#  Convergence of a simple discretization of
the finite Hilbert transformation

Martin Costabel  Univ. Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
<EMAIL_ADDRESS>
(Date: 04/09/2023)

###### Abstract.

For a singular integral equation on an interval of the real line, we study the
behavior of the error of a delta-delta discretization. We show that the
convergence is non-uniform, between order $O(h^{2})$ in the interior of the
interval and a boundary layer where the consistency error does not tend to
zero.

###### Key words and phrases:

singular integral equation, Hilbert transform, delta-delta discretization,
method of discrete vortices, discrete dipole approximation

###### 1991 Mathematics Subject Classification:

65R20, 45E05

##  1\. Introduction

Let $a,b\in{\mathbb{R}}$ with $a<b$. On the interval $\Omega=(a,b)$ we
consider the singular integral equation, abbreviated as
$(\lambda\mathbb{I}-A_{\Omega})u=f$,

(1.1) $\lambda u(x)-\mathop{\rm
p.v.}\int_{\Omega}\frac{u(y)}{i\pi(x-y)}dy=f(x)\,,\quad x\in\Omega\,.$

We discretize it in the simplest imaginable fashion. We choose
$N\in{\mathbb{N}}$ defining the mesh width $h=\frac{1}{N}$ and fix some origin
$a^{N}\in{\mathbb{R}}$, thus defining the infinite regular grid

$\Sigma^{N}=\\{x^{N}_{m}=a^{N}+mh\mid m\in{\mathbb{Z}}\\}$

and its finite counterpart $\Sigma^{N}\cap\Omega$ indexed by

$\omega^{N}=\\{m\in{\mathbb{Z}}\mid x^{N}_{m}\in\Omega\\}\,,$

and consider the system

(1.2) $\lambda u_{m}-\frac{1}{i\pi N}\sum_{n\in\omega^{N},m\neq
n}\frac{u_{n}}{x^{N}_{m}-x^{N}_{n}}=f(x^{N}_{m})\,,\quad(m\in\omega^{N})\,.$

This system can easily be programmed and solved with a couple of lines of
code, without any knowledge of analysis, and it produces surprisingly good
results, see Figure 1 for some examples for which the exact solution of the
integral equation is known. These examples are introduced below in Section 2.3
and further analyzed numerically in Section 4.

Figure 1. 3 examples: Exact solutions and their computed p/w constant
approximations

This simple delta-delta scheme is similar to Lifanov’s method of discrete
vortices [1], except that we take the same grid for the quadrature points and
the evaluation points, and we put zero on the diagonal.

There is also a similarity to the fully discrete Calderón calculus analyzed in
[5], although the differences, namely that we consider an open interval and
not a closed curve and that our integral operator is strongly singular, are
too important to try a similar analysis.

Note that the approximation scheme (1.2) for the integral equation (1.1) is
not a projection method (or Petrov-Galerkin scheme) in any meaningful sense,
although it has the form (except for the diagonal term) of a Galerkin scheme
with Dirac deltas as test and trial functions. This has the negative
consequence that some tools are not available that one would like to use in
order to generalize results proved for the model singular integral equation
(1.1) to equations with more general strongly singular kernels, a variable
multiplier $\lambda$ instead of a constant, or equations with lower order
terms. These tools use the persistence of stability under compact
perturbations of the operator, and this is known for projection methods and
also for more general discrete approximation schemes in the Stummel-Vainikko
sense, see for example [10]. But it seems that our simple scheme does not fit
into any of these frameworks.

The question of a convergence proof for the delta-delta approximation (1.2)
came up in the context of our recent research into the error analysis of
higher-dimensional simple discretization methods for strongly singular volume
integral equations related to the Discrete Dipole Approximation (DDA). The
latter has been a standard tool in computational electrodynamics for half a
century [7, 11, 2], with a wide range of applications from interstellar dust
clouds to nano-particles. Despite the popularity of DDA in computational
physics, there is very little known about its mathematical properties, but we
have now some results about its stability [4, 3]. In order to complete the
convergence proof, one needs estimates for the consistency error, and for
gaining insight on the behavior of this, the one-dimensional example (1.1),
(1.2) presented itself as a “toy problem”, where the error analysis should be
more transparent, while still showing some essential peculiarities of the more
complicated higher-dimensional situation. It turns out that this is indeed the
case, but that the results are interesting by themselves, and we present this
in the following.

##  2\. Stability

Numerical stability of the system (1.2), which we abbreviate as
$(\lambda\mathbb{I}-T^{N})U^{N}=F^{N}$, means that there exists a bound of
$U^{N}$ by $F^{N}$ uniform in $N$, that is, a uniform resolvent estimate in
some operator norm $\|(\lambda\mathbb{I}-T^{N})^{-1}\|\leq C_{S}$ for all $N$.

###  2.1. Stability in $\ell^{2}$

Such a stability estimate has been proved in [4], and although most of it is
based on quite well known arguments, for the sake of completeness we quote the
result and the main points of its proof here.

######  Proposition 2.1.

The matrix $T^{N}$ of the system (1.2) is selfadjoint with its eigenvalues in
${\mathcal{C}}=[-1,1]$. For $\lambda\in{\mathbb{C}}$ the discretization method
(1.2) is stable in the $\ell^{2}$ norm if and only if the integral operator
$\lambda\mathbb{I}-A_{\Omega}$ is boundedly invertible in $L^{2}(\Omega)$, and
this is equivalent to $\lambda\not\in{\mathcal{C}}$. For such $\lambda$, there
is an estimate for the operator norms

(2.1)
$\|(\lambda\mathbb{I}-T^{N})^{-1}\|_{{\mathcal{L}}(\ell^{2}(\omega^{N}))}\leq\mathrm{dist}(\lambda,{\mathcal{C}})^{-1}=\|(\lambda\mathbb{I}-A_{\Omega})^{-1}\|_{{\mathcal{L}}(L^{2}(\Omega))}\,.$

The arguments are based on Fourier analysis and on a Lax-Milgram or Galerkin
style use of the notion of numerical range. We recall the definition of the
numerical range $W(B)$ of a bounded linear operator $B$ in Hilbert space,
namely the range of values on the unit sphere of the sesquilinear form
associated with $B$

(2.2) $W(B)=\\{(u,Bu)|\|u\|=1\\}\,.$

This is a bounded set contained in the disk with radius $\|B\|$, convex by the
Toeplitz-Hausdorff theorem, and its closure contains the spectrum of $B$.
Writing for any $u$ of norm one and $z=(u,Bu)$ the estimate

$\mathrm{dist}(\lambda,W(B))\leq|\lambda-z|=|(u,\lambda
u-Bu)|\leq\|(\lambda\mathbb{I}-B)u\|\,,$

one immediately gets the resolvent estimate in the operator norm

(2.3) $\|(\lambda\mathbb{I}-B)^{-1}\|\leq\mathrm{dist}(\lambda,W(B))^{-1}\,.$

Considering that the restriction to a subspace does not increase the numerical
range, one gets the same resolvent estimate for operators defined from $B$ by
restricting the sesquilinear form to a subspace. This is the Lax-Milgram
argument for stability of Galerkin methods, and it applies here both to the
integral equation (1.1) and the discrete system (1.2).

The integral operator $A_{\Omega}$ in (1.1) and the system matrix $T^{N}$ in
(1.2) have this feature in common: They are both projections to bounded sets
of translation invariant (i.e. convolution) operators, and these can be
diagonalized by Fourier analysis.

The finite Hilbert transformation $A_{\Omega}$ is the restriction to $\Omega$
of the Hilbert transformation $A$ on ${\mathbb{R}}$, that is the convolution
with the kernel $K$ defined as the distribution

$K(x)=\frac{1}{i\pi}\mathop{\rm p.v.}\frac{1}{x}$

that has the Fourier transform

$\widehat{K}(\xi)=\mathrm{sign}\,\xi\quad(\xi\in{\mathbb{R}})\,.$

From the Plancherel theorem follows that the convolution operator $A$ acting
in the Hilbert space $L^{2}({\mathbb{R}})$ is equivalent to the operator of
multiplication with its symbol $\widehat{K}$ in $L^{2}({\mathbb{R}})$. It
follows in particular that $A$ is a selfadjoint involution and its spectrum
consists of two eigenvalues $\pm 1$ of infinite multiplicity.

This implies $W(A_{\Omega})\subset W(A)={\mathcal{C}}$ and the resolvent
estimate

$\|(\lambda\mathbb{I}-A_{\Omega})^{-1}\|_{{\mathcal{L}}(L^{2}(\Omega))}\leq\mathrm{dist}(\lambda,{\mathcal{C}})^{-1}\,.$

On the discrete side there is a parallel argument using Fourier series instead
of the Fourier transform. The system matrix $T^{N}$ is a finite section of a
(bi-)infinite Toeplitz matrix $T$ that, owing to

$\frac{1}{i\pi N}\frac{1}{x^{N}_{m}-x^{N}_{n}}=\frac{1}{i\pi(m-n)}\,,$

is independent of $N$,

(2.4) $T=\big{(}\frac{1}{i\pi(m-n)}\big{)}_{m,n\in{\mathbb{Z}}}\quad\mbox{
with zero on the diagonal.}$

Let $\sigma_{T}$ be its symbol (characteristic function) defined by the
Fourier series

$\sigma_{T}(\tau)=\sum_{m\in{\mathbb{Z}}}\frac{1}{i\pi
m}e^{im\tau}=\sum_{m=1}^{\infty}\frac{2\sin m\tau}{\pi
m}=\mathrm{sign}\,\tau-\frac{\tau}{\pi}\quad(\tau\neq
0)\,,\quad\sigma_{T}(0)=0\,.$

Then by the Plancherel theorem it follows that the discrete convolution
operator defined by $T$ acting on $\ell^{2}({\mathbb{Z}})$ is equivalent to
the operator of multiplication with $\sigma_{T}$ in $L^{2}(-\pi,\pi)$.
Therefore both its spectrum and the closure of its numerical range
$\overline{W(T)}$ are equal to ${\mathcal{C}}=[-1,1]$. Since $T^{N}$ arises
from $T$ by restriction to $\ell^{2}(\omega^{N})$, we get the inclusion
$W(T^{N})\subset{\mathcal{C}}$ and the uniform resolvent estimate in (2.1).

