Abstract
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.
Highlights
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 homogeneousDirichlet problem fulfills∆u = χ D in u=0 on Ω R \ ∂D, ∂Ω R (1) (2)where χ D is the characteristic function of the domain D ⊂ Ω R = { x ∈ R2 : | x | < R}
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]
We show that the rate O((− ln δ)− p ) of the modified Landweber method in Equation (6) under the logarithmic source condition in Equation (7) with 1 ≤ p ≤ 2 is obtained
Summary
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. 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]. An inverse potential problem can be formulated via a nonlinear operator equation 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.
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