Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.


Introduction
An inverse potential problem consists in determining the shape of an unknown domain D form measurements of the Neumann boundary values of u on ∂Ω, where the solution u of the homogeneous Dirichlet problem fulfills where χ D is the characteristic function of the domain D ⊂ Ω R = {x ∈ R 2 : |x| < R}. This inverse problem is a nonlinear severely ill-posed problem; see [1,2]. If a classical difference method is used for solving the inverse problem, the errors can grow exponentially fast as the mesh size goes to zero. Many regularizing methods are adopted to provide a stable solution of inverse potential problems, e.g., a second-degree method with frozen derivatives [3], level set regularization [4], the iteratively regularized Gauss-Newton method [5] and Levenberg-Marquardt method [1]. In this work, we consider a discrete version analoguous to the modified asymptotic regularization proposed by Pornsawad et al. [6] to recover the starlike shape of the unknown domain D.
In a general setting, an inverse potential problem can be formulated via a nonlinear operator equation where y is the normal derivative of u on the boundary, ∂u ∂ν | ∂Ω R , ν is the outer normal vector on ∂Ω R , the operator F : D (F) ⊆ X → Y is a nonlinear operator on domain D (F) ⊂ X, X and Y are Hilbert spaces, and the unknown x includes the information of the domain D ⊂ Ω R . For convenience in this article, the indices of inner products ·, · and norms · are neglected but they can always be identified from the context in which they appear. Due to the nonlinearity of Equation (3), we assume all over that Equation (3) has a solution x + which needs not to be unique. We have the disturbed data y δ with y δ − y ≤ δ (4) where δ > 0 is a noise level. If one solves Equation (3) by traditional numerical method, high oscillating solutions may occur. Thus, one needs a regularization to minimize the approximation and data error. One well-known continuous regularization is Showalter's method or asymptotic regularization [7], where an approximate solution is obtained by solving an initial value problem. Later, a second-order asymptotic regularization for the linear problem Ax = y was investigated in Zhang and Hofmann [8], where the optimal order is obtained under the Hölder type source condition and a conventional discrepancy principle as well as a total energy discrepancy principle. Recently, the study of modified asymptotic regularization is reported in Pornsawad et al. [6] where the termx − x δ (t), x δ (0) =x, is included to the method proposed by Tautenhahn [7], i.e., A discrete version analogue to Equation (5) is successfully developed in Pornsawad and Böckmann [9], where the whole family of Runge-Kutta methods is applied and one obtaines an optimal convergence rate under Hölder-type sourcewise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. It is well known that, for many applications such as the inverse potential problem and the inverse scattering problem [5], the Hölder type source condition in general is not fulfilled even if a solution is very smooth. It is applicable only for mildly ill-posed problems [1,10,11]. Therefore, the convergence rate analysis of an explicit Euler method presented by is considered in this article under the logarithmic source condition in Equation (7) and the properly scaled Fréchet derivative F (x + ) ≤ 1. The method in Equation (6) is a particular method of the iterative Runge-Kutta-type method [9], where τ n = α −1 n is the relaxation parameter obtained by discretization of conventional asymptotic regularization [7]. We define with p > 0 and the usual sourcewise representation where w is sufficiently small. The method in Equation (6) is also known as the modified Landweber method [12] which has the rate O( √ δ) under the Hölder-type source condition and general discrepancy principle. The convergence rate analysis under the logarithmic source condition in Equation (7) has been successfully studied by Hohage [5] for the iteratively regularized Gauss-Newton method and by Deuflhard et al. [13] for Landweber's iteration. Current studies of source condition may be found, e.g., in Romanov et al. [11], Bakushinsky et al. [14], Schuster et al. [15] and Albani et al. [16].
The purpose of this work is to present the convergence rate analysis of the iterative scheme of Equation (6) under the logarithmic source condition in Equation (7) with 1 ≤ p ≤ 2 and to recover the shape of an unknown domain D for an inverse potential problem (Equations (1) and (2)). Thus, in Section 2, a preliminary result is prepared. As usual, the Fréchet derivative of F needs to be scaled. Furthermore, we assume a nonlinearity condition of F in a ball B ρ (x 0 ) = {x ∈ X : x − x 0 ≤ ρ}, ρ > 0, which is given in Assumption 1. It is well known that, without the additional assumption on the nonlinear operator, the convergence rate cannot be provided. The following assumption has been used in many works [5,17], i.e., there exists a bounded linear operator R : Y → Y and Q : X → Y such that with nonnegative constants C R and C Q . However a weaker condition will be used in this work. This will be shown in Assumption 1. In Section 3, the convergence rate of the modified Landweber method under the logarithmic source condition is presented. Application of the modified Landweber method to an inverse potential problem is provided in Section 4.

