the hereditary $C^{*}$-subalgebra $D_{k}^{\perp}$ also has real rank zero
(Corollary 2.8 of [1]). Choose a non-zero projection $d_{k}\in D_{k}^{\perp},$
$k=1,2,...,m.$ Note that $d_{j}d_{k}=0=d_{k}d_{j}$ if $k\not=j.$

Let $F_{k}=\\{x\in B:xd,dx\in D_{k}\\}$ be the idealizer of $D_{k}$ in $B$ and
$E_{k}=\varphi(C(X))+D_{k}.$ Then $E_{k}\subset F_{k}$ and $d_{k}\in F_{k}.$
Let $\pi_{k}:F_{k}\to F_{k}/D_{k}$ be the quotient map. Then
$\pi_{k}\circ\varphi(f)=f(\zeta_{k})$ for all $f\in C(X).$ It follows that
$d_{k}\varphi(f)=\varphi(f)d_{k}=f(\zeta_{k})d_{k}$ for all $f\in C(X),$
$k=1,2,...,m.$ Define $d=\sum_{k=1}^{m}d_{k}.$ Then

$\displaystyle
d\varphi(f)=\varphi(f)d=\sum_{k=1}^{m}f(\zeta_{k})d_{k}\,\,\,{\rm and}\,\,\,$
(e 3.39)
$\displaystyle\varphi(f)=\sum_{k=1}^{m}f(\zeta_{k})d_{k}+(1-d)\varphi(f)(1-d)\,\,\,{\rm
for\,\,\,all}\,\,\,f\in C(X).$ (e 3.40)

∎

######  Lemma 3.3.

Let $\epsilon>0,$ $\Omega$ be a compact metric space, and ${\cal G}\subset
C(\Omega)$ be a finite generating set. There exists $\delta(n,\epsilon)>0$
satisfying the following: Suppose that $A$ is a unital $C^{*}$-algebra of real
rank zero and $X\subset\Omega$ is a non-empty compact subset and
$\varphi:C(X)\to A$ is a c.p.c. map such that

$\displaystyle\|\varphi(g_{1}|_{X}g_{2}|_{X})-\varphi(g_{1}|_{X})\varphi(g_{2}|_{X})\|<\delta\,\,\,\text{for\,\,\,all}\,\,\,g_{1},g_{2}\in{\cal
G},\,\,\,1\leq i,j\leq n\,\,\,\text{and}\,\,\,\|\varphi(f)\|\geq 1-\delta$

for any $f\in C(X)_{+}$ which has value $1$ on an open ball with the center
$x\in X$ (for some $x$) and the radius $\delta.$ Then, there are
$\xi_{1},\xi_{2},...,\xi_{m}\in X$ which is $\epsilon$-dense in $X,$ and
mutually orthogonal non-zero projections $p_{1},p_{2},...,p_{m}\in A$ such
that, for all $f\in{\cal G},$

$\displaystyle\|\sum_{k=1}^{m}f(\xi_{k})p_{k}+(1-p)\varphi(f)(1-p)-\varphi(f)\|<\epsilon.$
(e 3.41)

###### Proof.

We prove this by contradiction. So we assume the lemma is false. Then, we
obtain positive numbers $\epsilon_{0}>0,$ and $\epsilon_{1}>0,$ a sequence of
unital $C^{*}$-algebras $\\{A_{i}\\}$ of real rank zero, a sequence of compact
subsets $X_{i}\subset\Omega,$ a sequence of c.p.c. maps $L_{i}:C(X_{i})\to
A_{i}$ ($i\in\mathbb{N}$) and a sequence of positive numbers $\eta_{i}$ such
that $\eta_{i}\searrow 0$ such that

$\displaystyle\lim_{i\to\infty}\|L_{i}(fg|_{X_{i}})-L_{i}(f|_{X_{i}})L_{i}(g|_{X_{i}})\|=0\,\,\,{\rm
for\,\,\,all}\,\,\,f,g\in{\cal G}\,\,\,{\rm and}\,\,\,$ (e 3.42)
$\displaystyle\|L_{i}(h)\|\geq 1-\eta_{i}$ (e 3.43)

for any $h\in C(X_{i})_{+}$ which has value 1 on an open ball of radius
$\eta_{i}$ with center at a point $x\in X_{i},$ but

$\displaystyle\inf\\{\sup\\{\|\sum_{k=1}^{m_{i}}g(\xi_{i,k})p_{k,i}+(1-q_{i})L_{i}(g|_{X_{i}})(1-q_{i})-L_{i}(g|_{X_{i}})\|:g\in{\cal
G}\\}\\}\geq\epsilon_{0},$ (e 3.44)

where the infimum is taken among all mutually orthogonal non-zero projections
$\\{p_{i,1},p_{i,2},...,p_{i,m_{i}}\\}$ in $A_{i}$ with
$q_{i}=1-\sum_{k=1}^{m_{i}}p_{i,k}$ and all finite $\epsilon_{1}$-dense subset
$\\{\xi_{i,1},\xi_{i,2},...,\xi_{i,m_{i}}\\}$ of $X_{i}.$

Note that $(F(\Omega),d_{H})$ is compact. Therefore, by passing to a
subsequence, we may assume that there is a compact subset $X\subset\Omega$
such that

$\displaystyle\lim_{i\to\infty}{\rm dist}_{H}(X_{i},X)=0.$ (e 3.45)

Let $B=\prod_{i=1}^{\infty}A_{i},$ $\Phi:C(\Omega)\to B$ be defined by
$\Phi(f)=\\{L_{i}(f|_{X_{i}})\\}$ for $f\in C(\Omega),$ and $\Pi:B\to
B/\bigoplus_{i=1}^{\infty}A_{i}$ be the quotient map (note that
$\bigoplus_{i=1}^{\infty}A_{i}=\\{\\{a_{i}\\}:a_{i}\in A_{i}\,\,\,{\rm
and}\,\,\,\lim_{i}\|a_{i}\|=0\\}$). Then, by (e 3.42),
$\Psi^{\prime}=\Pi\circ\Phi^{\prime}:C(\Omega)\to
B/\bigoplus_{i=1}^{\infty}A_{i}$ is a homomorphism. Define
$L_{i}^{\prime}:C(\Omega)\to A_{i}$ by $L_{i}^{\prime}(f)=L_{i}(f|_{X_{i}})$
for $f\in C(\Omega).$ By (e 3.45), (e 3.43) and by Lemma 3.1,

$\displaystyle{\rm ker}\Psi^{\prime}=\\{f\in C(\Omega):f|_{X}=0\\}.$ (e 3.46)

Thus $\Psi^{\prime}$ induces an injective homomorphism $\Psi:C(X)\to
B/\bigoplus_{i=1}^{\infty}A_{i}$ (with $\Psi(f|_{X})=\Psi^{\prime}(f)$ for all
$f\in C(\Omega)$).

Let $\\{\zeta_{1},\zeta_{2},...,\zeta_{m}\\}\subset X$ be an
$\epsilon_{1}$-dense subset. Put $Q=B/\bigoplus_{i=1}^{\infty}A_{i}.$ Note
that $B=M(\bigoplus_{i=1}^{\infty}A_{i}),$ the multiplier algebra of
$\bigoplus_{i=1}^{\infty}A_{i}.$ So
$Q=M(\bigoplus_{i=1}^{\infty}A_{i})/\bigoplus_{i=1}^{\infty}A_{i}$ is a corona
algebra. By applying Lemma 3.2, we obtain mutually orthogonal non-zero
projections $d_{1},d_{2},...,d_{m}$ with $d=\sum_{k=1}^{m}d_{k}$ such that

$\displaystyle d\Psi(f)=\Psi(f)d=\sum_{k=1}^{m}f(\zeta_{k})d_{k}\,\,\,{\rm
and}\,\,\,$ (e 3.47)
$\displaystyle\Psi(f)=\sum_{k=1}^{m}f(\zeta_{k})d_{k}+(1-d)\Psi(f)(1-d)\,\,\,{\rm
for\,\,\,all}\,\,\,f\in C(X).$ (e 3.48)

Note that every projection in
$\prod_{i=1}^{\infty}A_{i}/\oplus_{i=1}^{\infty}A_{i}$ lifts to a projection
in $\prod_{i=1}^{\infty}A_{i}.$ So there is a projection $p_{1}\in B$ such
that $\Pi(p_{1})=e_{1}.$ An induction let us find mutually orthogonal non-zero
projections $p_{1},p_{2},...,p_{m}$ in $B$ (and $p=\sum_{k=1}^{m}p_{i}$) such
that $\Pi(p_{k})=d_{k}$ ($1\leq k\leq m$) and

$\displaystyle\Pi(\sum_{k=1}^{m}f(\zeta_{k})p_{k}+(1-p)\Phi(f)(1-p))$ (e 3.49)
$\displaystyle\hskip
28.90755pt=\sum_{k=1}^{m}f(\zeta_{k})d_{k}+(1-d)\Psi(f)(1-d)=\Psi(f)\hskip
14.45377pt\,\,\,{\rm for\,\,\,all}\,\,\,f\in C(X).$ (e 3.50)

Write $p_{k}=\\{p_{i,k}\\}_{i\in\mathbb{N}}$ and
$p=\\{q_{i}\\}_{i\in\mathbb{N}}$ in $B.$ Then, for all sufficiently large $i,$

$\displaystyle\|\sum_{k=1}^{m}g(\zeta_{k})p_{i,k}+(1-q_{i})L_{i}(g|_{X_{i}})(1-q_{i})-L_{i}(g|_{X_{i}})\|<\epsilon_{0}/2\,\,\,{\rm
for\,\,\,all}\,\,\,g\in{\cal G}.$ (e 3.51)

This contradict (e 3.44). The lemma then follows. ∎

######  Theorem 3.4.

Let $n\in\mathbb{N}$ and $\epsilon>0.$ There exists $\delta(n,\epsilon)>0$
satisfying the following: Suppose that $H$ is an infinite dimensional
separable Hilbert space and $T_{1},T_{2},...,T_{n}\in B(H)$ are self-adjoint
operators with $\|T_{j}\|\leq 1$ ($1\leq j\leq n$) such that

$\displaystyle\|T_{i}T_{j}-T_{j}T_{i}\|<\delta,\,\,\,\,1\leq i,j\leq n.$ (e
3.52)

Then, for any $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n})),$ there exists $v\in H$ with
$\|v\|=1$ such that

