Convergence of a series associated with the convexification method for coefficient inverse problems

04/12/2020 ∙ by Michael V. Klibanov, et al. ∙ UNC Charlotte Kansas State University 0

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^2 as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^1-norm for a sequence of L^2-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

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1 Introduction

In this section we formulate the inverse problem of interest, discuss some related works as well as our result on a convergence property for a truncated Fourier series approximation for the convexification method. Consider the scattering from a bounded penetrable object in ( or 3). Below Suppose that the scattering object is characterized by the bounded function which has a compact support. We are particularly interested in the case for which is nonnegative. This is typically the case for applications of non-destructive testing and explosive detection, see for instance [6, 19, 16] for a similar assumption. Consider the downward propagating incident plane wave where is the wavenumber. The scattering problem for the Helmholtz equation is described by

(1)
(2)

where is the total wave and the scattered wave satisfies the Sommerfeld radiation condition (2). The scattering problem (1)–(2) has a unique solution , see [9]. For we consider

Suppose that the scattering medium and the support of the coefficient are contained in , and that there is no intersection between these objects and . For positive constants , we consider the following inverse problem.

Inverse Problem. Given the backscatter Cauchy data

(3)
(4)

where the total wave is generated by the incident plane wave for , determine the function in (1) for .

Uniqueness theorem for this inverse problem can be proven using the so-called Bukhgeim-Klibanov method [6] if the right hand side of equation (1) is nonzero in , see, e.g. [4, 26, 17] for surveys on the development of this method. In addition, uniqueness theorem of an approximate problem can also be established, see, e.g. Theorem 3.2 in [14].

This inverse problem belongs to a wider class of coefficient inverse scattering problems that occur in various applications including non-destructive testing, explosive detection, medical imaging, radar imaging and geophysical exploration. We refer to [9] and references therein for theoretical and numerical studies on inverse scattering problems. To solve these inverse problems the conventional approach is optimization based methods, see, e.g. [2, 8, 11, 12, 10]. However, it is well-known that these methods may suffer from multiple local minima and ravines and their convergence analysis is also unknown in many cases. The qualitative methods [7, 15, 9] have been developed to avoid these drawbacks. However, these methods can reconstruct only geometrical information of the scatterer and typically require multi-static data which are not always available in practical applications.

To overcome the drawbacks of optimization-based approaches a recently new approach called globally convergent numerical methods (GCNMs) has been developed for solving coefficient inverse problems. We define that a numerical method for a nonlinear ill-posed problem converges globally if there is a rigorous guarantee that it delivers points in a sufficiently small neighborhood of the exact solution of this problem without any advanced knowledge of this neighborhood. Typically the GCNMs solve a coefficient inverse scattering problem using multifrequency data for a fixed direction of the incident plane wave. Most recently, it has been also studied for data associated with many locations of the point source but at a fixed single frequency [14]. We also want to mention that the GCNMs always consider non over-determined data. The main advantage of the GCNMs is the above indicated global convergence property that avoids the problem of multiple local minima. For theoretical results as well as numerical and experimental data study of the first type of GCNMs we refer to [4, 25, 30, 27, 29] and references therein.

The convexification method we are concerned with can be considered as the second type of the GCNMs. It was originated in 1995 and 1997 by Klibanov [19, 16] and continued since then in [5, 26, 20]. However, the numerical implementation of the method faced some obstacles until its recent improvement [1] in 2017. This work has been continued by a number of more recent publications [23, 22, 18, 24, 21, 14], which address both convergence analysis and numerical results. In particular, the verification of the convexification method on experimental data has been done in [23, 22]

. The main aim of the convexification method is to construct of a globally convex weighted Tikhonov-like functional with the Carleman Weight Function in it. As in the first type of the GCNMs the coefficient is eliminated from the scattering problem using a change of variables. Next, the inverse problem is formulated as the Cauchy problem for a system of quasilinear elliptic PDEs using a truncated Fourier expansion. The Cauchy boundary data are given only on part of the boundary. We use a weighted, nonlinear quasi-reversibility method to solve the Cauchy problem in which the weighted Tikhonov-like functional involves a Carleman Weight Function. This is the function which is involved as the weight function in the Carleman estimate of an associated PDE operator. The global convexity property of this functional is proven using the tool of Carleman estimates. And then the global convergence analysis of the method is established by proving convergence of the gradient projection method to the exact solution as the level of noise in the data tends to zero. We also refer to another version of the convexification 

[3, 28] in which it is assumed that the initial condition in a hyperbolic/parabolic PDE is nonzero in . This is unlike our case of the zero right hand side of equation (1).

In some recent previous works [14, 13, 23, 31] the convergence analysis of the convexification method has been studied under the assumption that a function generated by the total field is expanded by a truncated Fourier series. The Fourier basis we exploit for the convexification method is a special Fourier basis that has been recently introduced in [18]. This basis plays an important role in the study of the convexification method in [14, 13, 23, 31]. The most important property of this basis is that the set of derivatives of the basis is linearly independent and dense in the sense. It is exactly this property which allows us to prove the key Lemma 3 of this paper. Note that the trigonometric basis and most bases of orthogonal classical polynomials do not hold this property. In fact, the only basis we know that shares this property is the basis of Laguerre polynomials. However, unlike our basis which works on a finite interval, the Laguerre basis is defined on an infinite interval.

In this paper we prove that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in the sense as the truncation index of the truncated Fourier series tends to infinity. That approximate PDE contains both functions and for The original PDE is the equation obtained by eliminating the coefficient from scattering problem. In other words the more terms we have for the truncated Fourier series the better approximation we get for the equation obtained from elimination of the coefficient . The proof is done under some assumptions on the decay of the Fourier coefficients of the function generated by the total wave field. The key ingredient of the proof is to establish a convergence result in the -norm, rather than in the -norm, (see Lemma 3) for a sequence of -orthogonal projections on finite-dimensional subspaces spanned by the special Fourier basis elements.

We point out, however, that even though that residual tends to zero as tends to infinity, we still cannot prove that our solutions of those approximate PDEs tend to the correct one as tends to infinity. This is due to an ill-posed nature of our coefficient inverse problem. In other words, when working with that approximate PDE and assuming that is fixed, as it was done in [14, 13, 23, 31], we actually work within the framework of an approximate mathematical model. Nevertheless, numerical results are accurate ones. Furthermore, accurate reconstructions of complicated targets from experimental data were observed in [13]. In our opinion, the latter is the ultimate justification of our approximate mathematical models. See [14, 24] for detailed discussions of the issue of such approximate mathematical models.

The paper is structured as follows. The second section is dedicated to the formulation of the inverse problem as an approximate quasilinear elliptic PDE system. The convergence result for the truncated Fourier approximation is presented in Section 3.

2 A quasilinear elliptic PDE formulation using truncated Fourier series

The inverse problem can be approximately formulated as a Cauchy problem for a system of quasilinear elliptic PDEs. This formulation is one of the key ingredients for the convexification method. In this section we present this formulation for the convenience of readers. The idea is to eliminate the coefficient from the scattering problem and use truncated expansion of a special Fourier basis of . For the following Fourier basis of was introduced in [18]

We may obtain an orthonormal basis

by applying the Gram–Schmidt process to

. The orthonormal basis has the following properties, also, see [18]:
i) for all
ii) The matrix , where

is invertible with for and for .

Now making a change of variables

(5)

and substituting in (1) we have

(6)

We define as

(7)

where we refer to [31, 14, 22, 21] for discussion on the definition of the complex logarithm. Then substituting in (6) we obtain

(8)

Differentiation of (8) with respect to leads to

(9)

For some large we approximate and by

(10)

where

are the Fourier coefficients. Substituting these truncated series (10) in (9) we obtain

(11)

We multiply the above equation by and integrate with respect to over

(12)

We define two matrices

and an block matrix , where each block is an matrix given by

Setting

we rewrite (12) as a system of PDEs for as follows

(13)

Here if is an block matrix, where each block is an matrix, then for an matrix we define as

Now we can approximately reformulate the inverse problem as the Cauchy problem for the following system of quasilinear elliptic PDEs

(14)
(15)
(16)

where the boundary data and can be computed from the original data and in (3)–(4) using (5), (7) and (19). If is found by solving problem (14)–(16), the coefficient can be approximately recovered from (8).

To solve problem (14)–(16), a weighted globally strictly convex Tikhonov-like functional is constructed in each of the works [14, 13, 23, 31], where the weight is the Carleman Weight Function for the Laplace operator and the number is fixed. Then, the analytical study in each of these references results in global convergence theorems of the gradient projection method of the minimization of that functional. However, the latter is not a concern of this publication. Rather we focus on the convergence issue of the residual of the equation (11).

3 Convergence of the truncated Fourier series approximation

In this section we will prove that under certain assumption on the decay of the Fourier coefficients of the function and its derivatives the equation (9) is actually well approximated by the approximate equation (11) in the sense if is large enough.

From the property of the Fourier basis in the previous section we have

(17)

where is an invertible triangular matrix. Let . We define

(18)

From (17) we have and it is obvious that is dense in . Let be the orthogonal projection

where the coefficients are given by

(19)

We first recall a result in Fourier approximation.

Lemma 1.

For any , we have

Proof.

Let . Since is dense in , there is such that . Suppose that for some . Then for , we have and

This completes the proof. ∎

The next lemma aims to estimate the derivative of an element in .

Lemma 2.

For any we have

Proof.

We estimate

which completes the proof. ∎

Remark. The assumption (20) of the next lemma requires a fast decay of the Fourier coefficients that will be applied for the function in (9) and some of its derivatives in . This assumption is not really a strong restriction since is a smooth function in and its Fourier coefficients are indeed observed to have a fast decay in numerical simulations, see, e.g., [14, 31]. The numerical implementation of the convexification method in these cited papers only needs in the truncated Fourier series (10).

We now prove the key lemma for the result in this section.

Lemma 3.

Let such that , and assume that there are constants and such that, for all ,

(20)

Then we have

Proof.

Let . Then . From (17) we have and hence . Using Lemma 2 we estimate

Note that for and the infinite series in the last inequality above converges because of our assumption for the lemma. Since this is true for any , setting and using the fact we have

(21)

Consider the space

This is a Hilbert space with the inner product

We also have that is dense . Indeed, let and . Then there is such that . Taking we obtain

It is obvious that which satisfies the assumption of the lemma belongs to . Now from (21) we have

(22)

We can see that is also an orthogonal projection from to . Therefore using the assumption that we obtain

(23)

Now from (23) and our assumption that the series converges we obtain

We have just proven that the first infimum term in (22) tends to zero as . The second infimum term in (22) also tends to zero as thanks to Lemma 1. Finally it is obvious from the definition of the term in (22) tends to 0 as which completes the proof of the lemma.

The next theorem is the main result of this paper.

Theorem 4.

For , assume that

and furthermore, for , satisfies

(24)

for some positive function and and that the series in (20) is convergent. Let be the left hand side of (9) and define

Then and more importantly

Proof.

It is easy to see that following the assumption of the lemma. Rewriting it as

and making a subtraction from we obtain

We do some algebra for the nonlinear terms. Set

Then can be written as

We estimate in as

From the dominated convergence theorem and Lemma 3 we obtain

Similarly, for , we have

and therefore

Acknowledgement. The work of the first author was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044. The work of the second author was partially supported by NSF grant DMS-1812693.

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