With these simple arguments, we have proved Proposition 2.1, except for two
points: The assertion that stability implies that
$\lambda\not\in{\mathcal{C}}$, and the last equality in (2.1), where we would
only get an inequality.

For the first point we use the fact that whereas the system (1.2)

(2.5) $(\lambda\mathbb{I}-T^{N})U^{N}=F^{N}\,,$

when considered as an approximation scheme for the integral equation (1.1)
$(\lambda\mathbb{I}-A_{\Omega})u=f$, is not a projection method, it is indeed
a standard Galerkin scheme when considered as an approximation scheme for the
infinite system

(2.6) $(\lambda\mathbb{I}-T)U=F\quad\mbox{ in }\ell^{2}({\mathbb{Z}})\,.$

As such, it converges whenever it is stable; in detail: Denote by $P_{N}$ the
orthogonal projection operator in $\ell^{2}({\mathbb{Z}})$ onto the subspace
of sequences vanishing outside $\omega^{N}$, identified with
$\ell^{2}(\omega^{N})$, so that we can write
$(\lambda\mathbb{I}-T^{N})P_{N}=P_{N}(\lambda\mathbb{I}-T)P_{N}$. If we take
as the origin $a^{N}$ of the grid a fixed point inside $\Omega$, then

$\lim_{N\to\infty}P_{N}U=U\quad\mbox{ in $\ell^{2}({\mathbb{Z}})$ for any
$U$}.$

If $U^{N}$ and $U$ satisfy (2.5), (2.6) with $F^{N}=P_{N}F$, then we have the
identity

$U-U^{N}=\big{(}\mathbb{I}-(\lambda\mathbb{I}-T^{N})^{-1}P_{N}(\lambda\mathbb{I}-T)\big{)}(U-P_{N}U)\,.$

Now let $\lambda\in{\mathbb{C}}$ be such that the scheme (2.5) is stable,

$\|(\lambda\mathbb{I}-T^{N})^{-1}\|\leq C_{S}\,.$

Then $\|U-U^{N}\|\leq\big{(}1+C_{S}\|\lambda\mathbb{I}-T\|\big{)}\|U-P_{N}U\|$
(this is the quasi-optimality of the Galerkin scheme, Céa’s lemma), hence
$U^{N}\to U$, and because of $\|U^{N}\|\leq C_{S}\|F\|$, we find in the limit

$\|U\|\leq C_{S}\|F\|\,.$

This implies that $\lambda\mathbb{I}-T$ is invertible with the norm of the
inverse bounded by $C_{S}^{-1}$, hence $\lambda\not\in{\mathcal{C}}$, in fact
$\mathrm{dist}(\lambda,{\mathcal{C}})\geq C_{S}^{-1}$ .

For the second point, we recall that the finite Hilbert transformation and its
spectral theory is a well-studied classical object. In particular it can be
diagonalized by a generalized Fourier transformation [6] implying that as soon
as $\Omega$ is a proper subinterval of ${\mathbb{R}}$ (even a half line), its
action in $L^{2}(\Omega)$ is unitarily equivalent to the operator of
multiplication by $\sigma$ in $L^{2}(-1,1)$ with $\sigma(\xi)=\xi$. Both its
spectrum and the closure of its numerical range $\overline{W(A_{\Omega})}$ are
therefore equal to ${\mathcal{C}}$, which shows the last equality in (2.1).

###  2.2. Solution behavior

The simplicity of the statement of Proposition 2.1 is somewhat deceptive and
hides a more complicated situation: Whereas the solvability of the finite
system (1.2) is, of course, independent of the norm we choose in the space of
sequences, the latter only being used to describe the stability, this is quite
different for the solvability of the integral equation (1.1). Here, owing to
the singular behavior of the solution at the endpoints of $\Omega$, the
spectrum of the finite Hilbert transform depends on the function space. The
generalized eigenfunctions

$g_{\xi}(x)=\tfrac{1}{\pi}\sqrt{\tfrac{b-a}{2(x-a)(b-x)}}e^{\frac{i}{2\pi}\log\frac{\xi+1}{1-\xi}\log\frac{x-a}{b-x}}$

for which it is shown in [6] that they diagonalize the operator via the
Hilbert space isomorphism $f\mapsto F:L^{2}(a,b)\to L^{2}(-1,1)$ with the
transform pair

$F(\xi)=\tfrac{1}{\sqrt{1-\xi^{2}}}\int_{a}^{b}\overline{g_{\xi}(x)}f(x)dx,\quad
f(x)=\int_{-1}^{1}g_{\xi}(x)F(\xi)\tfrac{d\xi}{\sqrt{1-\xi^{2}}}$

such that $A_{\Omega}f$ is transformed to $\xi F(\xi)$, are genuine
eigenfunctions in $L^{p}(\Omega)$ with $p<2$. More precisely, one considers
the arcs of circles

$C_{\alpha_{0}}=\\{\lambda\in{\mathbb{C}}\mid\tfrac{\lambda+1}{\lambda-1}=e^{2\pi
i\alpha},\operatorname{Re}\alpha=\alpha_{0}\\}$

which connect the points $-1$ and $1$ inside the unit circle if
$\frac{1}{4}<\alpha_{0}<\frac{3}{4}$ and outside if
$\alpha_{0}\in(0,\frac{1}{4})\cup(\frac{3}{4},1)$. Then the spectrum of
$A_{\Omega}$ in $L^{p}(a,b)$ is the domain between $C_{1-\frac{1}{p}}$ and
$C_{\frac{1}{p}}$. Its interior consists of eigenvalues. For an eigenvalue
$\lambda$, define the exponent $\alpha$ by the above relation

$\alpha=\tfrac{1}{2\pi i}\log\tfrac{\lambda+1}{\lambda-1}\quad\mbox{ or
equivalently }\quad\lambda=-i\cot\pi\alpha,\qquad
1-\tfrac{1}{p}<\operatorname{Re}\alpha<\tfrac{1}{p}\,.$

Then the corresponding eigenfunction in $L^{p}(a,b)$ is

$u(x)=(x-a)^{-\alpha}(b-x)^{\alpha-1}.$

The spectrum in the dual space $L^{\frac{p}{p-1}}(\Omega)$ is the same set.
Thus for $p<2$ the solution of (1.1) exists, in general, but is not unique,
and for $p>2$ it is unique, but does not exist for all right hand sides. Only
for $p=2$, the two arcs of circle degenerate to the interval ${\mathcal{C}}$,
and for $\lambda\in{\mathcal{C}}$ the operator $\lambda\mathbb{I}-A_{\Omega}$
in $L^{2}(\Omega)$ is injective with dense range.

Thus, the delta-delta discretization manages to mimick the solution behavior
of the singular integral equation for the specific value $p=2$.

For $A_{\Omega}$, it is also known how to write the resolvent explicitly as a
singular integral operator with weight, involving multiplication by powers of
the distance to the points $a$ and $b$, see for example [8] or [9]. For the
discrete version $T^{N}$, such explicit formulas for the diagonalization and
resolvent do not seem to be available.

###  2.3. Exact solutions

Examples where both $u$ and $f$ are explicitly known functions can be obtained
from function-theoretic arguments. For $u\in L^{1}(a,b)$ define the Cauchy
integral for $z\in{\mathbb{C}}\setminus[a,b]$

$w(z)=\frac{1}{2\pi i}\int_{a}^{b}\frac{u(y)}{y-z}dy\,.$

Then $w$ is holomorphic in ${\mathbb{C}}\setminus[a,b]$, behaves as
$O(|z|^{-1})$ as $|z|\to\infty$, and satisfies on $\Omega=(a,b)$ the jump
relations

(2.7) $w_{+}-w_{-}=u\,;\qquad w_{+}+w_{-}=A_{\Omega}u,\quad\mbox{ where }\quad
w_{\pm}(x)=\lim_{\varepsilon\to 0+}w(x\pm i\varepsilon)\,.$

Therefore the integral equation (1.1) is equivalent to the jump condition

$(\lambda-1)w_{+}-(\lambda+1)w_{-}=f\,.$

Reciprocally, any function $w$ that is holomorphic outside $[a,b]$, vanishes
at infinity and has square integrable upper and lower traces on the branch cut
$[a,b]$ provides a solution $u$ to the integral equation (1.1) with right hand
side $f$, if $u$ and $f$ are defined from the traces $w_{\pm}$ via formulas
(2.7).

For example, the function $w_{0}(z)=\log\frac{z-a}{z-b}$ is (or can be chosen
to be) holomorphic in ${\mathbb{C}}\setminus[a,b]$, is $O(|z|^{-1})$ at
infinity, and has the upper and lower traces

$w_{0,+}(x)=\log\tfrac{x-a}{b-x}-i\pi\,,\quad
w_{0,-}(x)=\log\tfrac{x-a}{b-x}+i\pi$

Since these traces belong to $L^{2}(a,b)$, the jump $u_{0}(x)=-2\pi i$ solves
the integral equation (1.1) if we choose the right hand side $f_{0}(x)=-2\pi
i\lambda-2\log\frac{x-a}{b-x}$.

Another example is $w_{1}(x)=\sqrt{(z-a)(z-b)}-z+\tfrac{a+b}{2}$ that has the
right branch cut and behavior at infinity. Traces on $(a,b)$ are

$w_{1,+}(x)=i\sqrt{(x-a)(b-x)}-x+\tfrac{a+b}{2}\,,\quad
w_{1,-}(x)=-i\sqrt{(x-a)(b-x)}-x+\tfrac{a+b}{2}$

The jump $u_{1}(x)=2i\sqrt{(x-a)(b-x)}$ therefore solves the integral equation
(1.1) if we choose the right hand side
$f_{1}(x)=2i\lambda\sqrt{(x-a)(b-x)}+2x-a-b$.