$\displaystyle\max_{1\leq i\leq n}|{\rm
exp}_{T_{i}}(v)-\lambda_{i}|<\epsilon\,\,\,{\rm and}\,\,\,\max_{1\leq i\leq
n}\|(T_{i}-{\rm exp}_{T_{i}}(v)\cdot I)v\|<\epsilon.$ (e 3.53)

Moreover, for any $\mu=(\mu_{1},\mu_{2},...,\mu_{n})\in{\rm conv}(s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n}))),$ there is a vector $x\in H$ with
$\|x\|=1$ such that

$\displaystyle(\sum_{i=1}^{n}|\mu_{i}-{\rm
exp}_{T_{i}}(x)|^{2})^{1/2}<\epsilon,\,\,\,1\leq i\leq n.$ (e 3.54)

###### Proof.

We choose the generating set ${\cal G}=\\{e_{j}:0\leq j\leq n\\}$ of
$C({\mathbb{I}}^{n})$ (see Definition 2.3). Fix $0<\epsilon<1.$ Choose
$\eta_{1}:=\delta(\epsilon/16,n)>0$ given by Lemma 3.3 for $\epsilon/16$ (in
place of $\epsilon$), $\Omega={\mathbb{I}}^{n}$ and ${\cal G}$ chosen above.
Put $\eta=\min\\{\epsilon/4,\eta_{1}\\}.$ Let $\delta_{2}:=\delta(\eta/4,n)>0$
be given by Proposition 2.10 (see Proposition 2.11 of [11]) for $\eta/4$ (in
place of $\eta$). Let $\delta_{3}:=\delta(n,\delta_{2})>0$ be given by
Proposition 2.11 (see Proposition 2.15 of [11]) for $\eta/4$ (in place of
$\eta$). Put $\delta=\min\\{\epsilon/4,\eta/4,\delta_{2}/2,\delta_{3}\\}>0.$

Now assume that (e 3.52) holds for the above $\delta.$ By Proposition 2.10
(Proposition 2.11 of [11]) and by the choice of $\delta$ (and $\delta_{2}$),

$\displaystyle X:=n{\rm
Sp}^{\eta/4}((T_{1},T_{2},...,T_{n}))\not=\emptyset\,\,\,{\rm
and}\,\,\,Z:=s{\rm Sp}^{\eta/4}((T_{1},T_{2},...,T_{n}))\not=\emptyset.$ (e
3.55)

Applying Proposition 2.11, we also have

$\displaystyle X\subset Z\subset X_{\eta/2}\subset{\mathbb{I}}^{n}.$ (e 3.56)

By the definition of $\eta/4$-near spectrum, we obtain a unital c.p.c. map
$\varphi:C(X)\to A$ such that

$\displaystyle\|\varphi(e_{j}|_{X})-T_{j}\|<\eta/4,\,\,\,1\leq j\leq n，$ (e
3.57)
$\displaystyle\|\varphi(e_{i}e_{j}|_{X})-\varphi(e_{i}|_{X})\varphi(e_{j}|_{X})\|<\eta/4,\,\,\,1\leq
i,j\leq n\,\,\,{\rm and}\,\,\,$ (e 3.58) $\displaystyle\|\varphi(f)\|\geq
1-\eta/4$ (e 3.59)

for all $f\in C(X)_{+}$ which has value 1 on an open ball of radius $\eta/4$
with center $x\in X.$ By the choice of $\eta,$ applying Lemma 3.3, we obtain
an $\epsilon/4$-dense subset $\\{\xi_{1},\xi_{2},...,\xi_{m}\\}\subset X$ and
mutually orthogonal non-zero projections $p_{1},p_{2},...,p_{n}\in B(H)$ such
that

$\displaystyle\|\sum_{k=1}^{m}e_{j}(\xi_{k})p_{k}+(1-p)\varphi(e_{j}|_{X})(1-p)-T_{j}\|<\epsilon/4,\,\,\,1\leq
j\leq n.$ (e 3.60)

Next let $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n}))=Z.$ Then, by (e 3.56), there is
$k\in\\{1,2,...,m\\}$ such that

$\displaystyle\sqrt{\sum_{i=1}^{n}|\lambda_{i}-\xi_{k,i}|}<\eta/4<\epsilon/4,$
(e 3.61)

where $(\xi_{k,1},\xi_{k,2},...,\xi_{k,n})=\xi_{k}.$

Choose $v_{k}\in p_{k}(H)$ with $\|v_{k}\|=1.$ Then, by (e 3.60),

$\displaystyle|{\rm exp}_{T_{j}}(v_{k})-\xi_{k,j}|$ $\displaystyle=$
$\displaystyle|\langle
T_{j}v_{k},v_{k}\rangle-\xi_{k,j}|<\epsilon/4+|\sum_{i=1}^{m}\langle
e_{j}(\xi_{i})p_{i}v_{k},v_{k}\rangle-\xi_{k,j}|$ (e 3.62) $\displaystyle=$
$\displaystyle\epsilon/4+|\xi_{k,j}\langle
p_{k}v_{k},v_{k}\rangle-\xi_{k,j}|=\epsilon/4.$ (e 3.63)

Hence

$\displaystyle|{\rm
ep}_{T_{j}}(v_{k})-\lambda_{k,j}|<\epsilon/4+|\xi_{k,j}-\lambda_{k,j}|<\epsilon/4+\epsilon/4=\epsilon/2<\epsilon.$
(e 3.64)

Therefore, for each $1\leq j\leq n,$ by (e 3.60),

$\displaystyle\|(T_{j}-\xi_{k,j}\cdot I)v_{k}\|$ $\displaystyle<$
$\displaystyle\epsilon/4+\|(\sum_{i=1}^{m}e_{j}(\xi_{i})p_{i}+(1-p)\varphi(e_{j}|_{X})(1-p))v_{k}-\xi_{k,j}v_{k}\|$
(e 3.65) $\displaystyle=$
$\displaystyle\epsilon/4+\|(\sum_{i=1}^{m}e_{j}(\xi_{i})p_{i})v_{k}-\xi_{k,j}v_{k}\|$
$\displaystyle=$
$\displaystyle\epsilon/4+\|\xi_{k,j}v_{k}-\xi_{k,j}v_{k}\|=\epsilon/4.$ (e
3.66)

Hence, by (e 3.63),

$\displaystyle\|(T_{j}-{\rm exp}_{T_{j}}(v_{k})\cdot I)v_{k}\|$
$\displaystyle<$ $\displaystyle\epsilon/4+\|(T_{j}-\xi_{k,j}\cdot
I)v_{k}\|<\epsilon/4+\epsilon/4=\epsilon/2.$

Next, let $\mu=\sum_{l=1}^{N}a_{l}\zeta_{l},$ where $a_{l}\geq 0$ and
$\sum_{l=1}^{N}a_{l}=1$ and $\zeta_{l}\in s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n})).$ Since
$\\{\xi_{1},\xi_{2},...,\xi_{m}\\}$ is $\epsilon/4$-dense, we may assume that
there are ${\alpha}_{i}\geq 0$ with $\sum_{i=1}^{m}{\alpha}_{i}=1$ such that

$\displaystyle\|\mu-\sum_{i=1}^{m}{\alpha}_{i}\xi_{k}\|_{2}<\epsilon/4,$ (e
3.67)

where $\|\cdot\|_{2}$ is the Euclidian norm in $\mathbb{R}^{n}.$ Choose
$x=\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}v_{k}\in H.$ Since $v_{i}\perp v_{j},$
when $i\not=j,$ we have $\|x\|^{2}=\langle
x,x\rangle=\sum_{k=1}^{m}{\alpha}_{k}\|v_{k}\|^{2}=1.$ We then compute that,
by (e 3.66) and (e 3.67),

$\displaystyle{\rm exp}_{T_{j}}(x)=\langle
T_{j}x,x\rangle=\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}\langle
T_{j}v_{k},x\rangle\approx_{\epsilon/4}\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}\langle\xi_{k,j}v_{k},x\rangle$
(e 3.68) $\displaystyle=\sum_{k=1}^{m}{\alpha}_{k}\xi_{k,i}\langle
v_{k},v_{k}\rangle=\sum_{k=1}^{m}{\alpha}_{k}\xi_{k,i}\approx_{\epsilon/4}\mu_{i}.$
(e 3.69)

Thus (e 3.54) holds. ∎

Theorem 1.4 follows immediately from Theorem 3.4.

##  4 The case of compact commutators

######  Theorem 4.1.

Let $\epsilon>0$ and $n\in\mathbb{N}.$ There is $\delta(\epsilon,n)>0$
satisfying the following: Suppose that $H$ is an infinite dimensional
separable Hilbert space. Suppose that $T_{1},T_{2},...,T_{n}$ are in
$B(H)_{s.a.}$ with $\|T_{i}\|\leq 1$ ($1\leq i\leq n$) such that

$\displaystyle\|\pi_{c}(T_{i}T_{j})-\pi_{c}(T_{i}T_{j})\|<\delta,\,\,\,1\leq
i,j\leq n.$ (e 4.70)

Then, for any $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in s{\rm
Sp}^{\epsilon/4}((\pi_{c}(T_{1}),\pi_{c}(T_{2}),...,\pi_{c}(T_{n}))),$ there
exists $v\in H$ with $\|v\|=1$ such that

$\displaystyle\max_{1\leq i\leq n}|{\rm
exp}_{T_{i}}(v)-\lambda_{i}|<\epsilon\,\,\,\text{and}\,\,\,\max_{1\leq i\leq
n}\|(T_{i}-{\rm exp}_{T_{i}}(v)\cdot I)v\|<\epsilon.$ (e 4.71)

Moreover, for any $\mu=(\mu_{1},\mu_{2},...,\mu_{n})\in{\rm conv}(s{\rm
Sp}^{\epsilon/4}((\pi_{c}(T_{1}),\pi_{c}(T_{2}),...,\pi_{c}(T_{n}))),$ there
exists a vector $x\in H$ with $\|x\|=1$ such that

$\displaystyle(\sum_{i=1}^{n}|\mu_{i}-{\rm
exp}_{T_{j}}(x)|^{2})^{1/2}<\epsilon$ (e 4.72)

###### Proof.

Let $\delta_{0}>0$ (in place of $\delta$) be given by Proposition 2.10 (see
Proposition 2.11 of [11]) for $\epsilon/4$ (in place of $\eta$). Let
$\delta_{1}>0$ (in place of $\delta$) be given by Proposition 2.11
(Proposition 2.15 of [11]) for $\epsilon/16$ (in place of $\eta$). Put
$\eta=\min\\{\delta_{1}/2,\delta_{0}/2,\epsilon/32\\}>0.$ Choose
$\delta_{2}>0$ (in place of $\delta$) be given by Corollary 1.20 of [10] for
$\eta$ (in place of $\epsilon$). Put
$\delta=\min\\{\delta_{0},\delta_{1},\delta_{2}\\}.$

