A New Type of Ill-Posed and Inverse Problems for Parabolic Equations

TL;DR

This paper introduces a new class of ill-posed inverse problems for parabolic equations based on finite difference approximations of time derivatives, addressing open questions and developing a convergent numerical method for epidemic monitoring.

Contribution

It proposes the t-finite difference framework for parabolic inverse problems, solving longstanding open questions and providing a globally convergent numerical method for epidemic monitoring.

Findings

Introduced the t-finite difference framework (TFD) for parabolic inverse problems.

Addressed three long-standing open questions within the TFD framework.

Developed and proved the global convergence of a numerical method for epidemic monitoring.

Abstract

The time dependent experimental data are always collected at discrete grids with respect to the time t. The step size h of such a grid is always separated from zero by a certain positive number. The same is true for all computations, which are always done on discrete grids with their grid step sizes being not too small. These applied considerations prompt us to introduce a new type of Ill-Posed Problems and Coefficient Inverse Problems (CIP)for parabolic equations. In these problems the t-derivatives of corresponding parabolic operators are written in finite differences with the grid step size being separated from zero. We call this the "t-finite difference framework" (TFD). We address three long standing open questions within the TFD framework. Finally, a numerical method is developed for the CIP of monitoring epidemics. The global convergence of this method is proven.