A Multiscale Method for Model Order Reduction in PDE Parameter Estimation

TL;DR

This paper introduces a multiscale model order reduction technique using MSFV integrated into PDE-constrained optimization to efficiently solve large-scale parameter estimation problems in PDEs, reducing computational costs.

Contribution

The novel integration of MSFV into a PDE-constrained optimization framework with iterative basis updates and a new differentiation approach for reduced models.

Findings

Significant computational savings demonstrated in large-scale problems

Method benefits from parallelization

Effective reduction in PDE solve times

Abstract

Estimating parameters of Partial Differential Equations (PDEs) is of interest in a number of applications such as geophysical and medical imaging. Parameter estimation is commonly phrased as a PDE-constrained optimization problem that can be solved iteratively using gradient-based optimization. A computational bottleneck in such approaches is that the underlying PDEs needs to be solved numerous times before the model is reconstructed with sufficient accuracy. One way to reduce this computational burden is by using Model Order Reduction (MOR) techniques such as the Multiscale Finite Volume Method (MSFV). In this paper, we apply MSFV for solving high-dimensional parameter estimation problems. Given a finite volume discretization of the PDE on a fine mesh, the MSFV method reduces the problem size by computing a parameter-dependent projection onto a nested coarse mesh. A novelty in our work is the integration of MSFV into a PDE-constrained optimization framework, which updates the reduced space in each iteration. We also present a computationally tractable way of differentiating the MOR solution that acknowledges the change of basis. As we demonstrate in our numerical experiments, our method leads to computational savings particularly for large-scale parameter estimation problems and can benefit from parallelization.