Sequential bi-level regularized inversion with application to hidden reaction law discovery

TL;DR

This paper introduces a sequential bi-level regularization method for solving ill-posed inverse problems governed by nonlinear PDEs, improving convergence and applicability in reaction law discovery.

Contribution

It advances bi-level regularization by sequential initialization, enabling faster convergence, multi-scale effects, and inexact PDE solver usage, with applications to reaction-diffusion problems.

Findings

Accelerated convergence of the bi-level algorithm.

Effective discovery of nonlinear reaction laws.

Validation across multiple reaction-diffusion applications.

Abstract

In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear time-dependent partial differential equations (PDEs). In our recent work, we introduced a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequentially initializing the lower-level problem, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Moreover, by collecting the lower-level trajectory, we uncover an interesting connection to the incremental load method. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. We moreover prove that the proposed tangential cone condition is satisfied.