Other examples can be obtained by applying an entire analytic function $F$ to
the preceding example functions $w$, provided $F(0)=0$. For a third example,
we take the function $F(w)=e^{\alpha w}-1$ and apply it to the first example
above, that is, we choose an exponent $\alpha\in{\mathbb{C}}$ and define
$w_{2}(z)=e^{\alpha\log\frac{z-a}{z-b}}-1=\big{(}\frac{z-a}{z-b}\big{)}^{\alpha}-1$.
The traces on the branch cut $w_{2,\pm}(x)=(x-a)^{\alpha}(b-x)^{-\alpha}e^{\mp
i\pi\alpha}-1$ belong to $L^{2}(a,b)$ if
$\alpha\in(-\frac{1}{2},\frac{1}{2})$. The corresponding exact solution
$u_{2}$ and right hand side $f_{2}$ are given by

$u_{2}(x)=-2i\sin\pi\alpha\Big{(}\frac{x-a}{b-x}\Big{)}^{\alpha}\quad\mbox{
with
}A_{\Omega}u_{2}(x)=2\cos\pi\alpha\Big{(}\frac{x-a}{b-x}\Big{)}^{\alpha}-2\;\mbox{
and }f_{2}=\lambda u_{2}-A_{\Omega}u_{2}\,.$

Note that if $\lambda=i\cot\pi\alpha$, then $f_{2}$ is a constant function.

##  3\. Error estimates

###  3.1. Approximate solutions

For sake of simplicity we assume from now on that the length of $\Omega$ is a
multiple of $h$ and that the origin $a^{N}$ is chosen such that the boundary
points $a$ and $b$ are midpoints of the mesh.

(3.1) $b-a=Mh\,,\quad
M\in{\mathbb{N}}\,,\quad\omega^{N}=\\{1,\dots,M\\}\,,\quad
x_{1}^{N}=a+\tfrac{h}{2}\,,\quad x_{M}^{N}=b-\tfrac{h}{2}\,.$

When no confusion is possible, we omit superscripts $N$ and write the discrete
system as

(3.2) $\lambda u_{m}-\frac{1}{i\pi N}\sum_{\begin{subarray}{l}n=1\\\ n\neq
m\end{subarray}}^{M}\frac{u_{n}}{x_{n}-x_{m}}=f_{m}\,,\quad m\in\omega^{N}\,.$

In the spirit of Lifanov’s method of discrete vortices [1], the delta-delta
scheme can be considered as a Nyström-type quadrature method, where the
integral in (1.1) is replaced by its approximation by the midpoint rule. The
approximate solution $u^{(N)}$ satisfies the modified equation

(3.3) $\lambda
u^{(N)}(x)-\frac{1}{i\pi}\sum_{n\in\omega^{N}}\frac{1}{N}\frac{u^{(N)}(x_{n})}{x_{n}-x}=f(x)\,,$

and its evaluation in the gridpoints $x=x_{m}$ gives the system (3.2) for the
nodal values $u_{m}=u^{(N)}(x_{m})$.

Alternatively, one can also consider the delta-delta scheme as a one-point
quadrature approximation of a collocation method with piecewise constant trial
functions. This is the popular point of view for DDA, see for example the
error analysis in [12].

Let $\chi_{j}$ be the characteristic function of the interval
$I_{j}=[x_{j}-\frac{h}{2},x_{j}+\frac{h}{2}]$. We consider the space of
piecewise constant functions $S^{N,0}(\omega^{N})=\mathrm{span}\\{\chi_{j}\mid
j\in\omega^{N}\\}$.

Then the collocation scheme is: Find $u^{N}\in S^{N,0}(\omega^{N})$ such that

(3.4) $\lambda
u^{N}(x_{m})-\frac{1}{i\pi}\int_{\Omega}\frac{u^{N}(y)}{y-x_{m}}dy=f(x_{m})\,,\quad
m\in\omega^{N}\,.$

From this we get our system (3.2) by taking the $u_{n}$ as the coefficients of
$u^{N}$ in the basis of the $\chi_{n}$,

$u^{N}=\sum_{n\in\omega^{N}}u_{n}\chi_{n},$

and for the off-diagonal matrix elements $A_{\Omega}\chi_{n}(x_{m})$
approximating the integrals by a one-point quadrature rule:

$\int\frac{\chi_{k}(y)}{y-x_{m}}dy\;\sim\;\begin{cases}0&\text{ if $m=n$}\\\
\dfrac{1}{x_{n}-x_{m}}\,\dfrac{1}{N}&\text{ if $m\not=n$}\end{cases}$

After this replacement of the integrals, the system (3.4) becomes (3.2). For
the diagonal elements, note that the Cauchy principal value integral
$\int\frac{\chi_{n}(y)}{y-x_{n}}dy$ vanishes for symmetry reasons.

Notice that even if the coefficients $(u_{n})$ in (3.3) and (3.4) are the
same, the approximate solutions $u^{N}$ and $u^{(N)}$ are not the same:
$u^{N}$ in (3.4) is a piecewise constant function, whereas $u^{(N)}$, which
for $\lambda\neq 0$ is defined by (3.3) once the $u_{k}$ are known, is a sum
of the possibly smooth function $f$ and a rational function with simple poles
in the grid points.

###  3.2. Consistency error

We define the _consistency error_ as the vector $c^{N}[u]$ with components

(3.5)
$c_{m}^{N}[u]=\frac{1}{i\pi}\Big{(}\int_{a}^{b}\frac{u(y)}{y-x_{m}}dy-h\sum_{n\in\omega^{N}\setminus\\{m\\}}\frac{u(x_{n})}{x_{n}-x_{m}}\Big{)},\quad
m\in\omega^{N}\,.$

It can be written as $c^{N}[u]=(R_{N}A_{\Omega}-T^{N}R_{N})u$, where $R_{N}$
is the restriction operator that maps a function $u$ to its nodal values
$U=(u(x_{m}))_{m\in\omega^{N}}$.

Thus it is the quadrature error for the midpoint rectangle rule applied to the
Cauchy singular integral. This quadrature error has been estimated under
various assumptions on the function $u$ in [1, Section 1.3], from which we
borrow some ideas. We study here only the case where $u$ is Hölder continuous
up to the boundary, $u\in C^{\alpha}([a,b])$ for some $\alpha\in(0,1]$.

In the applications of the singular integral equation (1.1) studied in [1],
mainly from fluid dynamics, more general functions $u$ with singularities at
the boundary points $a$ and $b$ need to be considered. But since we want to
emphasize the analogy with the volume integral equation of the DDA method, and
solutions of the latter tend to be smooth up to (smooth points of) the
boundary, the simple situation of $u\in C^{\alpha}([a,b])$ will be sufficient
for our purpose.

An advantage of this simplification is that we can get convergence results in
an easy way from combining the consistency estimates of this section with the
stability estimates of the preceding section. Convergence results for the case
of solutions with singularities seem to require a much more complicated
analysis, see [1, Chapter 7].

We consider first the case where $u$ is constant.

######  Lemma 3.1.

Let

$s_{j}=\int_{a}^{b}\frac{dy}{y-x_{j}}-\sum_{k\in\omega^{N}\setminus\\{j\\}}\frac{h}{x_{k}-x_{j}}\,.$

Then for all $j\in\omega^{N}$

(3.6)
$|s_{j}|\leq\frac{h^{2}}{8}\big{|}\frac{1}{(x_{j}-a)^{2}}-\frac{1}{(b-x_{j})^{2}}\big{|}=|\tfrac{a+b}{2}-x_{j}|(b-a)\big{(}\frac{h}{2(x_{j}-a)(b-x_{j})}\big{)}^{2}\,.$

###### Proof.

We split

$s_{j}=\sum_{k\in\omega^{N}\setminus\\{j\\}}s_{jk}\quad\mbox{ with
}\;s_{jk}=\int_{I_{k}}\frac{dy}{y-x_{j}}-\frac{h}{x_{k}-x_{j}}\,.$

First we observe that there are cancellations: $s_{jk}+s_{jk^{\prime}}=0$ for
$k^{\prime}+k=2j$, because of the antisymmetry of the kernel. Therefore

$s_{j}=\sum_{k\in A_{j}}s_{jk}\quad\mbox{ with }\;A_{j}=\begin{cases}\\{k\mid
2x_{j}-a+\frac{h}{2}<x_{k}<b\\}&\mbox{ if
}x_{j}\leq\tfrac{a+b}{2}\,,\\\\[2.15277pt] \\{k\mid
a<x_{k}<2x_{j}-b-\frac{h}{2}\\}&\mbox{ if
}x_{j}\geq\tfrac{a+b}{2}\,.\end{cases}$

Then we apply the classical error representation for the midpoint rule

$\int_{-\frac{h}{2}}^{\frac{h}{2}}f(t)dt-
hf(0)=\int_{-\frac{h}{2}}^{\frac{h}{2}}\tfrac{1}{2}(\tfrac{h}{2}-|t|)^{2}f^{\prime\prime}(t)dt\,,$

valid for any $C^{2}$ function, to the function
$f(t)=\frac{1}{t+x_{k}-x_{j}}$. Consider the case $x_{j}\leq\tfrac{a+b}{2}$.
Then for the relevant indices $k$, we have $x_{k}>x_{j}$ and therefore
$f^{\prime\prime}(t)=2(t+x_{k}-x_{j})^{-3}>0$. Hence

$0<s_{jk}<\frac{h^{2}}{8}\int_{I_{k}}\frac{2dy}{(y-x_{j})^{3}}\quad\Longrightarrow\;0<s_{j}<\frac{h^{2}}{8}\int_{2x_{j}-a}^{b}\frac{2dy}{(y-x_{j})^{3}}=\frac{h^{2}}{8}\big{(}-\frac{1}{(b-x_{j})^{2}}+\frac{1}{(x_{j}-a)^{2}}\big{)}\,.$

Likewise, for $x_{j}\geq\tfrac{a+b}{2}$, the second derivative is always
negative, and we obtain

$0>s_{j}>\frac{h^{2}}{8}\int_{a}^{2x_{j}-b}\frac{2dy}{(y-x_{j})^{3}}=\frac{h^{2}}{8}\big{(}-\frac{1}{(b-x_{j})^{2}}+\frac{1}{(x_{j}-a)^{2}}\big{)}\,.$

∎

The lemma means that for $x_{j}$ in any compact subinterval of $(a,b)$, the
$s_{j}$ converge to zero as $O(h^{2})$. But when $x_{j}$ tends to $a$ or $b$
as $h\to 0$, then $s_{j}$ will have non-zero limits. For example for
$x_{1}=a+\frac{h}{2}$, one finds $s_{1}\to\log 2-\gamma$, where
$\gamma=0.5772..$ is Euler’s constant. For this case, the estimate (3.6) gives
$s_{j}\leq\frac{1}{2}$ in the limit.

