1.1. Linearised Calderón problem
Let be a bounded simply connected domain in for some . For an isotropic conductivity coefficient , with , and a Neumann boundary value (current density)
the conductivity equation in reads
where is the exterior unit normal of . The Lax-Milgram lemma yields a unique weak solution
for (1.1), corresponding to the interior electric potential. The Neumann-to-Dirichlet (ND) map is a compact self-adjoint operator in , associating applied current densities with boundary voltage measurements. In two dimensions and under the assumed boundary regularity, the ND map actually belongs to , the space of Hilbert–Schmidt operators on [19, Theorem A.2]. The forward map is Fréchet differentiable at with respect to complex-valued perturbations, with the corresponding derivative in .
We investigate the Fréchet derivative at the unit conductivity, i.e. . Let and be harmonic functions in with and , respectively, as their Neumann traces. Then satisfies
which we adopt as the definition of in the following. In our two-dimensional setting, one may actually consider more general (unbounded) perturbations.
extends to an operator in via (1.2).
The linearised Calderón problem consists of reconstructing a perturbation from the linearised data , in contrast to the nonlinear Calderón problem which seeks to reconstruct from . Calderón  proved that the linearised forward map associated to the Dirichlet-to-Neumann (DN) map is injective for perturbations. Dos Santos Ferreira–Kenig–Sjöstand–Uhlmann  proved the corresponding result for partial data; see also [34, 16, 33] for related works. In fact, Calderón’s original injectivity proof carries over to the case of general perturbations and the ND map, demonstrating that
implies the Fourier transform of (the zero-extension of)vanishes. In section 6 we provide an alternative elementary proof for the injectivity of on based on polynomial approximation.
Before presenting our two main results in the following subsection, let us showcase the main theorem on stability (Theorem 1.4) by a conceptually simple special case: The Hilbert–Schmidt structure that exhibits on extends to harmonic perturbations in the closed subspace
on which there is Lipschitz stability.
and there exists such that
for all .
1.2. Main results
Let us next present the details of our main results, which are given on the unit disk in . The involved spaces and estimates can be transferred to with the help of a Riemann mapping, as outlined in section 2. In particular, the constant in Corollary 1.2 satisfies
for any conformal mapping of onto , with denoting its complex derivative. To simplify the presentation, we denote the analysed operator by for both domains and .
Let for and comprise the orthonormal Zernike polynomial basis for . In the polar coordinates ,
We define a family of infinite-dimensional subspaces via
with the closure taken in , and set
Since it is known that (cf. Remark 4.1)
is an increasing family of closed subspaces approaching . Let , i.e. there is an sequence of coefficients such that
where is the orthogonal projection of onto , defined via the coefficients .
Our first main result gives an exact reconstruction method for recovering from the linearised data . Moreover, if , then the method only requires data associated to Neumann boundary values. In the following theorem, is the sign function with the convention and
are the complex Fourier orthonormal basis functions for .
Let be given as in (1.5). Then
for all and .
Let us carefully dissect how Theorem 1.3 can be implemented as a reconstruction method: To start with, , i.e. the coefficients , are reconstructed from the data . Next, , i.e. the coefficients , can be determined using the coefficients for and the data . This process can be continued so that is reconstructed from and the data for any . Hence, the whole of can be inductively reconstructed one orthogonal component at a time. Due to the material in section 2, this reconstruction method also generalises for via employing the aforementioned Riemann mapping . Indeed, if , one can define in and use (2.1) in section 2 to obtain
where . This demonstrates how the Neumann boundary values should be chosen on to reconstruct via Theorem 1.3. The original perturbation can subsequently be recovered as .
Next we shall focus on stability. If , then (1.5) becomes
Based on the representation (1.7), we define a special subset of :
Moreover, if the coefficients are real and the sign of (ignoring zero-coefficients) only depends on , then .
Our second main result shows that the Hilbert–Schmidt structure for extends to perturbations in and yields Lipschitz stability on . Below is the ’th harmonic number, with .
For each , with
On there is Lipschitz stability:
For the Lipschitz constant in (1.10) can be improved to .
In particular, Theorem 1.4 shows that on the infinite-dimensional spaces,
Notice that is the space of square-integrable harmonic functions on since , , are the standard harmonic basis functions for the unit disk. In consequence, equals up to a conformal mapping. Likewise, the spaces can be transferred to via a conformal mapping and, as outlined in section 2, this only modifies the constants in Theorem 1.4 by multiples that explicitly depend on the employed mapping; cf. (1.3). In particular, the -dependence remains as in Theorem 1.4. Moreover, the conformally transformed spaces form an orthogonal decomposition of in a weighted inner product (equivalent to the standard inner product) that also depends explicitly on the utilised conformal mapping. Stirling’s approximation reveals the asymptotic behaviour of the Lipschitz constant in (1.10) for a large :
which demonstrates an expected exponential growth in .
