Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems

TL;DR

This paper introduces an adaptive reduced basis trust region method to efficiently solve parabolic inverse problems, combining online adaptive model reduction with error certification within an IRGNM framework.

Contribution

It develops an online adaptive reduced-order modeling approach that accelerates PDE-based inverse problem solutions while ensuring approximation reliability.

Findings

Significant computational speed-up demonstrated in numerical tests.

Effective integration of adaptive model reduction with trust-region IRGNM.

Applicable to both time-dependent and independent reaction-diffusion systems.

Abstract

We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution [Kartmann et al. Adaptive reduced basis trust region methods for parameter identification problems. Comput. Sci. Eng. 1, 3 (2024)] for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use.