A variational technique of mollification applied to backward heat conduction problems

TL;DR

This paper introduces a mollification-based variational regularization method for solving severely ill-posed backward heat conduction problems with fractional Laplacian and time-dependent coefficients, demonstrating optimal convergence and robustness.

Contribution

It develops a simple, effective mollification-based regularization technique with proven convergence rates and an a-posteriori parameter choice rule for backward heat problems.

Findings

Achieves order-optimal convergence rates.

Demonstrates robustness through numerical examples.

Effective in image deblurring applications.

Abstract

This paper addresses a backward heat conduction problem with fractional Laplacian and time-dependent coefficient in an unbounded domain. The problem models generalized diffusion processes and is well-known to be severely ill-posed. We investigate a simple and powerful variational regularization technique based on mollification. Under classical Sobolev smoothness conditions, we derive order-optimal convergence rates between the exact solution and regularized approximation in the practical case where both the data and the operator are noisy. Moreover, we propose an order-optimal a-posteriori parameter choice rule based on the Morozov principle. Finally, we illustrate the robustness and efficiency of the regularization technique by some numerical examples including image deblurring.