An Uncertainty-Weighted Asynchronous ADMM Method for Parallel PDE Parameter Estimation

TL;DR

This paper introduces an uncertainty-weighted asynchronous consensus ADMM method for large-scale PDE parameter estimation, improving convergence speed and efficiency in parallel computing environments.

Contribution

It proposes a novel weighting scheme based on solution uncertainty and an asynchronous implementation to enhance the performance of consensus ADMM in PDE parameter estimation.

Findings

Improved early iteration progress with the weighting scheme.

Enhanced time-to-solution in 3D PDE problems.

Effective parallelization for multi-physics applications.

Abstract

We consider a global variable consensus ADMM algorithm for solving large-scale PDE parameter estimation problems asynchronously and in parallel. To this end, we partition the data and distribute the resulting subproblems among the available workers. Since each subproblem can be associated with different forward models and right-hand-sides, this provides ample options for tailoring the method to different applications including multi-source and multi-physics PDE parameter estimation problems. We also consider an asynchronous variant of consensus ADMM to reduce communication and latency. Our key contribution is a novel weighting scheme that empirically increases the progress made in early iterations of the consensus ADMM scheme and is attractive when using a large number of subproblems. This makes consensus ADMM competitive for solving PDE parameter estimation, which incurs immense costs per iteration. The weights in our scheme are related to the uncertainty associated with the solutions of each subproblem. We exemplarily show that the weighting scheme combined with the asynchronous implementation improves the time-to-solution for a 3D single-physics and multiphysics PDE parameter estimation problems.