Space-time finite element methods for the initial temperature reconstruction

TL;DR

This paper develops a space-time finite element approach combined with Tikhonov regularization to reconstruct initial temperature distributions in the backward heat equation, addressing its severe ill-posedness.

Contribution

It introduces a novel application of space-time finite element methods to the inverse heat problem with regularization, enabling stable reconstruction.

Findings

Effective reconstruction of initial temperature from terminal data

Stable solutions achieved through Tikhonov regularization

Application of finite element methods to ill-posed inverse problems

Abstract

This work is devoted to the reconstruction of the initial temperature in the backward heat equation using the space-time finite element method on fully unstructured space-time simplicial meshes proposed by Steinbach (2015). Such a severely ill-posed problem is tackled by the standard Tikhonov regularization method. This leads to a related optimal control for an parabolic equation in the space-time domain. In this setting, the control is taken as initial condition, whereas the terminal observation data serve as target. The objective becomes a standard terminal observation functional combined with the Tikhonov regularization. The space-time finite element method is applied to the space-time optimality system that is well-posed for a fixed regularization parameter.