Suppose that (e 4.70) holds for this $\delta.$ Recall that $B(H)/{\cal K}$ is
a purely infinite simple $C^{*}$-algebra. Then, by applying Corollary 1.20 of
[10], we obtain $s_{1},s_{2},...,s_{n}\in(B(H)/{\cal K})_{s.a.}$ such that

$\displaystyle s_{i}s_{j}=s_{j}s_{i}\,\,\,{\rm
and}\,\,\,\|s_{i}-\pi_{c}(T_{i})\|<\eta,\,\,\,1\leq i,j\leq n.$ (e 4.73)

By the choice of $\eta,$ applying Proposition 2.10, we conclude that $Z:=s{\rm
Sp}^{\epsilon/4}((\pi_{c}(T_{1}),\pi_{c}(T_{2}),...,\pi_{c}(T_{n})))\not=\emptyset.$
Let $C\subset B(H)/{\cal K}$ be the unital $C^{*}$-subalgebra generated by
$s_{1},s_{2},...,s_{n},1.$ Then there exists a unital injective homomorphism
$\psi:C(X)\to B(H)/{\cal K}$ for some compact subset
$X\subset{\mathbb{I}}^{n}$ such that $\psi(e_{i}|_{X})=s_{i},$ $1\leq i\leq
n.$ Then $X$ is an $\eta$-spectrum for
$(\pi_{c}(T_{1}),\pi_{c}(T_{2}),...,\pi_{c}(T_{n})).$

By Proposition 2.11 (with our choice of $\delta_{1}$),

$\displaystyle{\rm dist}_{H}(X,Z)<2(\epsilon/16)=\epsilon/8.$ (e 4.74)

Choose an $\epsilon/16$-dense subset
$\\{\zeta_{k}=(\zeta_{k,1},\zeta_{k,2},...,\zeta_{k,n}):1\leq k\leq m\\}$ of
$X.$ By applying Lemma 3.2, there are mutually orthogonal non-zero projections
$d_{1},d_{2},...,d_{m}$ with $d=\sum_{j=1}^{m}d_{j}$ such that

$\displaystyle(1-p)\psi(f)=\psi(f)(1-p)\,\,\,{\rm
and}\,\,\,\psi(f)=\sum_{j=1}^{m}f(\zeta_{j})d_{k}+(1-p)\psi(f)(1-p)$ (e 4.75)

for all $f\in C(X).$ By (e 4.73), there are $K_{i}\in{\cal K}$ ($1\leq i\leq
n$) and mutually orthogonal projections $p_{1},p_{2},...,p_{m}$ such that
$\pi_{c}(p_{j})=d_{j}$ ($1\leq j\leq m$) and

$\displaystyle\|\sum_{j=1}^{m}e_{i}(\zeta_{j})p_{j}+(1-p)T_{i}(1-p)+K_{i}-T_{i}\|<\eta,\,\,\,1\leq
i\leq n.$ (e 4.76)

Let $\\{E_{k,0}\\}$ be an approximate identity of $(1-p){\cal K}(1-p)$ and
$\\{E_{k,j}\\}$ be an approximate identity of $p_{j}{\cal K}p_{j}$ ($1\leq
j\leq m$) consisting of projections ($0\leq j\leq m$), respectively. Put
$E_{k}=E_{k,0}+\sum_{j=1}^{m}E_{k,j},$ $k\in\mathbb{N}.$ Then $\\{E_{k}\\}$
forms an approximate identity for ${\cal K}.$ We may choose $E_{k}$ such that

$\displaystyle\|(1-E_{k})K_{i}\|<\epsilon/8\,\,\,{\rm
and}\,\,\,\|K_{i}(1-E_{k})\|<\epsilon/8,\,\,\,i=1,2,...,n.$ (e 4.77)

Note that

$\displaystyle p_{j}-E_{k,j}=p_{j}(1-E_{k})\leq 1-E_{k},\,\,\,1\leq j\leq m.$
(e 4.78)

So we also have (by (e 4.76) and (e 4.77)), for $1\leq i\leq n,$

$\displaystyle\|\sum_{j=1}^{m}e_{i}(\zeta_{j})(p_{j}-E_{k,j})+(1-p)T_{i}(1-p-E_{k,0})-T_{i}(1-E_{k})\|<\eta+{\epsilon\over{8}}\,\,\,{\rm
and}\,\,\,$ (e 4.79)
$\displaystyle\|\sum_{j=1}^{m}e_{i}(\zeta_{j})(p_{j}-E_{k,j})+(1-p-E_{k,0})T_{i}(1-p-E_{k,0})-(1-E_{k})T_{i}(1-E_{k})\|<\eta+{\epsilon\over{8}}.$
(e 4.80)

Choose $v_{j}\in(p_{j}-E_{k,j})H$ with $\|v_{j}\|=1,$ $j=1,2,...,m.$ Then
$v_{k}\perp v_{j}$ if $k\not=j.$ Hence, by (e 4.78) and by (e 4.79) (and since
$\eta+\epsilon/8<5\epsilon/32$)

$\displaystyle
T_{i}v_{j}\approx_{{5\epsilon\over{32}}}\sum_{j=1}^{m}e_{i}(\zeta_{j})(p_{j}-E_{k,j})v_{j}=\zeta_{j,i}v_{j}.$
(e 4.81)

Next let $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in Z$ be given. We
may assume that, for some $\zeta_{j},$

$\displaystyle(\sum_{i=1}^{m}|\zeta_{j,i}-\lambda_{i}|^{2})^{1/2}<\epsilon/8+\epsilon/16=3\epsilon/16.$
(e 4.82)

By (e 4.81), we estimate that

$\displaystyle(\sum_{i=1}^{n}|{\rm
exp}_{T_{i}}(v_{j})-\lambda_{i}|^{2})^{1/2}<3\epsilon/16+(\sum_{i=1}^{n}|{\rm
exp}_{T_{i}}(v_{j})-\zeta_{j,i}|^{2})^{1/2}<3\epsilon/16+5\epsilon/32.$ (e
4.83)

Moreover, by (e 4.81), for $1\leq i\leq n,$

$\displaystyle\|(T_{i}-{\rm exp}_{T_{i}}(v_{j})\cdot
I)v_{j}\|<5\epsilon/32+\|\zeta_{j,i}v_{j}-{\rm
exp}_{T_{i}}(v_{j})v_{j}\|<\epsilon/2.$ (e 4.84)

Now let $\mu=\sum_{l=1}^{N}a_{l}\zeta_{l},$ where $a_{l}\geq 0$ and
$\sum_{l=1}^{N}a_{l}=1$ and $\zeta_{l}\in s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n})),$ $1\leq l\leq n.$ Since
$\\{\zeta_{1},\zeta_{2},...,\zeta_{m}\\}$ is $\epsilon/16$-dense, we may
assume that there are ${\alpha}_{i}\geq 0$ with $\sum_{i=1}^{m}{\alpha}_{i}=1$
such that

$\displaystyle\|\mu-\sum_{i=1}^{m}{\alpha}_{i}\zeta_{k}\|_{2}<\epsilon/16,$ (e
4.85)

where $\|\cdot\|_{2}$ is the Euclidian norm in $\mathbb{R}^{n}.$ Choose
$x=\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}v_{k}\in H.$ Since $v_{i}\perp v_{j},$
when $i\not=j,$ we have $\|x\|^{2}=\langle
x,x\rangle=\sum_{k=1}^{m}{\alpha}_{k}\|v_{k}\|^{2}=1.$ We then compute that,
by (e 4.81) and

$\displaystyle{\rm exp}_{T_{j}}(x)=\langle
T_{j}x,x\rangle=\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}\langle
T_{j}v_{k},x\rangle\approx_{5\epsilon/8}\sum_{k=1}^{m}\sqrt{{\alpha}_{k}}\langle\zeta_{k,j}v_{k},x\rangle$
(e 4.86) $\displaystyle=\sum_{k=1}^{m}{\alpha}_{k}\zeta_{k,i}\langle
v_{k},v_{k}\rangle=\sum_{k=1}^{m}{\alpha}_{k}\zeta_{k,i}\approx_{\epsilon/16}\mu_{i}.$
(e 4.87)

∎

The proof of Theorem 1.5: The condition that $T_{i}T_{j}-T_{j}T_{i}\in{\cal
K}$ in (e 3.52) means
$\|\pi_{c}(T_{i})\pi_{c}(T_{j})-\pi_{c}(T_{j})\pi_{c}(T_{i})\|=\delta$ for any
$\delta>0.$ Moreover ${\rm Sp}((T_{1},T_{2},...,T_{n}))\subset s{\rm
Sp}^{\epsilon/4}((T_{1},T_{2},...,T_{n}))$ for any $\epsilon>0.$ Hence, by
Theorem 4.1, (e 4.71) holds for any $\epsilon>0.$ Hence (e 1.11), (e 1.12) and
(e 1.13) all hold.

######  Remark 4.2.

Let $H$ be an infinite dimensional separable Hilbert space and
$T_{1},T_{2},...,T_{n}\in B(H)$ be self-adjoint operators.

(1) Even in a classical system, a system in which the observables (self-
adjoint operators $T_{1},T_{2},...,T_{n}$) commute, not every $n$-tuple
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n}),$ where each $\lambda_{i}$
is in the spectrum of $T_{i},$ can be measured simultaneously. The requirement
is that $\lambda\in{\rm Sp}((T_{1},T_{2},...,T_{n}))$ (see Definition 2.4). It
is well known that spectrum of an operator is difficult to compute and
unstable for small perturbations. When observables are not commuting, it is
even difficult to define “joint-spectrum”. However, for $\epsilon>0,$ $s{\rm
Sp}^{\epsilon}((T_{1},T_{2},...,T_{n}))$ can be tested (see (e 2.19)), and by
Proposition 2.10, it is not empty, when commutators are small. So the
$\epsilon$-synthetic-spectrum $s{\rm Sp}((T_{1},T_{2},..,T_{n}))$ is a
workable replacement.

