On regularization methods based on dynamic programming techniques

TL;DR

This paper explores the link between regularization for inverse problems and dynamic programming, introducing two new methods with proven stability and convergence, demonstrated through numerical experiments.

Contribution

It develops two novel regularization methods based on dynamic programming, providing theoretical analysis and convergence rates for solving ill-posed linear inverse problems.

Findings

Both methods are proven to be regularization techniques with convergence guarantees.

Numerical experiments confirm the theoretical convergence rates.

The methods effectively stabilize solutions to integral operator inverse problems.

Abstract

In this article we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of these methods is to obtain stable approximations to the solution of linear inverse ill-posed problems. We follow two different approaches and derive a continuous and a discrete regularization method. Regularization properties for both methods are proved as well as rates of convergence. A numerical benchmark problem concerning integral operators with convolution kernels is used to illustrate the theoretical results.