Another way to express this behavior is that for a given error threshold
$\epsilon$, the points $x_{j}$ where $|s_{j}|$ exceeds $\epsilon$ are confined
to a boundary layer of thickness $h/\sqrt{2\epsilon}$.

######  Proposition 3.2.

Let $0<\alpha\leq 1$. For $u\in C^{\alpha}([a,b])$ let $c_{j}^{N}[u]$ be
defined by (3.5). Then

(3.7) $|c_{j}^{N}[u]|\leq C\,\Big{(}h^{\alpha}|\log h|\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}+|u(x_{j})|\frac{h^{2}}{(x_{j}-a)^{2}(b-x_{j})^{2}}\Big{)}\,.$

Here the constant $C$ depends on $a,b$ but not on $h=\frac{1}{N}$ nor on $u$.
If, in addition, $u(a)=u(b)=0$ holds, then the estimate simplifies to

(3.8) $|c_{j}^{N}[u]|\leq C\,h^{\alpha}|\log h|\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\,.$

###### Proof.

Split

$\begin{split}i\pi c_{j}^{N}[u]&=c_{jj}+\sum_{k\neq
j}c_{jk}+u(x_{j})s_{j}\quad\mbox{ with }\\\
c_{jj}&=\int_{I_{j}}\frac{u(y)}{y-x_{j}}dy,\;c_{jk}=\int_{I_{k}}\big{(}\frac{u(y)-u(x_{j})}{y-x_{j}}-\frac{u(x_{k})-u(x_{j})}{x_{k}-x_{j}}\big{)}dy\end{split}$

and $s_{j}$ as defined and estimated in Lemma 3.1.
For $c_{jj}$ we find

$|c_{jj}|=\big{|}\int_{I_{j}}\frac{u(y)-u(x_{j})}{y-x_{j}}dy\big{|}\leq\int_{I_{j}}|y-x_{j}|^{\alpha-1}dy\,\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\leq
h^{\alpha}\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\,.$

For the term $u(x_{j})s_{j}$ we use the estimate from Lemma 3.1. It remains to
estimate $\sum_{k\neq j}c_{jk}$. Write

$c_{jk}=c_{jk}^{0}+c_{jk}^{1}\quad\mbox{ with
}c_{jk}^{0}=\int_{I_{k}}\frac{u(y)-u(x_{k})}{y-x_{j}}dy\,,\;c_{jk}^{1}=\int_{I_{k}}\big{(}u(x_{k})-u(x_{j})\big{)}\big{(}\tfrac{1}{y-x_{j}}-\tfrac{1}{x_{k}-x_{j}}\big{)}dy.$

Then

$\Big{|}\sum_{k\neq
j}c_{jk}^{0}\Big{|}\leq\big{(}\frac{h}{2}\big{)}^{\alpha}\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\Big{(}\int_{a}^{x_{j}-\frac{h}{2}}\frac{dy}{x_{j}-y}+\int_{x_{j}+\frac{h}{2}}^{b}\frac{dy}{y-x_{j}}\Big{)}\leq
C\,h^{\alpha}|\log h|\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\,.$

Finally,
$c_{jk}^{1}=\int_{I_{k}}\big{(}u(x_{k})-u(x_{j})\big{)}\tfrac{x_{k}-y}{(y-x_{j})(x_{k}-x_{j})}dy$
can be estimated as

$|c_{jk}^{1}|\leq\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}\int_{I_{k}}\frac{|x_{k}-y|}{|y-x_{j}|x_{k}-x_{j}|^{1-\alpha}}dy\leq
C\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}\,h^{2}\,|x_{k}-x_{j}|^{\alpha-2}=C\,\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}\,h^{\alpha}\,|k-j|^{\alpha-2}.$

For $\alpha<1$, the infinite series
$\sum_{k\in{\mathbb{Z}}\setminus\\{j\\}}|k-j|^{\alpha-2}$ converges, so that

$\Big{|}\sum_{k\neq j}c_{jk}^{1}\Big{|}\leq
C\,h^{\alpha}\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}\,.$

For $\alpha=1$, we would pick up an extra factor of $\log N=|\log h|$.
The estimate (3.7) is proved.
Finally, if $u(a)=u(b)=0$, then $|u(x_{j)}|\leq C\,\mathchoice{\|\hskip
0.6458ptu\hskip 0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}(x_{j}-a)^{\alpha}(b-x_{j})^{\alpha}$,
hence

$|u(x_{j})|\frac{h^{2}}{(x_{j}-a)^{2}(b-x_{j})^{2}}\leq
C\,h^{\alpha}\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}{\|u\|_{C^{\alpha}}}\big{(}\frac{h}{(x_{j}-a)(b-x_{j})}\big{)}^{2-\alpha}\,.$

The last factor is uniformly bounded, and this completes the proof of (3.8) ∎

###  3.3. Discrete error

If $u$ is the solution of the integral equation (1.1) with right hand side $f$
and $U^{N}=(u_{n})_{n\in\omega^{N}}$ is solution of the discrete system (1.2)
with $f_{m}=f(x_{m})$, then we define the _discrete error_ $E^{N}$ as vector
with the components $E_{m}=u(x_{m})-u_{m}$, $m\in\omega^{N}$. It is easy to
see that $E^{N}$ satisfies the discrete system with the consistency error as
right hand side,

(3.9) $(\lambda\mathbb{I}-T^{N})E^{N}=c^{N}[u]\,.$

By a direct application of our stability and consistency estimates we obtain
an estimate for the discrete error. We choose the situation where the
simplified consistency estimate (3.8) holds.

######  Theorem 3.3.

Assume $\lambda\in{\mathbb{C}}\setminus[-1,1]$. Let $u$ be solution of the
integral equation (1.1) with right hand side $f$ and
$(u_{n})_{n\in\omega^{N}}$ be solution of the discrete system (1.2) with
$f_{m}=f(x_{m})$. If we assume that $u\in C^{\alpha}([a,b])$ with
$\frac{1}{2}<\alpha\leq 1$ and $u(a)=u(b)=0$, then there is a constant $C$
independent of $N$ such that the discrete error satisfies

(3.10) $\mathchoice{\|\hskip 0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}\leq
C\,N^{\frac{1}{2}-\alpha}\log N\,.$

###### Proof.

From the error equation (3.9) and our stability estimate (2.1) we obtain

$\mathchoice{\|\hskip 0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}\leq
C\,\mathchoice{\|\hskip 0.6458ptc^{N}[u]\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}}{\|c^{N}[u]\|_{\ell^{2}(\omega^{N})}}{\|c^{N}[u]\|_{\ell^{2}(\omega^{N})}}{\|c^{N}[u]\|_{\ell^{2}(\omega^{N})}}\,.$

Now our consistency estimate (3.8) is an $\ell^{\infty}(\omega^{N})$ estimate,
so we lose a factor of $\sqrt{N}$:

$\mathchoice{\|\hskip 0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}}\leq|\omega^{N}|^{\frac{1}{2}}\mathchoice{\|\hskip
0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{\infty}(\omega^{N})$}}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}\leq
C\,N^{\frac{1}{2}}\mathchoice{\|\hskip 0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{\infty}(\omega^{N})$}}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}{\|c^{N}\|_{\ell^{\infty}(\omega^{N})}}\leq
C\,N^{\frac{1}{2}-\alpha}\log N\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\,.$

∎

###  3.4. Convergence in $L^{2}$

When the solution $u$ does not vanish on the boundary, we only have the
consistency estimate (3.7), and the consistency error $c^{N}[u]$ will not
converge to zero in $\ell^{\infty}$ and even less in $\ell^{2}$, hence the
discrete error will not converge to zero in $\ell^{2}$, either. Instead, we
can study the error of the piecewise constant approximation

$e^{N}=u-u^{N}\quad\mbox{ with }\quad
u^{N}=\sum_{k\in\omega^{N}}u_{k}\chi_{k}\,.$

Its $L^{2}(\Omega)$ norm can be bounded by

$\displaystyle\mathchoice{\|\hskip 0.6458pte^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
L^{2}(a,b)$}}^{2}}{\|e^{N}\|_{L^{2}(a,b)}^{2}}{\|e^{N}\|_{L^{2}(a,b)}^{2}}{\|e^{N}\|_{L^{2}(a,b)}^{2}}$
$\displaystyle=\sum_{k\in\omega^{N}}\int_{I_{k}}|u(y)-u_{k}|^{2}dy$
$\displaystyle\leq
2\sum_{k\in\omega^{N}}\Big{(}\int_{I_{k}}|u(y)-u(x_{k})|^{2}dy+\int_{I_{k}}|u(x_{k})-u_{k}|^{2}dy\Big{)}$
$\displaystyle\leq
2\sum_{k\in\omega^{N}}\Big{(}h^{2\alpha+1}\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}$}}^{2}}{\|u\|_{C^{\alpha}}^{2}}{\|u\|_{C^{\alpha}}^{2}}{\|u\|_{C^{\alpha}}^{2}}+h\,|E_{k}|^{2}\Big{)}$
$\displaystyle\leq C\,\big{(}h^{2\alpha}\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}+h\,\mathchoice{\|\hskip
0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{n})$}}^{2}}{\|E^{N}\|_{\ell^{2}(\omega^{n})}^{2}}{\|E^{N}\|_{\ell^{2}(\omega^{n})}^{2}}{\|E^{N}\|_{\ell^{2}(\omega^{n})}^{2}}\big{)}\,.$

Thus we get convergence as soon as we can show that
$N^{-\frac{1}{2}}\mathchoice{\|\hskip 0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}$}}}{\|E^{N}\|_{\ell^{2}}}{\|E^{N}\|_{\ell^{2}}}{\|E^{N}\|_{\ell^{2}}}$
tends to zero. By our $\ell^{2}$ stability estimates, this is equivalent to
the fact that $N^{-\frac{1}{2}}\mathchoice{\|\hskip 0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}$}}}{\|c^{N}\|_{\ell^{2}}}{\|c^{N}\|_{\ell^{2}}}{\|c^{N}\|_{\ell^{2}}}$
tends to zero. For this, as we have seen before, it would be _sufficient_ that
the consistency error $\mathchoice{\|\hskip 0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{\infty}$}}}{\|c^{N}\|_{\ell^{\infty}}}{\|c^{N}\|_{\ell^{\infty}}}{\|c^{N}\|_{\ell^{\infty}}}$
tends to zero.