(2) Let $s_{1}$ and $s_{2}$ be AMU states corresponding to joint expected
values $\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})$ and
$\mu=(\mu_{1},\mu_{2},...,\mu_{n}),$ respectively. Then the superposition
$s={s_{1}+s_{2}\over{\|s_{1}+s_{2}\|}}$ gives ${\rm
exp}_{T_{i}}(s)={\lambda_{i}+\mu_{i}\over{\|s_{1}+s_{2}\|}},$ $i=1,2,...,n.$
In fact, for any finitely many AMU states (or any states) $s_{k}:1\leq k\leq
m$ with joint expected values
$\zeta_{k}=(\zeta_{k,1},\zeta_{k,2},...,\zeta_{k,n})\in\mathbb{R}^{n},$ any
${\alpha}_{k}\geq 0$ ($1\leq k\leq m$) with $\sum_{k=1}^{m}{\alpha}_{k}=1,$
$s=\sum_{k=1}^{m}{\alpha}_{i}s_{i}$ is a state with joint expected value
$\zeta=\sum_{k=1}^{m}{\alpha}_{k}\zeta_{k}.$ This mixed state measures
$\sum_{k=1}^{m}{\alpha}_{k}\zeta_{k,j},$ $1\leq j\leq n,$ simultaneously.
However, if $\zeta\in s{\rm Sp}^{\eta}((T_{1},T_{2},...,T_{n}))$ for some
$\eta>0,$ it is not clear that $s$ should be called a cat-state (as $\zeta$
could be an eigenvalue).

(3) Let us assume that $\zeta\not\in s{\rm
Sp}^{\eta}((T_{1},T_{2},...,T_{n}))$ for some small $\eta>0.$ This cat-state
$s=\sum_{k=1}^{m}{\alpha}_{i}s_{i}$ is never a pure state as pure states
correspond to extremal states. The question is:

Could $\zeta$ be a joint expected value measured by a vector state
(simultaneously)?

The “Moreover” part of Theorem 3.4 (see also that of Theorem 4.1) states that
$\zeta$ could be approximately measured simultaneously by a vector state (but
not by a AMU state). This is an interesting feature since the $n$-tuple may
not be approximated by commuting ones.

(4) Under the assumption (e 1.10), the commutators could have a large norm.
However, (e 1.9) means that the commutators tend to zero along any orthonormal
basis of $H.$ In fact the commutators vanish in the corona algebra $B(H)/{\cal
K}$ (the standard quantum boundary). It is important to note that the
assumption (e 1.10) does not give us a compact perturbation of some $n$-tuple
commuting self-adjoint operators. More precisely, by the Brown-Douglas-
Fillmore Theory, there cannot be any self-adjoint operators
$S_{1},S_{2},...,S_{n}$ with $S_{j}S_{i}=S_{i}S_{j}$ such that

$\displaystyle T_{j}\,\,\,{\rm
is\,\,\,close\,\,to}\,\,\,S_{j}+K_{j},\,\,\,1\leq j\leq n$ (e 4.88)

for some compact operators $K_{j}$ unless a Fredholm index vanishes (see, also
Proposition 5.5 of [11]) . However, Theorem 1.5 shows that there are plenty of
AMU states.

Theorem 4.1 goes somewhat beyond both Theorem 1.4 and 1.5 even though the the
set of joint expected values given by AMU states might be smaller.

######  Example 4.3.

Let $S$ be the shift operator on $H=l^{2}.$ Put $A_{1}=(S+S^{*})/2$ and
$A_{2}=-(S-S^{*})/2i.$ Then $A_{1}$ and $A_{2}$ are selfadjoint operators and
$\|A_{1}A_{2}-A_{2}A_{1}\|=1.$ But $A_{1}A_{2}-A_{2}A_{1}\in{\cal K}.$ However
the pair $(A_{1},A_{2})$ is far away from commuting pairs and not even a
compact perturbation of commuting pairs as $S$ is not a compact perturbation
of any unitary. Nevertheless, by Theorem 1.5, for any
$\lambda=(\lambda_{1},\lambda_{2})$ with $|\lambda|=1,$ there are unit vectors
$v_{k}\in H$ ($k\in\mathbb{N}$) such that

$\displaystyle\lim_{k\to\infty}\max_{1\leq i\leq 2}|\lambda_{j}-{\rm
exp}_{A_{j}}(v_{n})|=0\,\,\,{\rm and}\,\,\,\lim_{k\to\infty}\|(A_{j}-{\rm
exp}_{A_{j}}(v_{n})\cdot I)v_{k}\|=0.$ (e 4.89)

Moreover, for any $\zeta=(\zeta_{1},\zeta_{2})\in\mathbb{R}^{2}$ with
$|\zeta|\leq 1,$ there exists a sequence of unit vectors $u_{m}\in H$ such
that

$\displaystyle\lim_{k\to\infty}\max_{1\leq i\leq 2}|\zeta_{j}-\langle
A_{j}u_{m},u_{m}\rangle|=0.$ (e 4.90)

## References

  * [1] L. G. Brown and G. K. Pedersen, $C^{*}$-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131–149.
  * [2] P. Friis, M. Rørdam, Almost commuting self-adjoint matrices – a short proof of Huaxin Lin’s theorem, J. Reine. Angew. Math. 479 (1996), 121–131.
  * [3] G. Gong and H. Lin Almost multiplicative morphisms and almost commuting matrices, J. Operator Th. 40 (1998), 217–275.
  * [4] G. Gong and H. Lin, Almost multiplicative morphisms and K-theory, Internat. J. Math. 11 (2000), 983–1000.
  * [5] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–931.
  * [6] P.R. Halmos, Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 67–76.
  * [7] M.B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys. 291 (2009), 321-345.
  * [8] L. Kachkovskiy and Y. Safarov, Distance to normal elements in $C^{*}$-algebras of real rank zero, J. Amer. Math. Soc., 29 (2016), 61-80.
  * [9] H. Lin, Almost commuting selfadjoint matrices and applications, Fields Inst. Commun., 13, American Mathematical Society, Providence, RI, 1997, 193–233. MR Featured Review: MR 1424963.
  * [10] H. Lin, Almost multiplicative morphism and some applications, J. Operator Theory, 37 (1997), 121–154.
  * [11] H. Lin, Almost commuting self-adjoint operators and measurements, arXiv:2401.04018.
  * [12] D. Mumford, Consciousness, robots and DNA, International Congress of Basic Science, https://www.bilibili.com/video/BV1hM4y1p7mi/?spm_id_from=333.337.search-card. all.click&vd_source=83a2062f9dd6cc6d66ac9ce8f763af51.
  * [13] D. Mumford, Numbers and the world—essays on math and beyond, American Mathematical Society, Providence, RI, 2023, xvi+241 pp. ISBN: [9781470470517]; [9781470474492]
  * [14] G. K. Pedersen, $SAW^{*}$-algebras and corona $C^{*}$-algebras, contributions to non-commutative topology, J. Operator Theory, 15 (1985), 15-32.

<EMAIL_ADDRESS>


# Identification of the heterogeneous conductivity in an inverse heat
conduction problem

Angel Ciarbonetti Corresponding author<EMAIL_ADDRESS>Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro
Científico Tecnológico CONICET-Santa Fe, Argentina  Sergio Idelsohn
<EMAIL_ADDRESS>CIMNE International Centre for Numerical Methods in
Engineering, Barcelona, Spain  Ruben D. Spies<EMAIL_ADDRESS>Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro
Científico Tecnológico CONICET-Santa Fe, Argentina  Departamento de
Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral,
Santa Fe, Argentina

###### Abstract

This work deals with the problem of determining a non-homogeneous heat
conductivity profile in a steady-state heat conduction boundary-value problem
with mixed Dirichlet-Neumann boundary conditions over a bounded domain in
$\mathbb{R}^{n}$, from the knowledge of the state over the whole domain. We
develop a method based on a variational approach leading to an optimality
equation which is then projected into a finite dimensional space.
Discretization yields a linear although severely ill-posed equation which is
then regularized via appropriate ad-hoc penalizers resulting a in a
generalized Tikhonov-Phillips functional. No smoothness assumptions are
imposed on the conductivity. Numerical examples for the case in which the
conductivity can take only two prescribed values (a two-materials case) show
that the approach is able to produce very good reconstructions of the exact
solution.

Keyword:Heat-conduction, Elliptic Boundary-value Problem, Inverse Problems,
Regularization, Tikhonov-Phillips, Thermal Materials Design

##  1 Introduction

The study, analysis and numerical solution of Inverse Heat Transfer Problems
(IHTP), has a significant role in a large number of engineering applications.
Aerospace, mechanical, chemical and nuclear engineering, materials design, are
just few of the many areas in which developing appropriate and efficient tools
for solving IHTP is of special interest.

In IHTP in general, one seeks to estimate unknown functions or parameters, for
example unknown flow conditions or physical parameters of the material, from
indirect measurements of the state, that is of the temperature field, which
more often than not, in practical cases is contaminated by noise and/or can be
acquired only at a few discrete points over the domain under study.

One of the engineering branches that has aroused the greatest interest in IHTP
is electronic engineering, where thermal dissipation problems inside
equipments or components are usually studied. These types of problems are
generally associated with thermal conduction processes (see [15], [16]). This,
like other growing branches in the last two decades, have generated a genuine
interest in deepening the study of the thermal problems associated with the
design of materials and thermal devices, seeking new material design
methodologies, as well as efficient and cost-effective solutions.

Different methodological approaches to deal with the problem of designing
thermal materials and devices in heterogeneous media having their roots in
mathematical modeling, numerical solutions and computational mechanics have
recently emerged. Among them, perhaps the most widely known is the
computational design of thermal materials in heterogeneous media. This
methodology arises from the application of optimization methods which were
originally designed for solving solid mechanics problems ([4], [25]). Although
this technique has shown to yield reasonable results (see for instance [10],
[23], [9]), it has some severe drawbacks both in regard to the uniqueness of
solutions (which appears to be highly dependent on the set of parameters used
in the optimizing algorithm), as well as from the post-processing requirements
of the obtained results.

Taking as inspirational base the optimization methods approach, we propose to
tackle the problem of design of thermal materials in heterogeneous media by
embedding it within the IHTP framework.

Undoubtedly the largest difficulty in dealing with IHTP problems is the fact
that they are severely ill-conditioned which is a direct consequence of the
violation of Hadamard’s third postulate ([11]): lack of continuous dependence
of the true solution on the data. Quite often there are additional
complications coming from the lack of good measurements of the temperature
field or the the fact they are only available at a few scarce points. That
makes extremely difficult to obtain good estimates of the heat flow which is
generally needed for solving most IHTP.

