A Reduced Basis Landweber method for nonlinear inverse problems

TL;DR

This paper introduces a novel approach combining the nonlinear Landweber method with adaptive online reduced basis techniques to efficiently solve high-dimensional parameter inverse problems in PDEs, demonstrated on heat conductivity reconstruction.

Contribution

It develops a new method that integrates adaptive online reduced basis updates with the nonlinear Landweber method for high-dimensional inverse PDE problems.

Findings

Reduces computational time for nonlinear inverse problems

Handles high-dimensional parameter spaces effectively

Successfully applied to heat conductivity reconstruction

Abstract

We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and want to couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.