But this is not _necessary_ : In fact, assume that the consistency estimate
(3.7) is satisfied. It implies

$|c_{j}^{N}|\leq C\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}{\|u\|_{C^{\alpha}([a,b])}}\Big{(}h^{\alpha}|\log
h|+\max\big{\\{}\frac{h^{2}}{(x_{j}-a)^{2}},\,\frac{h^{2}}{(b-x_{j})^{2}}\big{\\}}\Big{)}$

Now $x_{j}-a$ takes the values $\frac{h}{2}$, $\frac{3h}{2}$,
$\frac{5h}{2}$,…, and therefore the sum

$\sum_{j\in\omega^{N}}\big{(}\frac{h^{2}}{(x_{j}-a)^{2}}\big{)}^{2}$

is bounded independently of $N$ by a constant

$C^{\prime}=16\sum_{j=0}^{\infty}(2j+1)^{-4}<\infty\,.$

This implies finally

$h\mathchoice{\|\hskip 0.6458ptc^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}^{2}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}^{2}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}^{2}}{\|c^{N}\|_{\ell^{2}(\omega^{N})}^{2}}\leq
C\,\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}\Big{(}h^{2\alpha}|\log
h|^{2}+h\,C^{\prime}\Big{)}\,.$

We have proved the following error estimate.

######  Theorem 3.4.

Assume $\lambda\in{\mathbb{C}}\setminus[-1,1]$. Let $u$ be solution of the
integral equation (1.1) with right hand side $f$, let
$U^{N}=(u_{k})_{k\in\omega^{N}}$ be solution of the discrete system (3.2) with
$f_{j}=f(x_{j})$, and let $u^{N}=\sum_{k\in\omega^{N}}u_{k}\chi_{k}$ be the
corresponding piecewise constant function. If we assume that $u\in
C^{\alpha}([a,b])$, then there is a constant $C$ independent of $N$ such that
the error $e^{N}=u-u^{N}$ satisfies

$\mathchoice{\|\hskip 0.6458pte^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
L^{2}(a,b)$}}}{\|e^{N}\|_{L^{2}(a,b)}}{\|e^{N}\|_{L^{2}(a,b)}}{\|e^{N}\|_{L^{2}(a,b)}}\leq
C\,\big{(}N^{-\alpha}\log N+N^{-\frac{1}{2}}\big{)}\,.$

The last term $O(\sqrt{h})$ in this error estimate that prevents the estimate
to improve with regularity above $C^{\frac{1}{2}}$ is due to the boundary
layer and the choice of the $L^{2}$ norm for measuring the error. One could
choose $L^{p}$ for $1\leq p\leq\infty$ instead. In the same way as above, one
obtains

$\mathchoice{\|\hskip 0.6458pte^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
L^{p}(a,b)$}}}{\|e^{N}\|_{L^{p}(a,b)}}{\|e^{N}\|_{L^{p}(a,b)}}{\|e^{N}\|_{L^{p}(a,b)}}\leq
C\big{(}h^{\alpha}\mathchoice{\|\hskip 0.6458ptu\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle
C^{\alpha}([a,b])$}}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}{\|u\|_{C^{\alpha}([a,b])}^{2}}+h^{\frac{1}{p}}\,\mathchoice{\|\hskip
0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{p}(\omega^{n})$}}}{\|E^{N}\|_{\ell^{p}(\omega^{n})}}{\|E^{N}\|_{\ell^{p}(\omega^{n})}}{\|E^{N}\|_{\ell^{p}(\omega^{n})}}\big{)}\,.$

This would be best for $p=1$, but we are tied to $p=2$ because we depend on
our stability estimate.

##  4\. Numerical experiments

In this section, we fix arbitrarily the interval $[a,b]=[-0.15,1.35]$ and
choose $N$ as an odd multiple of $10$, so that $[a,b]$ is subdivided in $3N/2$
subintervals of length $h=\frac{1}{N}$. We use the $3$ examples discussed in
section 2.3, where we divide the first example by $-2\pi i$, so that
$u_{0}(x)=1$. In the second example, we normalize by dividing by $2i$, so that
$u_{1}(x)=\sqrt{(x-a)(b-x)}$. For the third example, we choose $\alpha=0.25$,
so that $u_{2}(x)=\Big{(}\frac{x-a}{b-x}\Big{)}^{\frac{1}{4}}/\sqrt{2}$. In
Figure 1, we saw the $3$ exact solutions with their computed piecewise
constant approximation, computed for $N=10$ with $\lambda=2$.

###  4.1. Behavior of consistency error and discrete error

Knowing the exact solution, we can define the consistency error $c^{N}$ as in
(3.5) and the discrete error $E^{N}$ as in Section 3.3. In Figure 2, we plot
for our 3 examples the absolute values of $c^{N}$ and $E^{N}$ as functions of
$x\in[a,b]$ for $N=10,50,250,1250$. In the logarithmic scale, one can see the
convergence in the interior of the interval with a power of $N$, whereas near
the boundary points there is either much slower convergence for example 2,
where $u_{1}(a)=u_{1}(b)=0$, or no convergence at all for example 1 where the
solution $u_{0}$ does not vanish at the boundary points, thus illustrating the
analysis shown in Section 3. Note that we have not presented an analysis for
the case of a singular solution such as the one in the third example, but the
numerical results show a similar behavior as for the other examples, although
the consistency and discrete errors, like the solution itself, tend to
infinity near $b$.

Figure 2. Exact solution u, rhs f, consistency error c and discrete error E

###  4.2. Convergence rates

In Figure 3 we plot several error norms as functions of $N=10\times 3^{j}$,
$j=0,...,6$ in a loglog scale: The $\ell^{2}(\omega^{N})$ norms of the
consistency error $c^{N}$ and the discrete error $E^{N}$ as well as the
$\ell^{2}$ norm of the restriction of $E^{N}$ to the interior subinterval
$[0,1.2]$ of $(a,b)$. Finally the $L^{2}(a,b)$ norm of the error $e^{N}$ of
the piecewise constant approximation, which we compare with the normalized
$\ell^{2}$ norm of $E^{N}$, $N^{-\frac{1}{2}}\mathchoice{\|\hskip
0.6458ptE^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}(\omega^{N})$}}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}{\|E^{N}\|_{\ell^{2}(\omega^{N})}}$.
What we observe for the three examples can be compared with the error
estimates in Proposition 3.2 and Theorems 3.3 and 3.4, taking into account
that we only have _global_ error estimates. Whereas we have seen in Section
3.2 that the consistency error $c^{N}$ behaves differently in the interior of
the domain and near the boundary, our technique of proof does not allow to
infer the same behavior for the discrete error $E^{N}$. Yet we see for both
examples 1 and 2 a $O(N^{-\frac{1}{2}})$ behavior for $\mathchoice{\|\hskip
0.6458ptE_{\rm int}^{N}\hskip
0.6458pt\|_{\raise-0.90417pt\hbox{$\scriptstyle\ell^{2}$}}}{\|E_{\rm
int}^{N}\|_{\ell^{2}}}{\|E_{\rm int}^{N}\|_{\ell^{2}}}{\|E_{\rm
int}^{N}\|_{\ell^{2}}}$, which would correspond to an order of $O(N^{-1})$ for
the $\ell^{\infty}$ norm and also for the $L^{2}$ error of the corresponding
piecewise constant approximation.

For the global $\ell^{2}$ norms of the errors $c^{N}$ and $E^{N}$ we see that
in example 1 they do not tend to zero, in accordance with the behavior near
the boundary of estimates (3.6) and (3.7). In example 2, they are of order
$O(N^{-\frac{1}{2}})$, which reflects again the boundary layer behavior for
the approximation of the function $u_{1}$, which is Hölder continuous of class
$C^{\frac{1}{2}}$ there.

The global $L^{2}$ norm of the error $e^{N}$ is seen to behave as
$O(N^{-\frac{1}{2}})$ in example 1, which corresponds to Theorem 3.4. In
example 2, the behavior is $O(N^{-1})$, which better than what Theorem 3.4
predicts for a $C^{\frac{1}{2}}$ solution. The explanation is that whereas the
Theorems 3.3 and 3.4 consider worst-case scenarios, the solution $u_{1}$ is in
fact $C^{\infty}$ except at the two boundary points, so that estimate (3.10)
may be seen to hold with $\alpha=1$.

Figure 3. Error norms, compared to $N^{-1/2}$ (Ex. 1), $N^{-1}$ (Ex. 2),
$N^{-1/4}$ (Ex. 3),

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# Benchmark Dataset Dynamics, Bias and Privacy Challenges in Voice Biometrics
Research

Casandra Rusti
University of Southern California
United States
<EMAIL_ADDRESS>This study was done while the authors collaborated on the Fair
EVA project (https://faireva.org) Anna Leschanowsky11footnotemark: 1
Fraunhofer IIS
Germany
<EMAIL_ADDRESS>Carolyn Quinlan11footnotemark: 1
University of Toronto
Canada
<EMAIL_ADDRESS>Michaela Pnacek(ova)11footnotemark: 1
York University
Canada
<EMAIL_ADDRESS>Lauriane Gorce11footnotemark: 1
Open North
Canada
<EMAIL_ADDRESS>Wiebke (Toussaint) Hutiri11footnotemark: 1
Delft University of Technology
The Netherlands
<EMAIL_ADDRESS>
###### Abstract

Speaker recognition is a widely used voice-based biometric technology with
applications in various industries, including banking, education, recruitment,
immigration, law enforcement, healthcare, and well-being. However, while
dataset evaluations and audits have improved data practices in face
recognition and other computer vision tasks, the data practices in speaker
recognition have gone largely unquestioned. Our research aims to address this
gap by exploring how dataset usage has evolved over time and what implications
this has on bias, fairness and privacy in speaker recognition systems.
Previous studies have demonstrated the presence of historical, representation,
and measurement biases in popular speaker recognition benchmarks. In this
paper, we present a longitudinal study of speaker recognition datasets used
for training and evaluation from 2012 to 2021. We survey close to 700 papers
to investigate community adoption of datasets and changes in usage over a
crucial time period where speaker recognition approaches transitioned to the
widespread adoption of deep neural networks. Our study identifies the most
commonly used datasets in the field, examines their usage patterns, and
assesses their attributes that affect bias, fairness, and other ethical
concerns. Our findings suggest areas for further research on the ethics and
fairness of speaker recognition technology.