Preliminary Results
In this section, preliminary results are prepared to provide the convergence analysis of the modified Landweber method. Lemma 1. Let A be a linear operator with A ≤ 1. For n ∈ N with n > 1, e 0 := f (λ)w with f given by Equation (7) and p > 0, there exist positive constants c 1 and c 2 such that with 0 < α i ≤ 1, i = 0, 1, 2, . . . , n.

Moreover, in
for some constantc 1 . Thus, the induction is complete. We prove Equation (14) by induction in the same manner as Equation (13).

Assumption 1.
There exist positive constants c L , c R , and c r and linear bounded operator R x : Y → Y such that, for x ∈ B ρ (x 0 ), the following condition holds where x + is the exact solution of Equation (3).

Lemma 2.
Let the Assumption 1 be assumed. Then, we have Proof. We note that the reverse triangle inequality and Equation (19) guarantee the estimates and Using the estimates in Equations (18), (20), (22), and (23) and the triangle inequality, we now have (17) and (18) in Assumption 1 be true. Then,

Proposition 2. Let the conditions in Equations
Proof. Define Using the mean value theorem with Equations (17) and (18), we obtain

Convergence Analysis
To investigate the convergence rate of the modified Landweber method under the logarithmic source condition, we choose the regularization parameter n according to the generalized discrepancy principle, i.e., the iteration is stopped after N = N(y δ ,δ) steps with where τ > 2−η 1−η is a positive number. In addition to the discrepancy principle, F satisfies the local Utilizing the triangle inequality yields to ensure at least local convergence to a solution x + of Equation (3) Assume that the problem in Equation (3) has a solution x + in B ρ 2 (x 0 ), y δ fulfills Equation (4), and F satisfies Equations (17) and (18). Assume that the Fréchet derivative of F is scaled such that F (x) ≤ 1 for x ∈ B ρ 2 (x 0 ). Furthermore, assume that the source condition in Equations (7) and (8) is fulfilled and that the modified Landweber method is stopped according to Equation (26). If w is sufficiently small, then there exists a constant K 2 depending only on p and w with and Proof. We give the abbreviation e n := x + − x δ n for the error of the nth iteration x δ n of Equation (6) and K := F (x + ). We can rewrite Equation ( 6) into the form Since e n := x + − x δ n and K := F (x + ), we present e n as Rewritting Equation (30), we have By recurrence and Equation (31), we obtain the closed expression for the error Moreover, it holds Next, for 0 ≤ n < N, using the discrepancy principle, triangle inequality, Equation (28), and τ > 2−η 1−η , we get Using Lemma 2, Proposition 2, and Equation (34), we obtain whereĉ 1 = c L 2 + K R 1−η , and we use the fact that 1 − α n ≤ 1. It holds that e n is decreasing independently of the source condition for 0 ≤ n < N; see Proposition 2.2 in Scherzer [12].
Because of Equations (26) and (28) we have We can rewrite Equation (57) as follows: Applying Equation (58) to Equation (55), we get In similar manner, Equation (56) can be written as Finally, we select w such that 1 + 1 Θ c * w +c p K 2 2 ≤ K 2 . This is always possible for sufficiently small w , [13]. Therefore, the induction is completed. Using Equation (36), we have e n ≤ K 2 ln n ln(n + e) and similarly, by using Equation (34), we have Thus, the assertion is obtained.

Theorem 2.
Under the assumptions of Theorem 1 and 1 ≤ p ≤ 2, we have with some constant c, C > 0.
For 1 ≤ p ≤ 2 we know that N −1/2 (ln N) p ≤ c 7 for some c 7 > 0, see Figure 3. Thus, the assertion can be obtained.  The nonlinear operator for an inverse potential problem is defined in the form Thus, the assertion can be obtained.

Application to an Inverse Potential Problem
It is well known that an inverse potential problem is severely ill-posed. It is the problem of determining the shape of an unknown domain D from measurements of the Neumann boundary values of u on ∂Ω R where the solution u fulfills Equations (1) and (2). In this work, Assumption 2.1 for the inverse potential problem cannot be presented. It fails even in the case of two concentric circles [2]. However, if we implement the method by representing the curve with a collocation basis, as will be seen in Proposition 3, the Fréchet derivative is reformulated. Without the verification of Assumption 1, we show a quite good performance of an approximated potential.
The nonlinear operator for an inverse potential problem is defined in the following form: where F : L 2 [0, 2π] → L 2 [0, 2π]. Moreover, the Fréchet derivative of the operator F is 0.5720 with the residual norm 28.2597 after 13 iterations and α n = 1 2 (1000 + n) −0.9 , δ = 0.001, and τ = 1700 provide the error 0.5925 with the residual norm 15.4140 after 14 iterations. Figures 4, 5b,d, and 6b show that the curve of ln x + − x δ n lies below a straight line with slope −p as suggested by Equation (29).