Discretized Tikhonov regularization for Robin boundaries localization

TL;DR

This paper presents a novel method combining linearization and Tikhonov regularization to accurately localize unknown boundary defects in materials using boundary measurements, with practical parameter choice and numerical validation.

Contribution

It introduces a discretized Tikhonov regularization approach for Robin boundary localization, including a parameter selection strategy and numerical demonstrations.

Findings

Effective boundary defect localization demonstrated

Regularization parameter chosen via balancing principle

Numerical tests confirm method efficiency

Abstract

We deal with a boundary detection problem arising in nondestructive testing of materials. The problem consists in recovering an unknown portion of the boundary, where a Robin condition is satisfied, with the use of a Cauchy data pair collected on the accessible part of the boundary. We combine a linearization argument with a Tikhonov regularization approach for the local reconstruction of the unknown defect. Moreover, we discuss the regularization parameter choice by means of the so called balancing principle and we present some numerical tests that show the efficiency of our method.