Subgradient-based Lavrentiev regularisation of monotone ill-posed problems

TL;DR

This paper introduces a subgradient-based Lavrentiev regularisation method for monotone ill-posed problems, enabling real-time solutions without adjoint operators, with proven convergence and practical applications in PDEs and integral equations.

Contribution

It presents a novel regularisation approach that avoids adjoint operators, suitable for time-causal problems, with a comprehensive theoretical framework and practical applications.

Findings

Established well-posedness in Banach spaces

Proved convergence rates under source conditions

Demonstrated effectiveness in PDE and integral equation problems

Abstract

We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + \alpha \partial \mathcal{R}(u) \ni f^\delta \end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs.