Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

Abstract

Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.

1. Introduction

The Cauchy problem for the Laplace equation in particular, and for other elliptic equations in general, occurs in the study of many practical problems in areas such as plasma physics [1], electrocardiology [2, 3], bioelectric field problems [4], nondestructive testing [5], magnetic recording [6], and the Cauchy problem for elliptic equations [7, 8]. These problems are known to be severely ill-posed [9], in the sense that the solution, if it exists, does not depend continuously on the Cauchy data in some natural norm (see, e.g., [5] and references therein). This is because the Cauchy problem is an initial value problem which represents a transient phenomenon in a time-like variable while elliptic equation describes steady-state processes in physical fields. A small perturbation in the Cauchy data, therefore, can affect the solution largely.

In this paper, we will concretely consider the following Cauchy problem for Laplace equation in a strip: where we want to determine for from the data with corresponding measured data function [9–11].

In [9], the authors constructed a regularization method based on the Meyer wavelet, but the convergence rate of the method was not obtained. In [10, 11], a modification method and a Fourier method for this problem were given, respectively, and some error estimates with satisfactory convergence rates were also proved. However, some very important and difficult problems in the theoretical study, that is, the optimal error bound, are not discussed. The major object of this paper is to give the optimal error bounds in theory for problem (1) by employing a regularized theory based on spectral decomposition. Meanwhile we provide two optimal regularized methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition method, which realize the optimal error bound.

The motivation of this paper is inspired by Tautenhahn, in [12], where he discussed a Cauchy problem for the elliptic equation in a bounded domain and used the eigenvalues of the elliptic operator to express the exact solution of the problem. However, this method does not suit problem (1) in an unbounded strip region. Instead of using eigenvalues we employ the technique of Fourier transform.

Let and denote the exact and measured data, respectively, which satisfy where denotes the -norm and the noise level is determined by the accuracy of the instruments. We assume the and other functions appearing in this paper with respect to variable belong to . Let denote the Fourier transform of function . We now analyze problem (1) in the frequency space. Taking Fourier transform for problem (1) with respect to the variable , we get The unique solution of (4) is [9–11] Due to Parseval formula, , and therefore (5) implies that must decay rapidly as . However, as the measurement data , we can not expect has the same decay in high frequency components; that is, small errors in high frequency components can blow up and completely destroy the solution for , noting that the factor in (5) increases exponentially as , so the problem (1) is severely ill-posed.

In order to obtain explicit stability estimate for problem (1), some “source condition” is needed. For this we introduce the Sobolev space according to , , where is the norm in . We require the a priori smoothness condition for problem (1) concerning the unknown solution according to

This paper is organized as follows: In Section 2 we briefly recount some preliminary results, which are the basis of the discussion for other sections. In Section 3 we give the optimal error bounds between the exact solution and its regularized approximation, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. In Section 4 we discuss two concrete regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, where both regularization methods realize the optimal error bounds.

2. Preliminary Result

We consider arbitrary ill-posed inverse problem [12–17] where is a linear injective bounded operator between infinite dimensional Hilbert spaces and with nonclosed range of . We assume that are available noisy data with . Any operator can be considered as a special method for solving (8), and the approximate solution of (8) is given by . However, the convergent rate of to can be arbitrarily slow without assuming additional quantitative a prior restrictions on the unknown solution , which is typical for ill-posed problem.

Assume we want to solve (8), we have the a priori information that the exact solution satisfies a source condition; that is, belongs to the source set where the operator function is well defined via spectral representation [13, 14]: whereis the spectral decomposition of , denotes the spectral family of the operator , and is a constant such that . In the case when is a multiplication operator, , the operator function attains the form

Let us assume that is an arbitrary mapping to approximately recover from . Then the worst case error for under the a priori information is [16, 17] This worst case error characterizes the maximal error of the method if the solution of problem (8) varies in the set . The best possible worst case error (or the optimal bound) is defined as where the minimum is taken over all methods . It can be shown (cf. [13, 18]) that the minimum in (14) is actually obtained and with the modulus of continuity defined by

In order to derive explicitly the optimal error bounds for the worst case error defined in (13) and obtain optimality results for special regularization methods, we assume that the function in (9) satisfies the following assumption.

Assumption 1 (see [13, 14]). The function in (14), where is a constant such that , is continuous and has the following properties:(i),(ii) is strong monotonically increasing on ,(iii) is strong convex.

Under Assumption 1, the next theorem gives us a formula for the optimal error bound.

Theorem 2 (see [13, 14]). Let be given by (14), let Assumption 1 be satisfied, and let , where denotes the spectrum of operator ; then

In the following we consider two special methods: the method of generalized Tikhonov regularization and the method of generalized singular value decomposition.

For the method of generalized Tikhonov regularization, a regularized approximation is determined by solving the minimization problem [13, 14, 16, 17] or equivalently, by solving the Euler equation and the following statement holds.

Theorem 3 (see [12, 13]). Let be given by (14), let Assumption 1 be satisfied, let be two times differentiable, let be strong convex on , and . If the regularization parameter is chosen optimally by then, for the Tikhonov regularized solution defined by (18) or (19), the optimal error estimate holds.

The regularized approximation based on the method of generalized singular value decomposition is given by

For this method the following result holds [13, 14].

Theorem 4. Let be given by (9), let Assumption 1 be satisfied, let be two times differentiable, let be strong convex on , and . If the regularization parameter is chosen optimally by then for the regularized solution defined by (22) the optimal error estimate (21) holds.

3. Optimal Error Bounds for Problem (1)

Let us formulate the problem (1) for identifying from unperturbed data as operator equation with a linear operator ; then the equivalent operator equation of (24) in the Fourier domain is given by where is the (unitary) Fourier operator that maps any function into its Fourier transform . From (5) and (25), we have where is a multiplication operator. It is easy to know that the operator is self-adjoint, so is given by

Due to Parseval formula, is equivalent to , where withNote that the source condition (9) for problem (1) can be written as and then its equivalent form in Fourier frequency space is given by

Due to the equivalence of conditions (30) and (7), we know the conditions (28) and (31) are equivalent and we have the following result.

Proposition 5. For operator equation (25), the set given by (7) is equivalent to the general source set given by (30) provided is given (in parameter representation) by

Proof. From (5) we have which gives and then the inequality is equivalent to Note that the set given by (28) is equivalent to the set given in (31), we know (35) is equivalent to So, we obtain that the operator function in (30) has the representation Together with (27), is given by (32) in its parameter representation. The proof is complete.

The function defined by (32) possesses the following properties.

Proposition 6. The function defined by (32) is continuous and satisfies the properties:
(i) .
(ii) is strong monotonically increasing.
(iii) is strong monotonically increasing and possesses the parameter representation: (iv) is strong monotonically increasing and possesses the parameter representation: (v) For the inverse function of the following holds: for any fixed .
(vi) The function defined by (38) is strong convex for , and , .

Proof. The continuity of function is obvious.
(i) From (32) we have It is easy to know that for ; that is, the function is decreasing and , so, Moreover, when , , we know is increasing and , From (42) and (43), we have .
(ii) From (32), we know Together with (41), we have It is easy to know that and are all even functions about variable ; therefore is also so. Here it is only to consider the case . Note that the function is strong monotonically increasing; together with (32) we know for . Moreover, a straightforward computation gives So, we can easily see for , and , ; that is, when , and , , is strong monotonically increasing.
(iii) From (ii) we know that is strong monotonically increasing for , and , . Therefore, when , and , , is also strong monotonically increasing. From (32), it is easy to know that has the parameter representation So, the parameter expression (38) of function holds.
(iv) According to (iii), is also strong monotonically increasing and its parameter representation (39) can be obtained from (38) immediately.
(v) From (39) we have and consider It is easy to see that Inserting (49) and (50) in (39), we have Indeed, if we denote it is easy to prove that So, the representation (40) holds.
(vi) We know that the function is strong convex if and only if . Denoting with , from (38) we have where It is easy to know that is even function about variable ; therefore we only need to consider the case .
1st Case (). In this case, . Then is equivalent to , that is, A straightforward computation shows that (56) is equivalent to From (57) we have Note that the function is monotonically increasing and is monotonically decreasing, and we obtain that for all . So, when , the function is strong convex if that is, Note that the function is increasing about for ; by numerical computation we know So, when , (60) holds naturally. With (57) and together, we know that is strong convex for , and .
2nd Case (). An elementary calculation shows for all and . From above, (vi) is proved.
The proof is finished.

Now we formulate our main result of this section concerning the best possible worst case error defined by (17) for identifying solution of problem (1) from noisy data under the condition (2) and where the set is given by (7). We denote with given as in (14). Due to Parseval identity and the equivalence of which are given by (28) and (31), respectively, we have

Applying Theorem 2 and Proposition 6 we have the following result.

Theorem 7. Let , and then consider the following.
(i) In case and , the following holds: (Hölder stability).
(ii) In case and , the following holds: (logarithmic stability).
(iii) In case and , the following holds:

Proof. From (32), we have It is easy to know the function takes maximum at , and , so . Due to Theorem 2 and (40) given in Proposition 6, we know if .
(i) For and , (Hölder stability).
(ii) In case and , (logarithmic stability).
(iii) In case and , The proof is complete.

4. Optimal Regularization Methods

In this section we consider two special regularization methods, apply them to problem (1), and show how to choose the regularization parameter such that it guarantees the optimal error bounds given by (65)–(67).

The method of generalized Tikhonov regularization (18) consists in the determination of a regularized approximation by solving the minimization problem:or, equivalently, it is the solution of Euler equation:

Applying Theorem 3 and Proposition 5 to problem (1), we obtain the following result.

Theorem 8. Consider the operator equation (24) and assume its unknown solution given by (7). If holds, then the method of generalized Tikhonov regularization (72) or (73) is optimal on provided that the regularization parameter is chosen optimally by where is the unique solution of the equation (i) In the case and the following holds: (ii) In the case and , the following holds: Furthermore, the optimal error estimate holds, where is given by (65)–(67), respectively.

Proof. From Theorem 3, it follows that the optimal regularization parameter is given by (20) with given by (32), which is equivalent to withthat is, is the unique solution of (75).
(i) In the case and , note that for , and for , and from (78) and (50), we have(ii) In the case and , the following holds: The proof is finished.

Now we consider the method of generalized singular value decomposition. Due to (22), the regularized approximation is given by

Applying Theorems 4 and 7 to problem (1), we have the following.

Theorem 9. Consider the operator equation (24) and assume its unknown solution given by (7). If holds, then the method of generalized singular value decomposition (82) is optimal on provided that the regularization parameter is chosen optimally by where is the unique solution of (75).
(i) In the case and , the following holds: (ii) In the case ,, the following holds:Furthermore, the optimal error estimate holds, where is given by (65)–(67), respectively.

Proof. From Theorem 4, we know that the optimal regularization parameter is given by (23) with given by (32), which is equivalent to where is the unique solution of (75).
(i) In the case and , note that for , and for , and from (83) and (75), we have(ii) In the case and , from (75) and (83), we have Furthermore, the optimal error estimate holds, where is given by (65)–(67), respectively. This completes the proof.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partly supported by the Fundamental Research Funds for the Central Universities No. 27R1410016A.