Several variational models widely used for the numerical solution of IHTP
generally introduce simplification hypotheses on the constitutive models
and/or on the boundary conditions. For example in [32], the authors consider
conductivity to be a polynomial form of the temperature. Similarly in [22],
the model used for the IHTP problem assumes a linear dependence between
conductivity and temperature. Although these simplifying hypotheses allow the
successful implementation of conventional schemes for obtaining good numerical
solutions, they are far from reflecting most real world problems.

In our case, the IHTP is associated with identifying an unrestricted
heterogeneous conductivity profile in two dimensions from measurements of the
temperature in a steady-state regime. From the materials design approach, an
arbitrary field of temperatures and boundary conditions can be initially
imposed so that they be consistent with the constraints of the expected design
solution.

Heterogeneity adds a considerable degree of difficulty to an IHTP, however,
that assumption corresponds to the problems of greatest practical interest,
both in the analysis of thermographic images as in the material design problem
in which we focus this contribution.

Implementation and numerical resolution of any IHTP will, at some point,
require solving an optimization problem combined with some type of
regularization to deal with ill-posedness.

In [14] the authors deal with the problem of identifying the conductivity
profile from the temperature map. A simplifying assumption consisting of a
direct dependence on conductivity and temperature is made together with pure
Neumann-type boundary conditions. Under these hypotheses the authors solve the
problem using a conjugate gradient approach. A similar approach using
conjugate gradient is used in [26] where the authors solve the inverse problem
of finding the geometry of a defect or inclusion from an IR image.

In this work, a novel approach for solving the identification of an
heterogeneous conductivity profile based on a variational approach of the PDE
model followed by appropriate discretization and regularization is introduced.
The model allows for mixed Dirichlet-Neumann boundary conditions, and no
restrictive assumptions are made on the conductivity profile. The approach is
numerically tested with 2D examples although the setting is not restricted to
two dimensions.

###  1.1 A brief historical mathematical tracking of the problem

In this article we consider the problem of determining the elliptic
coefficient profile function in an homogeneous elliptic boundary value
problem.

Several authors have worked on this type of problems before, as they appear in
several areas and concrete applied problems such as electrical conductivity
problems, oil resevoir and ground water flow problems ([3], [5], [17], [19],
[31]) among others.

In a 1980 article ([7]), A. P. Calderón considered the following problem. Let
$\Omega\subset\mathbb{R}^{n}$, $n\geq 2$ be a bounded domain with Lipschitz
boundary $\partial\Omega$ and define

$L_{>0}^{\infty}(\Omega)\;\doteq\;\left\\{k\in
L^{\infty}(\Omega)\;:\;\exists\,\epsilon>0\textrm{ such that
}k(x)\geq\epsilon>0,\;\forall x\in\Omega\right\\}.$ (1)

Further, for $k\in L_{>0}^{\infty}(\Omega)$ let $L_{k}$ be the differential
operator

$L_{k}(u)\;=\;\nabla.(k\nabla u),$ (2)

with domain $D(L_{k})\doteq H^{1}(\Omega)$ and the quadratic form
$Q_{k}:D(Q_{k})\subset L_{\infty}(\partial\Omega)\to\mathbb{R}_{0}^{+}$,
defined by the Dirichlet integral

$Q_{k}(\varphi)=\int_{\Omega}k(x)\left(\nabla
u_{\varphi}(x)\right)^{2}\;dx\;=\;\int_{\partial\Omega}\varphi(x)k(x)\frac{\partial
u_{\varphi}}{\partial\nu}\;ds,$ (3)

where $\nu$ denotes the outward normal to $\partial\Omega$ and $u_{\varphi}\in
H^{1}(\Omega)$ is the solution of the Dirichlet boundary value problem

$\begin{cases}L_{k}(u)=0,&x\in\Omega,\\\
u=\varphi,&x\in\partial\Omega.\end{cases}$ (4)

It is timely to mention here that for $\varphi$ in the space of traces of
functions of $H^{1}(\Omega)$, the IBVP (4) has in fact a unique solution $u\in
H^{1}(\Omega)$ (see [27]).

The problem considered by Calderón was to decide wether the function $k$ is
uniquely determined by the quadratic form $Q_{k}$ and, if that is true, try to
compute $k$ in terms of $Q_{k}$. He was able to show that, with an appropriate
norm defined in the space of quadratic forms, the mapping

$M:k\longrightarrow Q_{k}$ (5)

is bounded and analytic in $L_{>0}^{\infty}(\Omega)$. Moreover, although the
general problem escaped its proof at that time, Calderón was able to show that
for the linearized problem, the answer is affirmative, that is,
$d\,M|_{k=const.}$ is an injective mapping. In addition he showed that if the
function $k$ is “close enough” to being a constant, then $k$ is “nearly”
determined by its quadratic form $Q_{k}$ and derived a bound for the
$L^{\infty}$-norm of the error (see [7] for details).

The previous result for “nearly constant” functions was later extended and
formalized by Sylvester and Uhlmann ([28]) only for the case $n=2$.

In 1984, Kohn and Vogelius ([18]) showed that if $k$ is real analytic, then it
can be uniquely determined from the knowledge of its Dirichlet integral
$Q_{k}$. This result was later extended to piecewise real analytic functions
$k$ in 1985 by the same authors ([20]).

In 1987, Sylvester and Uhlmann ([29]) showed that for functions $k$
sufficiently smooth, the quadratic form $Q_{k}$ does indeed uniquely determine
the function $k$. More precisely, they showed that the mapping $M$ in (5) is
injective over $C^{\infty}(\overline{\Omega})\cap L_{>0}^{\infty}(\Omega)$.

The case of determining $k$ in (4) from information about $u$ on the whole
domain $\Omega$ was studied by several authors. It is clear that some
assumptions on $k$ and/or on $u$ must be required. Thus, for instance, it is
evident that $k$ cannot be uniquely determined in any subregion of $\Omega$
where $\nabla u=0$. However, if $|\nabla u|>0$ everywhere on $\Omega$, once
the values of $k$ are given on a hypersurface transversal to $\nabla u$, the
method of characteristics will yield a unique solution. Several authors
obtained similar uniqueness results under weaker assumptions on $\nabla u$
under diverse assumptions on $k$ (e.g. [1], [2], [6], [24]).

There are several articles devoted to the problem of recovering $k$ from
information about $u$. However all of them assume some degree of smoothness on
$k$ (at least differentiability), which is almost never true in practical
applications, where at best, only piecewise smoothness and jump
discontinuities are to be expected, and quite often the available data
consists only of noisy measurements at some discrete points. Although the
mathematical theory of elliptic equations with discontinuous principal
coefficients is well known ([27], [21]), there is not much done on the inverse
problem of recovering $k$ in these cases.

In this article we develop a method for approximating the perhaps
discontinuous principal coefficient function based on a regularized
variational approach. This approach takes appropriate care of the strong ill-
posedness of the inverse problem which has been already carefully reported by
several authors (see for instance [30], [8]), and is characteristic in all
inverse heat conduction problems. Several numerical examples are presented
that show that the method is able to yield very good approximations of $k$
even in the case of discontinuous profiles.

##  2 Methodology

###  2.1 Preliminaries

Let $\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$) a bounded open set with smooth
boundary
$\Gamma=\partial\Omega\,=\,\overline{\Gamma}_{1}\cup\overline{\Gamma}_{N}$,
with $\Gamma_{1}\cap\Gamma_{N}=\emptyset$, $c,k,f,g,h\in L^{2}(\Omega)$ with
$0<\gamma_{1}\leq k(x)\leq\gamma_{2}$ and $c\geq 0$, and consider the
following problem, as described in Figure 1:

${\cal{P}}={\cal{P}}(k,c,f,g,h):\;\begin{cases}-{\textrm{div}}(k(x)\nabla
u(x))+c(x)u(x)=f(x),&x\in\Omega,\qquad(a)\\\
u(x)=g(x),&x\in\Gamma_{D},\hfill(b)\\\ k(x)\,\nabla
u(x)\cdot\vec{n}=h(x),&x\in\Gamma_{N}.\hfill(c)\end{cases}$ (6)

Figure 1: Schematic representation of problem ${\cal{P}}(k,c,f,g,h)$

Assume further that $g\in C(\Gamma_{D})$ and define the affine subset of
$H^{1}(\Omega)$

$H^{1}_{\Gamma_{D},g}(\Omega)\;=\;\left\\{v\in
H^{1}(\Omega)\;:\;v|_{\Gamma_{D}}=g\right\\}$ (7)

Multiplying equation $(a)$ in (6) by $v\in H^{1}_{\Gamma_{D},0}(\Omega)$ and
integrating we obtain:

$\displaystyle 0$ $\displaystyle=-\int_{\Omega}\textrm{div}(k\nabla
u)v\;dx+\int_{\Omega}cuv\;dx-\int_{\Omega}fv\;dx$
$\displaystyle=\int_{\Omega}\langle k\nabla u,\nabla
v\rangle\;dx-\int_{\partial\Omega}\langle k\nabla u,\vec{n}\rangle
v\;ds+\int_{\Omega}cuv\;dx-\int_{\Omega}fv\;dx$
$\displaystyle=\int_{\Omega}\left(\langle k\nabla u,\nabla
v\rangle+cuv\right)\;dx-\int_{\Omega}fv\;dx-\int_{\Gamma_{D}\cup\Gamma_{N}}\langle
k\nabla u,\vec{n}\rangle v\;ds$ $\displaystyle=\int_{\Omega}\left(\langle
k\nabla u,\nabla v\rangle+cuv\right)\;dx-\int_{\Omega}fv\;dx$
$\displaystyle\qquad-\int_{\Gamma_{N}}hv\;ds\qquad\qquad{\textrm{(since
$v|_{\Gamma_{D}=0}$ and $k\nabla u\cdot\vec{n}|_{\Gamma_{N}}=h$})}$
$\displaystyle\doteq F(u,v),$

where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in
$L^{2}(\Omega)$ and $\vec{n}$ is the outer normal to $\partial\Omega$. Hence,
the variational formulation of problem ${\cal{P}}$ is the following.