##  1 Introduction

Speaker recognition is widely used in voice biometrics in the private and
public sectors, e.g, to verify the identity of banking clients [42, 3] or
employees [43], and to secure an expanding network of voice assistants and
voice-based Internet of Things devices through which people interact with
digital services [37]. The large scale deployment of speaker recognition
systems has been facilitated by the adoption of deep learning approaches that
have greatly improved technology performance [2]. However, using data-
intensive, deep learning systems in the development of biometric systems can
lead to bias and discrimination, characterized by disparate error rates across
demographic groups [8, 21, 13, 19]. Nonetheless, despite rapid progress and
widespread adoption of speaker recognition technology, bias, discrimination
and fairness remain largely unexplored in voice biometrics.

The biometrics community has noted the importance of evaluating bias. For
example, the NIST Face Recognition Vendor Test now includes an evaluation of
bias across demographic groups [28], and several studies have evaluated error
rate disparities across groups in face recognition models [8, 13, 20]. Model
bias however presents only one of several sources of bias in the development
of deep learning systems [41] and training and evaluation datasets should be
additionally interrogated [31]. Bias in training datasets reflects downstream
in the learned models, while bias in evaluation datasets skews evaluation
outcomes, channels future development efforts and makes it impossible to
assess if models are biased [41]. Motivated by prior research on dataset
evaluations [32, 17, 29] and development [36, 23], this paper presents the
first study on the impact of training and evaluation datasets on bias and
other ethical concerns in the voice biometrics domain.

This study aims to explore dataset usage dynamics in order to gain insights
into community adoption of datasets and potential cultural shifts in data
practices that are likely to impact the development of speaker recognition
technology. We address the following questions to examine how the adoption of
deep learning has shifted data practices and to extrapolate how data and
evaluation practices may impact fairness and privacy concerns in voice
biometrics:

  1. 1.

Which datasets are used for training and evaluation in speaker recognition
research?

  2. 2.

How has dataset usage changed over the period from 2012 to 2021?

  3. 3.

What are the attributes of the most used datasets?

  4. 4.

What are the implications of the above questions for bias, fairness, privacy
and other ethical challenges in speaker recognition?

While prior dataset studies have been published in speaker recognition [6,
39], they focused primarily on describing available corpora and did not
interrogate how data practices impact ethical and societal outcomes. This
paper presents insights on development practices in the speaker recognition
domain, and contributes to the emerging body of work on bias, fairness and
privacy challenges in biometrics. We start by reviewing related literature in
Section 2 before describing our research approach in Section 3, and presenting
results in Section 4. We consolidate our findings and reflect on the study in
Section 5.

##  2 Related Work

The section covers prior research on bias in face recognition systems,
measures for evaluating demographic bias in biometric verification systems,
and the impact of bias in data and datasets on machine learning models.

###  2.1 Bias in Biometrics

Existing bias literature on biometrics mainly focuses on measuring disparate
error rates across demographic groups in face recognition systems. Studies
have proposed various measures, such as statistical methods and Fairness
Discrepancy Rate, for assessing the demographic bias of biometric verification
systems [20, 13]. Meanwhile, others have provided checklists for measuring
race bias in face recognition emphasizing the need to consider data-driven
factors and scenario modelling including accounting for sub-population
distributions, algorithm quality, the representation of and conditions
captured by images, threshold selection and appropriate considerations around
demographic pairing [8].

In the voice biometrics domain, Hutiri and Ding present an empirical and
analytical examination of bias in the machine learning development workflow of
speaker verification benchmarks [19]. They identify historical, representation
and measurement bias during the data gathering stage, and learning,
evaluation, aggregation and deployment bias during model building. In
particular, pairing of trials in speaker recognition benchmarks can result in
evaluation datasets of variable difficulty across demographic groups [18]
showing similarities to bias evaluation for face recognition [8].

###  2.2 Bias in Data and Datasets

Machine learning models are impacted by bias in data and datasets, including
historical, representation, measurement, and evaluation bias [36, 41]. As
datasets form the basis for training, evaluating, and benchmarking models,
dataset evaluations are important to interrogate bias in machine learning
systems [31]. Prior research has found that the dominant developer culture,
which emphasizes rapid progress and ever-larger models, can lead to
representation bias in datasets and inadequate dataset documentation [31].
Similarly, evaluation failures can result from implementation variations,
errors in test set construction, overfitting, and inadequate baselines [23].

This study takes inspiration from recent dataset evaluations in the visual
domain [32, 29, 17]. We particularly draw on the work of Raji and Fried, who
studied the detailed make-up of evaluation datasets and benchmarks used for
face recognition [32]. They surveyed over 100 face datasets to consider the
selection of evaluation tasks, benchmark data, evaluation criteria and
metrics. Their approach revealed insights into the historical evolution of
face recognition, driven by evaluation challenges of the NIST, with funding
from intelligence agencies. The advancement of speaker recognition research
shares many common attributes with that of face recognition. In contrast to
Raji and Fried’s study, we however examine dataset usage in the speaker
recognition research community over a period of time.

##  3 Research Approach

Next, we describe our approach to studying the dynamics of speaker recognition
dataset adoption and use in the research literature over the past decade. The
study focuses on peer-reviewed research published over a ten year period from
2012 - 2021 at the Interspeech conference, one of two main international
conference venues for academic and industrial speech research. We included all
papers from the International Speech Communication Association (ISCA)
archive111https://www.isca-speech.org/archive/ that contained the search terms
speaker recognition, speaker verification, speaker identification or speaker
authentication in their title or abstract. This query resulted in 696 papers,
which we analyzed further. We excluded 25 papers that were overview papers,
studied speaker recognition by humans or applications rather than model
development, or did not explicitly mention which datasets were used to train
and evaluate their models. Our final analysis thus includes 671 papers. Over
the decade that we analysed, the number of papers on speaker recognition
published at Interspeech has doubled, as can be seen in Figure 1.

Figure 1: Histogram of papers published at Interspeech over the past decade
and analyzed in this study.

All papers were tagged with the training and evaluation datasets that they
used. As datasets were not always named consistently, some assumptions were
made. For example, studies that used datasets from the NIST Speaker
Recognition Evaluations (SREs) - regularly and typically annually released for
benchmarking purposes - often rely on several of these datasets for training
and refer to them as a range (e.g. NIST 2004 - 2008). In these cases, we
assumed that every dataset in the range was used for training or evaluation,
as indicated by the authors. Overall, we encountered many naming
inconsistencies. Especially evaluation datasets were sometimes only referred
to on the highest level (e.g. VoxCeleb), without specifying which dataset
version, subset or evaluation protocol was used. Whenever possible, we
standardized dataset and subset names and otherwise tagged papers by their
training and evaluation dataset family. The dataset family name was created by
manually cleaning the dataset names, and then extracting the first word in the
name as the family name. It is common in the speech domain to refer to
training datasets as development data or corpora, and to evaluation datasets
as test data. In this paper, we use the terms training and evaluation
datasets, unless we refer to the names of specific datasets.

##  4 Speaker Recognition Dataset Dynamics over a Decade of Use

This section examines which datasets the speaker recognition research
community has adopted over the past decade, how the use of datasets for
training and evaluation purposes has changed over time, and which attributes
characterize the most frequently used datasets. This provides insight into
data practices and usage dynamics and their implications on bias, fairness and
ethics in the speaker recognition domain.

###  4.1 Community Adoption of Speaker Recognition Datasets

Figure 2: Histogram showing the count of most used training dataset families
between 2012 and 2021.

Over the past decade, a wide range of datasets has been used for training and
evaluating speaker recognition systems. In total, we counted 185 unique
training and 164 unique evaluation dataset families. Despite this variety, a
small number of dataset families has dominated speaker recognition research.
Figure 2 shows the frequency counts of the top 30 dataset families used for
training and evaluation respectively in the analyzed papers. As papers could
use more than one dataset to train and evaluate systems, the total count of
dataset families exceeds the number of analyzed papers.

The NIST Speaker Recognition Evaluation (SRE) corpora dominate both training
and evaluation. These corpora were not unique datasets in their own right, but
rather collections and subsets of other datasets, predominantly Switchboard
and Mixer. We have kept the NIST SRE labels distinct from Switchboard and
Mixer to stay true to the naming conventions used by researchers. Moreover,
the NIST SREs typically required specific settings for training and evaluation
that did not necessarily include the entire datasets. The NIST SREs were both
users and drivers of these dataset collections, as annual evaluation
challenges required new datasets to evaluate speaker recognition technology in
ever more difficult settings [12]. Table 3 shows that Mixer and Switchboard
datasets are commonly used, in the NIST SREs, among the top ten training
datasets.

Dataset | times used
---|---
NIST SRE 04 (Switchboard, Mixer) | 136
NIST SRE 05 (Mixer) | 133
NIST SRE 06 (Mixer) | 127
NIST SRE 08 (Mixer) | 96
VoxCeleb 2 | 76
VoxCeleb 1 | 52
Switchboard (version not specified) | 48
Switchboard Cellular 2 | 43
VoxCeleb 1 - dev | 43
NIST SRE 10 (Switchboard, Mixer, etc.) | 40
Table 1: Top 10 most frequently used training datasets. 33footnotemark: 3

After the NIST SREs, Switchboard occurs second most frequently for training,
but surprisingly is only rarely used for evaluation. We elaborate on this
later in Section 4.3. After the NIST SRE corpora, VoxCeleb and ASVspoof have
been popular datasets for training and evaluation. ASVspoof are datasets
released by the Automatic Speaker Verification and Spoofing Countermeasures
Challenge444https://www.asvspoof.org/, which was launched in 2015 to address
growing concerns of security breeches in speaker verification technology due
to voice spoofing and deepfakes. In later sections, we elaborate on VoxCeleb
and its rise to popularity.

