Convergence Rates of a Regularized Newton Method in Sound-Hard Inverse Scattering

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

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A Regularized Newton Method with Correction for Unconstrained Nonconvex Optimization

Journal of Mathematics Research ◽

10.5539/jmr.v7n2p7 ◽

2015 ◽

Vol 7 (2) ◽

pp. 7 ◽

Cited By ~ 2

Author(s):

Heng Wang ◽

Mei Qin

Keyword(s):

Newton Method ◽

Convergence Property ◽

Hessian Matrix ◽

Local Error ◽

Cubic Convergence ◽

Local Error Bound ◽

Lipschitz Continuous ◽

Nonconvex Function ◽

Global Convergence Property ◽

Regularized Newton Method

In this paper, we present a modified regularized Newton method for minimizing a nonconvex function whose Hessian matrix may be singular. We show that if the gradient and Hessian of the objective function are Lipschitz continuous, then the method has a global convergence property. Under the local error bound condition which is weaker than nonsingularity, the method has cubic convergence.

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The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2018-0093 ◽

2019 ◽

Vol 27 (4) ◽

pp. 539-557

Author(s):

Barbara Kaltenbacher ◽

Andrej Klassen ◽

Mario Luiz Previatti de Souza

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Newton Method ◽

Noise Level ◽

Numerical Experiments ◽

Convergence Rates ◽

A Priori ◽

Discrepancy Principle ◽

A Posteriori ◽

Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

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