$VF(\cal{P})$: Find $u$ in $H^{1}_{\Gamma_{D},g}(\Omega)$ such that $F(u,v)=0$
for all $v\in H^{1}_{\Gamma_{D},0}(\Omega)$, i.e. such that

$\int_{\Omega}\left(\langle k\nabla u,\nabla
v\rangle+cuv\right)\;dx\;=\;\int_{\Omega}fv\;dx+\int_{\Gamma_{N}}hv\;ds,\qquad\forall
v\in H^{1}_{\Gamma_{D},0}(\Omega).$ (8)

Now define the continuous symmetric bilinear form $B_{k,c}:H^{1}(\Omega)\times
H^{1}(\Omega)\to\mathbb{R}$ by

$B_{k,c}(u,v)\;\doteq\;\int_{\Omega}\left(\langle k\nabla u,\nabla
v\rangle+cuv\right)\;dx.$ (9)

Then, if $c>0$, $B_{k,c}$ defines an inner product on $H^{1}(\Omega)$ (note
also that if $c>0$ then $B_{k,c}$ is $H^{1}(\Omega)$-elliptic) with associated
norm

$\|u\|^{2}_{B_{k,c}}\;\doteq\;\int_{\Omega}\left(k\|\nabla
u\|^{2}+c|u|^{2}\right)\;dx.$ (10)

Define also the energy functional
$J:H^{1}_{\Gamma_{D},g}(\Omega)\to\mathbb{R}$ by

$\displaystyle J(v)\;$
$\displaystyle\doteq\;\frac{1}{2}B_{k,c}(v,v)-\int_{\Omega}fv\;dx-\int_{\Gamma_{N}}hv\;ds$
$\displaystyle=\frac{1}{2}\int_{\Omega}\left(\langle k\nabla v,\nabla
v\rangle+c\,v^{2}\right)\;dx-\int_{\Omega}fv\;dx-\int_{\Gamma_{N}}hv\;ds.$
(11)

In the next result we show that the directional derivatives of this energy
functional are characterized by the functional $F$ defined above.

######  Lemma 2.1.

For any $u\in H^{1}_{\Gamma_{D},g}(\Omega)$ and any $v\in
H^{1}_{\Gamma_{D},0}(\Omega)$ there holds

$\frac{d}{dt}J(u+tv){\Large|}_{t=0}\;=\;F(u,v).$

###### Proof.

First note that if $u\in H^{1}_{\Gamma_{D},g}(\Omega)$ and $v\in
H^{1}_{\Gamma_{D},0}(\Omega)$ then $u+tv\in H^{1}_{\Gamma_{D},g}(\Omega)$ for
all $t\in\mathbb{R}$. Moreover, $J(u+vt)$ is clearly differentiable with
respect to $t$ and

$\displaystyle\frac{d}{dt}J(u+tv)\;$
$\displaystyle=\;\frac{d}{dt}\left[\frac{1}{2}B_{k,c}(u+tv,u+tv)-\int_{\Omega}(u+tv)\,f\;dx-\int_{\Gamma_{N}}(u+tv)\,h\;ds\right]$
$\displaystyle=\frac{d}{dt}\left[\frac{1}{2}\int_{\Omega}\left(\langle
k\nabla(u+tv),\nabla(u+tv)\rangle+c(u+tv)^{2}\right)\;dx-\int_{\Omega}fu\;dx\right.$
$\displaystyle\qquad\quad\left.-t\int_{\Omega}fv\;dx-\int_{\Gamma_{N}}hu\;ds-t\int_{\Gamma_{N}}hv\;ds\right]$
$\displaystyle=\int_{\Omega}\left(\langle k\nabla u,\nabla v\rangle+kt|\nabla
v|^{2}+cuv+tv^{2}\right)\;dx$
$\displaystyle\qquad-\int_{\Omega}fv\;dx-\int_{\Gamma_{N}}hv\;ds.$ (12)

Hence

$\displaystyle\frac{d}{dt}J(u+tv){\Large|}_{t=0}\;$
$\displaystyle=\;\int_{\Omega}\left(\langle k\nabla u,\nabla
v\rangle+cuv\right)\;dx-\int_{\Omega}fv\;dx-\int_{\Gamma_{N}}hv\;ds$
$\displaystyle=F(u,v).$ (13)

∎

The next theorem constitutes a fundamental result for the approximation
framework that we shall develop later. It relates the solution of the
variational problem $VF(\cal{P})$ with the minimizer of the energy functional
$J$.

######  Theorem 2.2.

The variational formulation problem $VF(\cal{P})$ in (8) does have a unique
solution $u^{*}$ in $H^{1}_{\Gamma_{D},g}(\Omega)$. Moreover, such a solution
is characterized by the unique minimizer of the energy functional $J$ defined
in (2.1), i.e.

$u^{*}\;=\;\operatorname*{arg\,min}_{u\in
H^{1}_{\Gamma_{D},g}(\Omega)}\;J(u).$ (14)

###### Proof.

It is easy to show that the energy functional $J$ in (2.1) is strictly convex
and coercive over $H^{1}_{\Gamma_{D},g}(\Omega)$ (for this we need $c$ to be
strictly positive), which implies the existence and uniqueness of a global
minimizer $u^{*}\in H^{1}_{\Gamma_{D},g}(\Omega)$. Hence, for any $v\in
H^{1}_{\Gamma_{D},0}(\Omega)$ there must hold

$0\;=\;\frac{d}{dt}\,J(u^{*}+tv){\Large|}_{t=0}\;=\;F(u^{*},v),$

where the last equality follows from Lemma 2.1. Thus $F(u^{*},v)=0$ for all
$v\in H^{1}_{\Gamma_{D},0}(\Omega)$ which implies that

$\int_{\Omega}\left(\langle k\nabla u^{*},\nabla
v\rangle+cu^{*}v\right)\;dx=\int_{\Omega}fv\;dx+\int_{\Gamma_{N}}hv\;ds\qquad\forall
v\in H^{1}_{\Gamma_{D},0}(\Omega),$

and therefore $u^{*}$ is a solution of problem $VF(\cal{P})$ given in (8).

Now let us prove that the solution of problem $VF(\cal{P})$ is unique. Let
$u^{*}$ be a solution of problem $VF(\cal{P})$ and $w\in
H^{1}_{\Gamma_{D},g}(\Omega)$ be arbitrary. Then

$\displaystyle J(w)$
$\displaystyle-J(u^{*})\;=\;\frac{1}{2}B_{k,c}(w,w)-\int_{\Omega}fw\;dx-\int_{\Gamma_{N}}hw\;ds-\frac{1}{2}B_{k,c}(u^{*},u^{*})$
$\displaystyle\qquad+\int_{\Omega}fu^{*}\;dx+\int_{\Gamma_{N}}hu^{*}\;ds$
$\displaystyle=\frac{1}{2}\left(B_{k,c}(w,w)-B_{k,c}(u^{*},u^{*})\right)-\left(\int_{\Omega}f\underset{\in
H^{1}_{\Gamma_{D},0}(\Omega)}{\underbrace{(w-u^{*})}}\;dx+\int_{\Gamma_{N}}h\underset{\in
H^{1}_{\Gamma_{D},0}(\Omega)}{\underbrace{(w-u^{*})}}\;ds\right)$
$\displaystyle=\frac{1}{2}\left(B_{k,c}(w,w)-B_{k,c}(u^{*},u^{*})\right)$
$\displaystyle\quad-\int_{\Omega}\left(\langle k\nabla
u^{*},\nabla(w-u^{*})\rangle+cu^{*}(w-u^{*})\right)\;dx\quad{\textrm{(since
$u^{*}$ solves $VF(\cal{P})$)}}$
$\displaystyle=\frac{1}{2}\left(B_{k,c}(w,w)-B_{k,c}(u^{*},u^{*})\right)-B_{k,c}(u^{*},w-u^{*})$
$\displaystyle=\frac{1}{2}B_{k,c}(w,w)-\frac{1}{2}B_{k,c}(u^{*},u^{*})-B_{k,c}(u^{*},w)+B_{k,c}(u^{*},u^{*})$
$\displaystyle=\frac{1}{2}\left(B_{k,c}(w,w)+B_{k,c}(u^{*},u^{*})-2B_{k,c}(u^{*},w)\right)$
$\displaystyle=\frac{1}{2}B_{k,c}(w-u^{*},w-u^{*})$
$\displaystyle=\frac{1}{2}\|w-u^{*}\|_{B_{k,c}}^{2}.$

It then follows that $J(w)\geq J(u^{*})$ for all $w\in
H^{1}_{\Gamma_{D},g}(\Omega)$ and $J(w)=J(u^{*})$ if and only if $w=u^{*}$.

Hence problem $VF(\cal{P})$ has a unique solution $u^{*}\in
H^{1}_{\Gamma_{D},g}(\Omega)$ which is the unique minimizer of the energy
functional $J(\cdot)$ over $H^{1}_{\Gamma_{D},g}(\Omega)$. ∎

######  Remark 2.1.

Note that within this variational formulation, the condition $c>0$ (strictly)
is necessary for uniqueness.

######  Remark 2.2.

The condition $k(x)\geq\gamma_{1}>0$ is strictly necessary for $J$ to be
accretive. The condition $k(x)\leq\gamma_{2}<\infty$ can be replaced by $k(x)$
bounded above by a positive function in $L^{1}(\Omega)$.

###  2.2 The inverse problem

Now, given all the model parameters $\Omega$, $\Gamma_{D}$, $\Gamma_{N}$, $c$,
$f$, $g$ and $h$, and a prescribed temperature distribution $\hat{u}\in
H^{1}_{\Gamma_{D},g}(\Omega)$ we consider the problem of finding the
corresponding distributed conductivity field $k(\cdot)$ such that
$u^{*}=\hat{u}$. That is, we intend to “invert” problem
${\cal{P}}={\cal{P}}(k,c,f,g,h)$ with respect to $k$.