Figure 3: Histogram of number of datasets used for evaluation by each paper
that we reviewed. Most papers use a single dataset.

Overall, papers used a far greater number of datasets for training than for
evaluation. Figure 3 shows that the majority of papers in our study used only
a single dataset for evaluation. Papers that evaluated on more datasets rarely
used more than three. These numbers are low, and reminiscent of evaluation
failures in machine learning more broadly [23]. While speaker recognition
development on limited corpora that target specific and evolving tasks over an
extended period of time may have been justified to advance the field prior to
the adoption of deep neural networks [39], the same practices today will lead
to overfitting. Consequently, we ought to question the validity of claimed
advances.

###  4.2 Dataset Dynamics over a Decade of Use

Next, we examine changes in dataset usage, and by extension community adoption
of datasets over the past decade. Where the NIST SREs, Switchboard and Mixer
datasets featured prominently in our previous analysis where we considered
aggregate counts over the past decade, examining dataset usage year-on-year
reveals that their popularity has declined dramatically. In their stead,
VoxCeleb now dominates speaker recognition training and evaluation. Figures 4
and 5 illustrate these dataset dynamics by visualizing the proportional use of
datasets in training and evaluation. These figures show densities and should
be considered together with Figure 1, which shows the growth in publications
and consequently also dataset usage over the decade. Thus, since 2017 more
papers have been published in speaker recognition, and a greater proportion of
these studies uses VoxCeleb to train and evaluate their models.

Figure 4: Distribution (%) of dataset use for speaker recognition training
(VoxCeleb datasets purple, NIST SREs turquoise & green).

Particularly striking is the extent to which VoxCeleb1 dominates speaker
recognition evaluations. The dataset is disjoint from its successor,
VoxCeleb2. A popular pairing is thus to use the larger VoxCeleb2 dataset for
training, and VoxCeleb1 for evaluation. In 2020 and 2021 VoxCeleb1 made up
over half of all evaluation dataset usage. More so, VoxCeleb1-test, a small
subset of 40 predominantly male, US speakers whose name starts with an E [19]
is used in a significant proportion of evaluations. As mentioned previously,
papers may use more than one dataset for evaluation. We thus investigate the
number of papers relying solely on VoxCeleb, or on its even more limited
subset VoxCeleb1-test for evaluation in Table 2. This additional analysis
reveals that VoxCeleb1 is not only popular for evaluation, but that a
significant proportion of studies evaluate their approaches on a single
VoxCeleb dataset only (2019: 17%, 2020: 21%, 2021: 12%), or even more
restricted on VoxCeleb1-test only (2019: 12%, 2020: 13%, 2021: 8%).

Year |  | Published
---
papers
| 1 evaluation
---
dataset only
| 1 VoxCeleb
---
dataset only
| VoxCeleb1
---
\- test only
2012 | 37 | 26 (70%) | - | -
2013 | 62 | 46 (74%) | - | -
2014 | 46 | 33 (72%) | - | -
2015 | 65 | 55 (85%) | - | -
2016 | 53 | 39 (74%) | - | -
2017 | 66 | 47 (71%) | 1 | -
2018 | 64 | 40 (63%) | 4 (6%) | 2 (3%)
2019 | 89 | 59 (66%) | 15 (17%) | 11 (12%)
2020 | 103 | 75 (73%) | 22 (21%) | 13 (13%)
2021 | 86 | 55 (64%) | 10 (12%) | 7 (8%)
Table 2: Most speaker recognition studies evaluate their models on a single
dataset, with a large number only using one VoxCeleb dataset, or even just
VoxCeleb1-test. Figure 5: Distribution (%) of dataset use for speaker
recognition evaluation (VoxCeleb datasets purple, NIST SREs turquoise &
green).

###  4.3 Dataset Attributes

The past two sections have highlighted that the Switchboard, Mixer and
VoxCeleb datasets, together with the NIST Speaker Recognition Evaluations
(SRE) have dominated the development of speaker recognition technologies, each
in its own era. Therefore, we now investigate attributes of these dataset
families and their influence on bias, fairness and privacy of the technology
today.

Background and Motivation for Corpora Collection

The collection and release of the Switchboard corpora started in the 1990s and
continued through the early 2000s. In total, seven datasets of two-sided
English language telephone conversations were released. The dataset
collections were funded by the US Defense Advanced Research Projects Agency
(DARPA) and the US Department of Defense. The Linguistic Data Consortium (LDC)
was primarily responsible for data collection and management. According to the
LDC, these datasets were intended for ”research, development, and evaluation
of automatic systems for speech-to-text conversion, talker identification,
language identification and speech signal detection purposes” [16].

The Mixer and Transcript Reading (short Mixer) corpora succeeded the
Switchboard corpora, as the collection protocol of the latter became
complicated, time-consuming and expensive. Moreover, telephone behaviour of
people changed as cellphones became popular [12]. The Mixer project aimed to
support various speaker recognition tasks in multi-lingual and cross-channel
settings with channel referring to the medium used for speech recordings,
e.g., microphone types. Mixer was created by the LDC in collaboration with the
Lincoln Laboratory, the US NIST and the Speaker Identification research
community.

The VoxCeleb datasets were a timely response to an increasing appetite in the
speaker recognition community to test and develop their approaches in more
challenging real-world (i.e. ”in the wild”) settings. VoxCeleb1 was released
in 2017 by the Visual Geometry Group (VGG) at the University of Oxford, with
the goal of creating a large scale, text-independent speaker recognition
dataset that mimics unconstrained, real-world speech conditions [27]. A key
driver for this was to explore the use of deep neural networks (DNNs), which
had gained traction in computer vision, for speaker recognition tasks. A year
later, VGG released VoxCeleb2 to expand the original data collection.

Attributes and Usage555We did not purchase a membership or licenses Therefore,
for the analysis of attributes we relied on openly accessible information
provided by the LDC and papers on these corpora. We did not have access to the
Switchboard and Mixer datasets for an in-depth analysis of attributes.

Most of the Switchboard datasets contain more than 2 000 recordings with over
100 hours of speech. The Switchboard Credit Card dataset is an exception with
only 35 recordings and 227 minutes of speech data. At the time of collection,
these datasets were considered rich data sources for training speaker
recognition models. Demographic metadata in Switchboard includes information
on age, sex, years of completed education, country of birth, city and state
where raised. The gender distribution in Switchboard is generally balanced on
a speaker level, but not reported for the number of conversation samples.
Switchboard Cellular Part 2 Audio reported gender demographics across
recordings and has an overall balanced split in male and female representation
across the dataset versions (with some years skewed 5-10% towards one gender).
The recordings are annotated for channel quality (e.g. echo, crosstalk,
static) and background noise. Switchboard Cellular Part 2 Audio also annotates
the speaker’s environment (indoors, outdoors, moving vehicle). The earlier
Switchboard datasets record landline conversations with a variety of
telephones, and the later datasets cellphone conversations. Including
different handsets was always considered important to capture channel
variability, which has a major influence on signal processing. Our analysis in
Section 4.2 shows that Switchboard was predominantly used for training, with
relatively limited use in evaluation. The reason for this is that speakers
appear across multiple recordings in different dataset releases. However, it
is not possible to connect speakers between the various releases. Using
Switchboard for training and evaluation thus has potential for data leakage,
which diminishes the quality of an evaluation.

Mixer recorded significantly more data than Switchboard, with individual
datasets capturing between 5 000 and 20 000 calls, resulting in tens of
thousands of hours of speech. The early phases of the Mixer corpora focused on
multi-lingual data collection. Focus later shifted towards multi-channel set-
ups. Publicly available information for the Mixer corpora mostly contains
details on the multi-channel set-up and recording devices. Descriptions of
speaker metadata 666https://catalog.ldc.upenn.edu/docs/LDC2020S03/readme.txt
and https://catalog.ldc.upenn.edu/docs/LDC2013S03/readme.txt for Mixer 4, 5,
and 6 give insights into the demographics that were collected which include
sex, year of birth, education, occupation, ethnicity, height and weight,
smoking and information about his or her family. Given this detailed metadata,
the Mixer collection has been used for age estimation [35], smoker
identification [24, 25] and for predicting speaker demographics from word
usage [15]. These studies have found that the number of recordings in Mixer
decreases drastically with age, and that non-smokers and female speakers are
overly represented. Based on these observations, an extensive dataset
evaluation of the Mixer corpora would be beneficial to identify sources of
bias and their influence on the development of speech-related technology.

The VoxCeleb datasets were scraped from celebrity YouTube videos to capture a
large number of audio clips where people speak in unconstrained settings.
VoxCeleb1 consists of 153 516 speech utterances from 1 251 speakers. VoxCeleb2
contains 1 128 246 utterances from 6 112 speakers. The creators of the dataset
promote its use for speaker identification and verification, speech separation
(i.e. diarization), talking face synthesis, cross-modal transfer between face
and voice (i.e. making inferences about somebody’s face based on their voice,
and vice versa), emotion recognition and face generation. The only metadata
available are gender and nationality labels. The dataset descriptions are not
transparent about how gender labels were obtained, but it is likely that they
came from VGGFace1 and 2 [30, 7], which provided the candidate list of
speakers to include in VoxCeleb. The nationality labels were inferred from
speakers’ countries of citizenship, as obtained from Wikipedia. The motivation
for doing this was to assign a label that is indicative of a speaker’s accent
[26]. The authors claim that the datasets are gender balanced, with 55% and
61% male speakers in VoxCeleb1 and 2 respectively.

