Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

TL;DR

This paper analyzes the stability of backward problems for degenerate viscous Hamilton-Jacobi equations and introduces numerical methods for data identification, supported by theoretical stability results and numerical experiments.

Contribution

It provides the first stability analysis for nonlinear degenerate viscous Hamilton-Jacobi equations and develops numerical algorithms for data reconstruction in such settings.

Findings

Conditional stability established via Carleman estimates.

Effective numerical algorithms demonstrated for noisy data.

Numerical tests validate the proposed methods.

Abstract

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.