For simplicity and clarity in the presentation of the numerical results only
the 2D-case will be considered, i.e. we will take $n=2$, and we will assume
$f=h\equiv 0$. Thus given a prescribed temperature field $\hat{u}(x,y)\in
H^{1}_{\Gamma_{D},g}(\Omega)$ we want to find $k=k(x,y)$ such that $\hat{u}$
be the unique solution of problem ${\cal{P}}(k,c,f,g,h)$. Then, according to
Theorem 2.2, $k(x,y)$ must satisfy:

$\displaystyle 0\;$ $\displaystyle=\;\frac{d}{dt}J(\hat{u}+tv){\Large|}_{t=0}$
$\displaystyle=\int_{\Omega}\left(\langle k\nabla\hat{u},\nabla
v\rangle+c\hat{u}v\right)\;dx\,dy,\qquad\forall v\in
H^{1}_{\Gamma_{D},0}(\Omega).$ (15)

##  3 Implementation

###  3.1 Discretizing the optimality condition and regularization

Let $\Omega_{i}$, $1\leq i\leq L$, be a partition of $\Omega$ by open sets,
i.e. such that $\Omega_{i}\subset\Omega$ is open for all $i$,
$\displaystyle\cup_{i=1}^{L}\overline{\Omega_{i}}=\overline{\Omega}$ and
$\Omega_{i}\cap\Omega_{j}=\emptyset$ if $i\neq j$. For each $i$ let
$(x_{i},y_{i})\in\Omega_{i}$ and for any function $q(x,y)$ defined on $\Omega$
let us denote with $q_{i}$ the value of $q$ at the point $(x_{i},y_{i})$, i.e.
$q_{i}\doteq q(x_{i},y_{i})$. With this notation, equation (2.2) can then be
approximated by

$0=\sum_{i=1}^{L}\left[k_{i}\left(\hat{u}_{x,i}v_{x,i}+\hat{u}_{y,i}v_{y,i}\right)+c_{i}\hat{u}_{i}v_{i}\right]\,m(\Omega_{i}),\qquad\forall
v\in H^{1}_{\Gamma_{D},0}(\Omega),$ (16)

where the subscripts $x$ and $y$ denote the respective partial derivatives and
$m(\Omega_{i})$ is the Lebesgue measure of $\Omega_{i}$. Assuming that the
partition is regular so that $m(\Omega_{i})$ is constant, equation (16) is
equivalent to

$\sum_{i=1}^{L}k_{i}\left(\hat{u}_{x,i}v_{x,i}+\hat{u}_{y,i}v_{y,i}\right)=-\sum_{i=1}^{L}c_{i}\hat{u}_{i}v_{i},\qquad\forall
v\in H^{1}_{\Gamma_{D},0}(\Omega).$ (17)

Consider now a finite, arbitrary set of functions

$v^{r}\in H^{1}_{\Gamma_{D},0}(\Omega),\qquad 1\leq r\leq R.$ (18)

Then (17) implies that

$\sum_{i=1}^{L}k_{i}\left(\hat{u}_{x,i}v^{r}_{x,i}+\hat{u}_{y,i}v^{r}_{y,i}\right)=-\sum_{i=1}^{L}c_{i}\hat{u}_{i}v^{r}_{i},\qquad\forall\,1\leq
r\leq R.$ (19)

By defining

$a_{r\ell}\;\doteq\;\hat{u}_{x,\ell}\,v^{r}_{x,\ell}+\hat{u}_{y,\ell}\,v^{r}_{y,\ell}\;\textrm{
and
}f_{r}\;\doteq\;-\sum_{i=1}^{L}c_{i}\,\hat{u}_{i}\,v^{r}_{i},\;1\leq\ell\leq
L,\;\;1\leq r\leq R,$

we can write (19) as follows:

$\sum_{\ell=1}^{L}a_{r\ell}k_{\ell}\;=f_{r},\qquad\forall\,1\leq r\leq R.$
(20)

Let now $\mathbf{A}$ be an $R\times L$-dimensional matrix,
$K\in\mathbb{R}^{L}$ and $F\in\mathbb{R}^{R}$ whose elements are $a_{r\ell}$,
$k_{\ell}$ and $f_{r}$, respectively. Then, equation (20) can be simply
written in matrix form as:

$\mathbf{A}\,K=F.$ (21)

Note that we still need to impose the condition that all components of the
vector $K$ be bounded between the values $\gamma_{1}$ and $\gamma_{2}$. In
what follows the idea is to solve (21) in the least squares sense, weakly
imposing this restriction through a penalizer.

With the above in mind we consider a generalized Tikhonov-Phillips functional
of the form:

$J_{\alpha,W}(K)\;\doteq\;\|\mathbf{A}K-F\|^{2}+\alpha\,W(K),$ (22)

where $\alpha>0$ is a regularization parameter and the penalizer $W(K)$ must
be appropriately constructed in such a way as to deter non-admissible values
as well as any undesired property of the conductivity profile $k(x,y)$.

###  3.2 Assumptions for the numerical implementations

In what follows we take $\Omega=(0,1)\times(0,1)$ and the Dirichlet and
Newmann boundaries are the vertical and horizontal boundaries of $\Omega$,
respectively, i.e. $\Gamma_{D}=\\{0,1\\}\times(0,1)$ and
$\Gamma_{N}=(0,1)\times\\{0,1\\}$ (see Figure 2).

Also, for the numerical experiments that follow, the Dirichlet boundary
conditions will be prescribed as $g(x)=T_{1}$ on $\\{0\\}\times(0,1)$ and
$g(x)=T_{2}$ on $\\{1\\}\times(0,1)$ where $T_{1}$ and $T_{2}$ are prescribed
constant temperature values, with $T_{1}>T_{2}$. Furthermore, we shall assume
that $k(x,y)$ can take only two possible values, say $k_{L}$ and $k_{U}$, with
$0<k_{L}<k_{U}<\infty$. Physically this implies that only two different
materials having those conductivities can be present in the domain $\Omega$.
Clearly this assumption is extremely important to design appropriate forms for
$W(K)$. It is timely to mention however that all the previous formulation
leading up to equations (21) and (22) only requires the assumption $k\in
L_{>0}^{\infty}(\Omega)$.

Figure 2: Schematic representation of the problem setting and its
discretization

###  3.3 On the penalizer $\mathbf{W(K)}$

Under the above assumption that at each point $(x,y)\in\Omega$, $k(x,y)$ can
only take one of the values $k_{L}$ or $k_{U}$, the penalizer $W(K)$ must be
designed so that it deters each and every component of the vector $K$ to take
any but one of those two values. A simple way to do that is as follows.

Let $p:\mathbb{R}\to\mathbb{R}$ be the monic quadratic polynomial that
vanishes at $k_{L}$ and at $k_{U}$, i.e.
$p(z)=(z-k_{L})(z-k_{U})\;=\;z^{2}-(k_{L}+k_{U})z+k_{L}k_{U}$. Next we define
$W_{1}:\mathbb{R}^{L}\to\mathbb{R}^{+}_{0}$ as

$W_{1}(K)\;\doteq\;\|p(K)\|^{2}_{\mathbb{R}^{L}},$ (23)

where $p$ acting on the vector $K$ must be understood as its component-wise
action, i.e. $\left(p(K)\right)_{i}=p(K_{i})$. Clearly, then, $W_{1}(K)\geq 0$
and $W_{1}(K)=0$ if and only if each and every component of $K$ takes one of
the two values $k_{L}$ or $k_{U}$.

A different approach is to design an ad-hoc penalizer which, besides promoting
all the components of the vector $K$ to take only the values $k_{L}$ or
$k_{U}$, provides data-driven information about where to take one or the other
value. With that in mind let $b_{U}\in\mathbb{R}^{L}$ be a binary vector whose
$i^{th}$ component takes the value $1$ if and only if the gradient of
$\hat{u}$ at the point $(x_{i},y_{i})$ is “large”, i.e. if
$\|\nabla\hat{u}(x_{i},y_{i})\|>\gamma$, where $\gamma$ is a certain
prescribed threshold value, and define the penalizer $W_{2}(K)$ as

$W_{2}(K)\doteq\|b_{U}\odot(K-k_{L}\,\mathbf{1})\|^{2},$ (24)

where $\odot$ denotes elementwise product and $\mathbf{1}\in\mathbb{R}^{L}$ is
a vector of ones. Hence, at a point where $\|\nabla\hat{u}\|$ is large (as
dictated by the threshold parameter $\gamma$), this penalizer will discourage
the corresponding component of $K$ to assume any value except $k_{L}$.

In the definitions (23) and (24), the $\mathbb{R}^{L}$-norm can be replaced by
any other norm. Finally, many other options for the functional $W(K)$ exist if
other restrictions on the function $k(x,y)$ are to be enforced, for instance,
certain regularity properties (which clearly would not be appropriate for our
case at hand).

###  3.4 Examples, numerical experiments and discussions

####  3.4.1 Case I

We first solved the forward problem $\cal{P}$ under the previous assumptions
with $T_{1}=322.0\;[K]$, $T_{2}=283.0\;[K]$, $c(x,y)=1.0=$ constant, and
$k(x,y)$ as shown in Figure 3. For this we used a standard discretization by
finite element method, using biquadratic interpolation elements S2 with
8-nodes for computing $\hat{u}(x,y)$, $\hat{u}_{x}(x,y)$ and
$\hat{u}_{y}(x,y)$. The same scheme was used to solve all forward problems in
the coming cases. The resulting temperature distribution $\hat{u}(x,y)$, whose
discretized values are then used as inputs for the inverse problem, is shown
in Figure 4 b).

Figure 3: Conductivity $k(x,y)$ used for solving the forward problem in Case I
Figure 4:  a) Sketch of the distretized domain used to solve the forward
problem, for case I. The finite element mesh S2 used is regular with elements
size $h=1/200$. b) Temperature distribution $\hat{u}(x,y)$ for $k(x,y)$ as in
Figure 3.

Setting 1: For this case we picked $\alpha=0$ (non-penalized case) in (22) and
the set of functions $v^{r}\in H^{1}_{\Gamma_{D},0}(\Omega),\;1\leq r\leq R$,
was chosen as an appropriate single index reordering of the set of functions:

$v^{m,n}(x,y)\;\doteq\;x^{m}(1-x)^{n},\qquad\textrm{for }1\leq m,n\leq M,$
(25)

where $M\in\mathbb{N}$ is appropriately chosen. Note that $v^{m,n}\in
H^{1}_{\Gamma_{D},0}(\Omega)$ as required in (2.2). Thus $R=M^{2}$ and
$v^{r}_{i}\doteq v^{m,n}(x_{i},y_{i})=x_{i}^{m}(1-x_{i})^{n}$ for all $1\leq
i\leq L$, $1\leq r\leq R$ and $1\leq m,n\leq M$. The general idea behind our
approach is to somehow determine “how much” information about $k(x,y)$ can be
obtained from the set of functions (25) as a subset of
$H^{1}_{\Gamma_{D},0}(\Omega)$ through the optimality equation (2.2) which
characterizes $k(x,y)$ or, more precisely, through its discrete counterpart
(17).