It should be mentioned that Zernike polynomials have previously been used in a numerical study on the stability of the linearised Calderón problem in .
1.3. Comments on the main results
We emphasise that our results are inherently two-dimensional: The Sobolev embeddings allowing to define as an operator in via (1.2) are sharp and do not hold in higher dimensions. Moreover, is not a Hilbert–Schmidt operator in higher dimensions, not even for a bounded perturbation [19, Appendix A]. Interestingly, the proofs for both parts of Theorem 1.4 can be reduced to known bounds on the trigamma function.
Let us then briefly discuss why (1.9) does not seem to follow from previous results. In two dimensions, [19, Theorem A.2] shows that if for some , then is in fact a Hilbert–Schmidt operator in , and there exists (independent of ) such that
As usual, the -symbol refers to a zero-mean condition. For we thus have
implying Hilbert–Schmidt continuous dependence of on . However, defining via (1.2) for requires the associated Neumann boundary values to be in in order to enable the use of suitable Sobolev embeddings. Therefore, one does not know a priori that is a Hilbert–Schmidt operator for , but the Hilbert–Schmidt property and the associated continuous dependence must indeed be separately proved for to establish Theorem 1.4.
To our knowledge, Theorem 1.4 is the first infinite-dimensional Lipschitz stability result in the linearised Calderón problem, which nicely complements the long tradition for Lipschitz stability results in finite-dimensional settings for the nonlinear Calderón problem; see e.g. [22, 1, 2, 3] for some recent general results. This suggests that it may be beneficial to consider stability in terms of ND maps, instead of DN maps, due to the more desirable topological properties of the former. It is also worth mentioning that our proofs do not rely on the standard arguments involving complex geometric optics solutions. Indeed, it seems to be the Hilbert space structures of perturbations and Hilbert–Schmidt operators that enable deducing Lipschitz stability for the infinite-dimensional spaces, in comparison to the optimal logarithmic bounds in terms of operator norms in the infinite-dimensional linearised and nonlinear Calderón problems [4, 5, 29, 25, 31, 27, 26].
In the related problem of detection and shape reconstruction of inclusions/obstacles, Lipschitz stability has recently been proved for general classes of polygonal and polyhedral inclusions [9, 10, 7]; for inclusions of small volume fraction this has already been known for some time . Moreover, the injectivity for certain classes of unbounded coefficients, strictly contained in , has been shown in the nonlinear Calderón problem in two dimensions [8, 12, 30]. The exact reconstruction of inclusions defined by -Muckenhoupt coefficients has also recently been proved in an arbitrary spatial dimension .
The standard spaces and involved in the linearisation of the Calderón problem, with the ND map as the data, are non-separable Banach spaces, with dual spaces that are poorly characterised for the use in practical computations where adjoints are needed. In contrast, resorting to perturbations and the Hilbert–Schmidt operator topology allows a simple characterisation for the adjoint of . To demonstrate this, let be the conformally transformed onto the domain . Both and are separable Hilbert spaces with explicit orthonormal bases. Let be the orthogonal projection of onto , let be an orthonormal basis for , and set in (1.2). A simple calculation reveals that the adjoint is given by
Hence, even if the perturbation is bounded, e.g. it belongs to one of the spaces , the Hilbert–Schmidt topology enables practical optimisation methods for reconstruction. Note also that Lipschitz stability leads to favourable convergence rates for iterative algorithms; see e.g. .
Due to the bound in (1.10), has a closed range for the closed subspace . This is a desirable property for the series reversion approximations introduced in , although it remains an open problem to adapt the method of  for perturbations. Be that as it may, the original motivation for this work was the question of whether there exists an infinite-dimensional space of perturbations satisfying the assumptions in .
1.4. Article structure
This article is organised as follows. Section 2 outlines how the problem setting can be reduced to the unit disk domain . Sections 3, 4, and 5 present the proofs for Proposition 1.1, Theorem 1.4, and Theorem 1.3, respectively. In section 6 we present a simple alternative proof for the injectivity of on .
