Carleman estimates for the regularization of ill-posed Cauchy problems

TL;DR

This survey discusses how Carleman estimates can be used to regularize ill-posed Cauchy problems for linear PDEs, including convergence rates and extensions to nonlinear inverse problems.

Contribution

It highlights the generation of Tikhonov functionals via unbounded operators for PDEs where Carleman estimates hold, and discusses extensions to nonlinear inverse problems.

Findings

Tikhonov functionals can be generated by unbounded operators for PDEs with Carleman estimates

Convergence rates of minimizers are established using Carleman estimates

Extensions to nonlinear inverse problems are discussed

Abstract

This is a survey, which is a continuation of the previous survey of the author about applications of Carleman estimates to Inverse Problems, J. Inverse and Ill-Posed Problems, 21, 477-560, 2013. It is shown here that Tikhonov functionals for some ill-posed Cauchy problems for linear PDEs can be generated by unbounded linear operators of those PDEs. These are those operators for which Carleman estimates are valid, e.g. elliptic, parabolic and hyperbolic operators of the second order. Convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are discussed as well.