We first solved (22) with $\alpha=0$. Hence, no restrictions on $k(x,y)$ were
imposed. Also, we used $M=5$ in (25) which results in $R=25$ test functions in
(18). We computed the best approximate solution (i.e. the least squares
solution of minimal norm) of $AK=F$ as $K^{\dagger}=A^{\dagger}F$ where
$A^{\dagger}$ denotes the Moore-Penrose generalized inverse of $A$ ([8]). The
obtained (discretized) $k(x,y)$ is plotted in Figure 5. We observe how the
basic geometric distribution of the two possible values of $k(x,y)$ is already
recovered using only the 25 functions $v^{m,n}(x,y)$ in (25) with $M=5$, even
without using any penalization in equation (22). It is timely to point out
here that the severe ill-posedness of the problem is clearly reflected on the
matrix $A$ whose condition number turns out to be approximately equal to
$2.5\times 10^{18}$ (here we used $L=10^{4}$ resulting from a $100\times 100$
regular grid on $\Omega$. It is reasonable to think that a better restoration
of $k(x,y)$ could be obtained by taking $\alpha>0$ and minimizing the
functional $J_{\alpha,W}$ given in (22) using the penalizer $W_{1}(K)$ defined
in (23).

Figure 5: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 4, using a non-penalized least squares approach.

Setting 2: For this example all parameters and functions are the same as in
Setting 1, except that here we used $\alpha>0$ in (22) and the penalizer was
chosen as $W(K)=W_{1}(K)$ given by (23). The appropriate value of $\alpha>0$
was picked by means of the L-curve method ([13], [12], [8]). Note that finding
the minimizer of (22) corresponds to finding the Tikhonov-Phillips
$W_{1}$-regularized solution of (21). The obtained conductivity $k(x,y)$ is
shown in Figure 6. As it can be seen, the introduction of the penalizer
$W_{1}$, as expected, induces $k$ to take only values around $K_{L}=1.0$ and
$K_{U}=300$. Although the correct geometric profile of $k(x,y)$, shown in
Figure 3, can be clearly appreciated in this figure, the obtained result is
far from being satisfactory. It is highly desirable to come up with a better
penalizer that be able to reconstruct the conductivity distribution as close
as possible.

Figure 6: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 4, via the penalized least squares approach (22) with $W(K)=W_{1}(K)$
given as in (23).

Setting 3: Here we proceed to use the penalizer $W_{2}(K)$ which, unlike
$W_{1}$, includes data-driven information about the local size of
$\|\nabla\hat{u}(x,y)\|$. Hence, $k(x,y)$ is now obtained as the Tikhonov-
Phillips $W_{2}$-regularized solution of problem (21), that is, as the global
minimizer of (22) when the penalizer $W(K)$ is given by $W_{2}(K)$ as defined
in (24). The value of the threshold parameter $\gamma$ needed to define
$b_{U}$ in (24) was set to $0.01*M$, where $M\doteq\max_{1\leq i\leq
L}\|\hat{u}(x_{i},y_{i})\|$ while the regularization parameter $\alpha$ was
again chosen by means of the L-curve method. The restored conductivity profile
$k(x,y)$ is shown in Figure 7. As it can be observed, although some small
artifacts still appear, a much better approximation of the true profile is
obtained.

Figure 7: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 4, using a penalized least squares approach with $W(K)=W_{2}(K)$
defined as in (24)

We now proceed to present the results obtained for three other different
configurations of the conductivity profile $k(x,y)$. For the sake of brevity,
we shall only show the results obtained under Setting 3 of Case I, which is
clearly by far the best approach for solving the inverse problem at hand.

####  3.4.2 Case II

For this case the conductivity profile is as depicted in Figure 8. Here again
we computed $\hat{u}(x,y)$ by first running the forward problem with
$T_{1}=318.15\;[K]$, $T_{2}=288.15\;[K]$, $c(x,y)=1.0=$ constant. The
resulting temperature distribution $\hat{u}(x,y)$, whose discretized values
are used as inputs for the inverse problem, is shown in Figure 9 b).

Figure 8: Distributed values of the conductivity $k(x,y)$ used for solving the
forward problem in Case II
Figure 9:  a) Sketch of the distretized domain used to solve the forward
problem for Case II. The finite element mesh S2 used is regular with elements
size $h=1/200$. b) Temperature distribution $\hat{u}(x,y)$ for $k(x,y)$ as in
Figure 8, Case II.

Here, the set of functions $v^{r}\in H^{1}_{\Gamma_{D},0}(\Omega),\;1\leq
r\leq R$, was chosen as in Setting 1 of Case I. We proceeded to compute the
generalized Tikhonov-Phillips solution of problem (21) when the penalizer is
given by $W_{2}(K)$ as defined in (24). The threshold parameter was now chosen
as $\gamma=0.0125\,M$, where, as before, $M\doteq\max_{1\leq i\leq
L}\|\hat{u}(x_{i},y_{i})\|$ while the regularization parameter $\alpha$ was
again chosen by means of the L-curve method. The restored conductivity profile
$k(x,y)$ is shown in Figure 10. Here again, despite a few artifacts, we can
observe how our method is able to satisfactory reconstruct the conductivity
distribution profile.

Figure 10: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 9, using a penalized least squares approach with $W_{2}(K)$ defined as
in (24), Case II.

####  3.4.3 Case III

For this case the conductivity profile is as depicted in Figure 11. Here again
we computed $\hat{u}(x,y)$ by first running the forward problem with
$T_{1}=373.15\;[K]$, $T_{2}=353.15\;[K]$, $c(x,y)=1.0=$ constant, $k_{U}=100$
and $k_{L}=0.7$. The resulting temperature distribution $\hat{u}(x,y)$ is
shown in Figure 12 b).

Figure 11: Distributed values of the conductivity $k(x,y)$ used for solving
the forward problem in Case III
Figure 12:  a) Sketch of the discretized domain used to solve the forward
problem, for case III. The finite element mesh S2 used is regular with
elements size $h=1/200$. b) Temperature distribution $\hat{u}(x,y)$ for
$k(x,y)$ as in Figure 11, Case III.

As in the previous case, the set of functions $v^{r}\in
H^{1}_{\Gamma_{D},0}(\Omega),\;1\leq r\leq R$, was chosen as in Setting 1 of
Case I. We proceeded to compute the generalized Tikhonov-Phillips solution of
problem (21) when the penalizer is given by $W_{2}(K)$ as defined in (24). The
value of the threshold was now chosen as $\gamma=0.0125\,M$, where, as before,
$M\doteq\max_{1\leq i\leq L}\|\hat{u}(x_{i},y_{i})\|$ while the regularization
parameter was again chosen by means of the L-curve method. The restored
conductivity profile $k(x,y)$ is shown in Figure 13. Once again we can observe
how our method is able to satisfactory reconstruct the conductivity
distribution profile.

Figure 13: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 12 b), using a penalized least squares approach with $W_{2}(K)$ defined
as in (24), Case III.

####  3.4.4 Case IV

We end up presenting a case where the conductivity profile has a very
irregular shape, as depicted in Figure 14. The goal is to verify if the
methods is able to detect these irregular boundaries in $k(x,y)$. Here again
we computed $\hat{u}(x,y)$ by first running the forward problem with
$T_{1}=308.15\;[K]$, $T_{2}=298.15\;[K]$, $c(x,y)=1.0=$ constant, $k_{U}=125$
and $k_{L}=20$. The resulting temperature distribution $\hat{u}(x,y)$ is shown
in Figure 15 b).

Figure 14: Distributed values of the conductivity $k(x,y)$ used for solving
the forward problem in Case IV.

As in the previous cases, we proceeded to compute the generalized Tikhonov-
Phillips solution of problem (21), as the global minimizer of (22) when the
penalizer is given by $W_{2}(K)$ as defined in (24). The value of the
threshold in this case was chosen as $\gamma=0.3153\,M$, with
$M\doteq\max_{1\leq i\leq L}\|\hat{u}(x_{i},y_{i})\|$ while the regularization
parameter was again chosen by means of the L-curve method. The restored
conductivity profile $k(x,y)$ is shown in Figure 16. Once again, we can
observe how our method is able to quite satisfactory reconstruct the
conductivity distribution profile.

Figure 15:  a) Sketch of the discretized domain used to solve the forward
problem, for case IV. The finite element mesh S2 used is regular with elements
size $h=1/200$. b) Temperature distribution $\hat{u}(x,y)$ for $k(x,y)$ as in
Figure 14, Case IV.
Figure 16: Reconstruction of $k(x,y)$ obtained using $\hat{u}(x,y)$ as in
Figure 15, using a penalized least squares approach with $W_{2}(K)$ defined as
in (24), Case IV.

## Conclusions and future work

In this article we introduced a method for solving the inverse problem of
estimating the principal coefficient in a steady state elliptic diffusion
equation. The method is based on the discretization of an optimality equation
derived from a variational approach which leads to a least squares problem
based on a fidelity term. The method is followed by regularization consisting
of adding an appropriate penalizer, encoding prior-information about the
conductivity profile, leading to a the problem of finding the minimizer of a
generalized Tikhonov-Phillips functional.

Several numerical experiments with different types of discontinuous
distributions for the leading coefficient function $k(x,y)$ were presented,
showing that, with the appropriate penalizer, the method yields excellent
reconstructions of the true (unknown in practical problems) function.

There is clearly much room for improvements and generalizations. First we find
it timely to point out that several numerical experiments have shown that
adding more functions in $H^{1}_{\Gamma_{D},0}(\Omega)$ to the the set in (25)
does not improve the reconstructions in none of the cases. That is due to the
severe ill-posedness of the problem.

As it was shown, the success of the method is highly dependent on the
appropriate choice of the regularization term. The penalizer $W_{2}$ defined
in (24) works very well for the case in which $k(x,y)$ takes only two
prescribed values. However, in that case, although experiments showed that the
results are somewhat robust with respect to the choice of $\gamma$, designing
rigorous analytic ways for estimating an optimal value of that threshold
parameter $\gamma$ is highly desirable.

A detailed analysis of the results and images show that the small artifacts
obtained in the reconstruction of the function $k(x,y)$ are always associated
to places where $|\nabla u|=0$. Although this is reasonable since at those
points $k$ cannot be restored, more research is necessary to come up with a
way to avoid those artifacts, perhaps taking into account a-priori information
about the expected conductivity.

In regard to the design of appropriate penalizers, no general recipe is
expected to be found which will work for arbitrary functions $k(x,y)$. It is
not completely clear, for instance, what would be a good choice for the
penalizer $W(\cdot)$ in the case in which three (or more) possible
conductivity values are present. Efforts in all of these directions are
currently under way.

## Acknowledgements

This work was supported in part by Consejo Nacional de Investigaciones
Científicas y Técnicas, CONICET, through grants PIP 2021-2023 number
11220200100806CO and PUE-IMAL number 22920180100041CO, by Agencia Nacional de
Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación,
througn grant PICT-2019-2019-01933 and by Universidad Nacional del Litoral,
through grant CAI+D 2020 Nro. 50620190100069LI.

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