Perhaps the biggest difference between VoxCeleb and the other two datasets is
that it used to be freely available for download777VGG has recently removed
the public download link and added a privacy note to their website.
Switchboard and Mixer require a subscription to the LDC ($3 850 for
universities, $27 500 for corporations) or must be purchased. Licensing costs
for an individual dataset range between $100 and $300, however, not all
datasets can be licensed without an LDC membership. This made the VoxCeleb
datasets the first large scale, freely available datasets for speaker
recognition.

Influence of Collection Method on Bias

For Switchboard and Mixer participants were recruited to meet the language
requirements of the study. They received financial compensation for
participating (this changed over time, with start of the Mixer collection
compensation changed from a fixed compensation per call to a per-call
incentive and completion bonuses [11]). While the free phone call was a strong
incentive to participate when the data collection started, after the turn of
the millennium it lost its appeal [12]. Between the releases of the
Switchboard and Mixer datasets, the team of data collectors had continuity.
This has led to consistency in the data collection method. However, it also
implies that individual biases that shaped the design of the datasets can have
gone unnoticed. For example, it is unclear how categories for labels such as
ethnicity or ”country where raised” were chosen, and what rules were applied
to ensure consistent labelling. Such labeling choices can lead to measurement
bias, which affects the validity of predictions made by speaker recognition
systems. The target population for the Switchboard data collection were native
English speakers in the American South. Most participants were college
students, resulting in a dataset with a young demographic.

English had always been a dominant language in speaker recognition. The first
three phases of the Mixer project thus focused on collecting multilingual data
from bilingual speakers. 16% of Mixer calls in Phases 1 and 2 are in Arabic,
Mandarin, Russian or Spanish [9]. The Mixer 3 collection also aimed at
supporting language recognition, and had more than 2900 participants making
calls in 19 different languages [12]. The defense backing of the Mixer
datasets is evident in the languages that were selected for the project, and
their connection to US national security and military interests. The different
Mixer corpora include conversations in Arabic, Egyptian, Farsi, Bengali,
Hindi, Urdu, Tamil, 4 dialects of Chinese (also Mandarin), Japanese, Korean,
Tagalog, Thai, Vietnamese, German, Italian, Russian, 3 dialects of English
(including American English), Spanish and Canadian French. Mixer 4, 5 and 6
featured a wider variety of channels and recording scenarios. The variety of
languages and accents decreased as a consequence and the collections focused
on native speakers of American English only [4]. The data collectors attempted
to balance dialects by recruiting 25% of participants from Philadelphia, 25%
from Berkeley, and specifically from Texas, Georgia, Illinois, and New York
[4]. On-site recordings for Mixer 4 and 5 were carried out at two different
locations, the LDC in Philadelphia, Pennsylvania, and at the International
Computer Science Institute (ICSI) in Berkeley, California. Recruitment for
Mixer 6 was done at the LDC [4, 5], thus decreasing the likelihood of
collecting speech samples from speakers of various dialects and increasing the
risk of representation bias.

The VoxCeleb datasets were constructed with a fully automated data processing
pipeline from audio-visual media scraped from YouTube [27, 26]. Both data
pipelines consist of the same processing steps: first select a list of
candidate speakers, then download videos from YouTube, apply face tracking,
identify active speakers, verify identities from faces, remove duplicates, and
finally find associated nationality metadata on Wikipedia. The candidate
speakers for the datasets were sourced from VGGFace1 [30] and VGGFace2 [7]
respectively. A previous study highlighted that this automated processing
pipeline reinforces popularity bias from search results in candidate
selection, and directly translates bias in facial recognition systems into the
speaker recognition domain [19]. Moreover, celebrities, especially actors and
singers, have a high degree of control over their voice and accent, and should
not be assumed to represent ordinary conversational speech.

More than Bias: Privacy Threats and Ethical Questions

The Switchboard and Mixer dataset collections passed an institutional ethical
review888Guidelines of the Institutional Review Board of the University of
Pennsylvania and the LDC kept personal identifying information separate from
the recordings [10, 4]. Nonetheless, the two corpora would today be considered
as posing significant privacy risks to study participants. The privacy risks
stem from two sources, firstly the content of the conversations and secondly
the rich metadata, which makes it possible with today’s data processing
techniques to retrospectively correlate personal attributes with voice
characteristics. The amount of personal information stored in the metadata is
quite extensive and makes it possible to use these datasets for various tasks
that include the identification of personal information from speech data in
future. At the time of data collection, the ethical consequences and privacy
concerns due to the extent of personal information contained in the voice may
not have been clear to researchers. However, a decade of progress in speech
science has changed that [38]. We are not aware that any efforts have been
made to address the presence of personally identifiable information and
sensitive attributes in the recordings, the potential impact on data subjects
as well as the risks of re-identification. In today’s data-driven society,
privacy and anonymity of data subject are vital concerns that require
attention and proper measures.

During data collection, participants were asked to discuss a specific topic
with an automated operator on a phone call, but to withhold personal
information. Yet, the topics provided for discussion included political,
cultural, social and religious topics999In the European Union the General Data
Protection Regulation (GDPR) considers personal data revealing ethnic origin
or religious beliefs as particularly sensitive and allows processing only on
certain legal bases [1]. Another category of interviews took the form of
informal conversations, adapted from sociolinguistic interview modules. Here
subjects were encouraged to describe events of the past [5]. This form of
interview creates the illusion of an informal setting, making it more likely
that participants share personal information with interviewers [33]. While
participants were not forced to discuss the topic provided, most of them
followed the suggestion [4]. Moreover, participants might have shared
sensitive information during phone calls or interviews, increasing the risk of
re-identification. For instance, Mixer 5 interviews covered family and
personal history, raising the likelihood of participants shareing personal
stories with similarly sensitive information as collected in the metadata. To
our knowledge calls were not redacted to exclude personal information.

The VoxCeleb datasets present very different privacy concerns. As has been the
case with other web-scraped datasets, the dataset creators did not obtain
consent from data subjects. Initially, they justified their approach by
claiming that these datasets were ”open-source” media. More recently, they
have added a privacy notice to their website, calling on a data protection
exemption of the University of Oxford based on Article 14(5)(b) of the UK
GDPR, which allows data processing for scientific or historic purposes.
Considering the highly sensitive nature of voice data, the military and
security foundations of speaker recognition, and the wide-scale applications
in the surveillance industry, it seems prudent to interrogate whether this
exception ought to apply to voice data collected for speaker recognition
purposes.

##  5 Discussion

The NIST SREs, Switchboard, and Mixer datasets have significantly influenced
speaker recognition research over two decades. An important focus of this
research has been on addressing audio processing challenges and reducing
intra-speaker variability to ensure robustness of voice biometrics technology
[44, 34, 40]. The NIST’s evaluation-driven research agenda evolved alongside
technology advancements, considering varied task environments, collection
devices, background noise, and room acoustics. However, inter-speaker
differences related to demographics or other speaker-dependent attributes are
still considered secondary. Building robust systems to address inter-speaker
variability is crucial to avoid biased and discriminatory systems from being
deployed in critical applications, such as financial systems and voice-
activated emergency response.

This study highlights how the shift to deep neural networks in speaker
recognition has led to changes in research and data practices. Our analysis
clearly illustrates the dominance of the VoxCeleb datasets for training and
evaluation. These datasets met the demand of researchers to study speaker
recognition systems with deep neural networks in unconstrained, ”in-the-wild”
settings. A subtle value-shift seems to prioritize studying challenging ”in-
the-wild” settings, potentially at the cost of developing systems that cater
to diverse users. These shifts resemble observations made by [31] about data
practices in machine learning research.

###  5.1 Recommendations

Speaker recognition technology offers benefits but also poses potential risks
and harms depending on its deployment and use. Given recent advances in voice
cloning [22], anti-spoofing research should consider bias and fairness, to
ensure that all demographic groups are adequately protected. When used for
voice-based authentication and access control, it is crucial to ensure the
technology works for all users. To do this, representative evaluation datasets
are necessary.

To promote fairness and reduce bias in voice biometric datasets curation,
valuable insights can be drawn from prior work in facial recognition [32].
Diverse and representative datasets that accurately reflect the demographics
of the population being served are needed. Factors such as age, gender, race,
and ethnicity should be considered when selecting people for dataset
collection. Furthermore, representation should be ensured at the speaker and
utterance level to ensure equitable evaluation across demographic groups [18].
Additionally, dataset collection procedures and dataset attributes should be
documented carefully, for example by adopting datasheets [14] for voice
biometric datasets. More research is needed to understand application-specific
requirements and how to incorporate these into evaluation protocols.

Beyond performance disparities, speaker recognition systems contribute to a
hidden and pervasive surveillance infrastructure that enables governments and
corporations to identify citizens and extract sensitive personal information
from their voice. From a surveillance perspective, speaker recognition
technology poses privacy risks to citizens. Striving for more representative
datasets or detecting and mitigating bias can unintentionally increase harm to
citizens rather than reduce it. Continued research efforts are needed to
enable private and privacy-preserving voice processing.

###  5.2 Limitations

We acknowledge limitations in our study and research approach. Our focus was
on studying dataset dynamics, and we did not consider evaluation protocols and
metrics, which are also important in speaker recognition evaluations. By
analyzed peer-reviewed publications from the Interspeech conference, we
assumed that dataset dynamics in the research domain are indicative of
adoption and attitudes towards datasets in the voice biometrics community. We
speculate that these research practices extend to industry and may indicate
potential bias in deployed applications. Further studies and technology audits
are necessary to evaluate bias in speaker recognition to enable
accountability, transparency, and auditability in speaker recognition.

##  6 Conclusion

Our research provides a comprehensive overview of the evolution of speaker
recognition datasets used for training and evaluation over the past decade.
Through our analysis of the adoption, dynamics, and attributes of these
datasets, we have identified potential issues related to bias, fairness, and
ethical concerns in speaker recognition technology. These findings emphasize
the importance of ongoing investigation into dataset attributes and usage,
particularly in light of current research practices. Moreover, our study
highlights the need for further exploration of how data and evaluation
practices may impact the ethics and fairness of speaker recognition
technology. These insights contribute to our understanding of how data
practices in this field may influence the future development of voice
technologies, raising awareness about potential biases or fairness concerns.

##  7 Acknowledgements

This research was supported by a Mozilla Foundation grant.

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