A Unified 2D/3D Large Scale Software Environment for Nonlinear Inverse Problems

TL;DR

This paper presents a flexible, scalable software environment for large-scale nonlinear inverse problems constrained by PDEs, combining mathematical fidelity with high performance to facilitate research and industrial applications.

Contribution

It introduces a hierarchical, modular software design that balances mathematical clarity with computational efficiency and includes a novel preconditioner for 3D Helmholtz equations in distributed systems.

Findings

Effective scaling from small to large problems with minimal code changes

Reduces cognitive load by reflecting underlying mathematics

Demonstrates high performance in 2D and 3D test problems

Abstract

Large scale parameter estimation problems are among some of the most computationally demanding problems in numerical analysis. An academic researcher's domain-specific knowledge often precludes that of software design, which results in inversion frameworks that are technically correct, but not scalable to realistically-sized problems. On the other hand, the computational demands for realistic problems result in industrial codebases that are geared solely for high performance, rather than comprehensibility or flexibility. We propose a new software design for inverse problems constrained by partial differential equations that bridges the gap between these two seemingly disparate worlds. A hierarchical and modular design allows a user to delve into as much detail as she desires, while exploiting high performance primitives at the lower levels. Our code has the added benefit of actually reflecting the underlying mathematics of the problem, which lowers the cognitive load on user using it and reduces the initial startup period before a researcher can be fully productive. We also introduce a new preconditioner for the 3D Helmholtz equation that is suitable for fault-tolerant distributed systems. Numerical experiments on a variety of 2D and 3D test problems demonstrate the effectiveness of this approach on scaling algorithms from small to large scale problems with minimal code changes.