2. Reduction to the unit disk
In this section, we demonstrate that it is sufficient to replace by in the forthcoming proofs by virtue of the Riemann mapping theorem. To this end, let be a Riemann mapping with the inverse . The complex derivatives of and , when interpreted as mappings between subsets of , are denoted by and . Due to the assumed boundary regularity for , the Kellogg–Warschawski theorem (see [32, Theorem 3.6 & Exercise 3.3.5] and [35, Theorem 12]) ensures that and have extensions, with non-vanishing first derivatives, to the closures of the respective domains. In particular, and are bounded on and , respectively. The absolute value restricted to is the Jacobian determinant of the boundary transformation , while is the Jacobian determinant for the domain transformation by virtue of the Cauchy–Riemann equations. The analogous conclusions naturally apply to as well.
which means that if either of the two integrals exists, then the same also applies to the other integral with the same value; section 3 below demonstrates that the right-hand side of (2.1) is well defined for and . Moreover, admits a characterisation as a harmonic function on with the Neumann boundary value
Let and denote norms of and on the domain boundaries. The maximum modulus principle entails
Thus we have
as well as
The Hilbert–Schmidt norms
are also equivalent (including the case of both being infinite). Recall that we denote the studied operator by the same symbol on both domains for simplicity. As this last equivalence requires a bit of work, we present a brief proof in what follows.
Let be the standard complex Fourier orthonormal basis for given in (1.6), meaning that is an orthonormal basis for . Setting defines an orthonormal basis for in the weighted inner product with the weight . Consequently, is an orthonormal basis for in the standard inner product.
where is interpreted as a multiplication operator, and in the final step a standard norm inequality for products of operators is used; see e.g. [36, Theorem 7.8(c)]. Since projects onto the orthogonal complement of the constant function,
By also performing the analogous calculation in the opposite direction, we finally deduce
where the identities and have also been utilised.
3. Proof of Proposition 1.1
Based on (2.1), (2.2), and (2.3) in section 2, it is sufficient to prove the result on . Let and let and be harmonic functions on with the Neumann boundary values , respectively. According to [28, Chapter 2, Remark 7.2], with continuous dependence on the Neumann data, i.e.
Using the continuous embedding (e.g. [14, Corollary 4.53]), we may estimate as follows:
Combining this with (3.1) concludes the proof. ∎
Denote by the solution to (1.1) for a (nonconstant) coefficient and with replaced by . The mapping
is bounded for if ; cf. e.g. [20, Corollary 22.214.171.124]
, which can be modified for the quotient space. It thus follows from complex interpolation theory for Hilbert spaces (e.g.[28, Chapter 1, Theorems 5.1, 7.7, 9.6, 13.2]) that the solution map remains bounded for , i.e. from to , for . By the above analysis, this demonstrates that the Fréchet derivative in the unit disk belongs to for a Lipschitz continuous coefficient . Based on the material in section 2, this result also extends for a domain with a Lipschitz continuous conductivity coefficient.
4. Proof of Theorem 1.4
To shorten the notation, we denote the standard inner product by and the Hilbert–Schmidt norm on by in the following.
4.1. On the Zernike polynomials
Recall the Zernike polynomials , , , given as functions of the polar coordinates in (1.4). On the weighted space
with the inner product
the radial Zernike polynomials corresponding to the same angular frequency are orthogonal (see e.g. the original paper by Zernike ),
This immediately leads to the orthonormality of ; we have included a comment on the density of the span of the Zernike polynomials in as Remark 4.1 below, because this matter is not often explicitly discussed in the literature.
Monomials can be expanded as
in terms of the Pochhammer symbols defined as ; see e.g. [24, Equation (35)].
The density of
in is typically not explicitly discussed, and since the indices for the radial and angular parts are not independent, this might not be immediately obvious. We briefly argue this fact, ensuring that is indeed an orthonormal basis for . Consider in polar coordinates for some and . Since for by (4.2), the approximation of in by the Zernike polynomials can be straightforwardly reduced to approximating in by polynomials in . However, is dense in by Lemma 6.1 in section 6. As
is dense in , the argument is complete.
4.2. Infinite matrix coefficients for
is a certain unitary matrix, a direct calculation reveals
It is thus enough to concentrate on the case in what follows.
Let , i.e. there is an sequence of coefficients such that
Furthermore, if , then where . Hence, (4.2) gives
Substituting with some yields
Inserting (4.3) finally gives the sought-for infinite matrix coefficients
4.3. Boundedness of on in the Hilbert–Schmidt norm
We proceed to prove (1.9). Assume that for some , i.e. whenever . Because obviously
we may estimate for as follows: