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

04/12/2020
by   Michael V. Klibanov, et al.
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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/19/2020

Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data

This paper is concerned with the inverse scattering problem which aims t...
research
02/04/2020

Convexification for a 1D Hyperbolic Coefficient Inverse Problem with Single Measurement Data

A version of the convexification numerical method for a Coefficient Inve...
research
02/19/2020

A numerical reconstruction algorithm for the inverse scattering problem with backscatter data

This paper is concerned with the inverse scattering problem which aims t...
research
04/23/2021

Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data

To compute the spatially distributed dielectric constant from the backsc...
research
08/03/2019

The tangential cone condition for some coefficient identification model problems in parabolic PDEs

The tangential condition was introduced in [Hanke et al., 95] as a suffi...
research
03/01/2023

Trust your source: quantifying source condition elements for variational regularisation methods

Source conditions are a key tool in variational regularisation to derive...
research
06/20/2022

The Carleman convexification method for Hamilton-Jacobi equations on the whole space

We propose a new globally convergent numerical method to solve Hamilton-...

Please sign up or login with your details

Forgot password? Click here to reset