MFEM Workshop 2025 | A Scalable Interior-Point Gauss-Newton Method

The MFEM (Modular Finite Element Methods) project provides high-order mathematical calculations for large-scale scientific simulations. Held on September 10-11, 2025, the fifth annual MFEM community workshop brought together users and developers for a review of software features and the development roadmap, a showcase of technical talks and applications, student lightning talks, Q&A, and a visualization contest.

LLNL's Tucker Hartland presented a scalable method for large-scale PDE- and bound-constrained optimization. Such problems are a means to learn unknown aspects of PDE-based models. It is assumed that such model uncertainty is mathematically manifest in an unknown spatially distributed parameter field, ρ(x). Bound-constraints ρ(x)≥ρℓ(x) are a natural means to introduce additional knowledge of an unknown parameter field, e.g., nonnegativity of a diffusivity parameter field. Bound-constraints are, however, the source of additional computational challenges as they introduce complementarity conditions into the nonlinear optimality system. We utilize a robust, full-space, interior-point method to solve the optimization problem. In order to avoid the computational costs required to regularize the inertia of the linearized optimality system matrix, use a Gauss-Newton search direction. He discussed two related preconditioned Krylov-subspace solution strategies for said linear system. He showed that the number of preconditioned Krylov-subspace iterations is independent of not only discretization but also the ill-conditioning that notoriously plagues interior-point linear systems. He concluded with parallel scaling results on a nonlinear elliptic and linear parabolic PDE- and bound-constrained optimization example problems. The results were generated with a native implementation of the computational framework that makes extensive use of MFEM, a scalable C++ finite element library.

Learn more about MFEM at https://mfem.org/. LLNL-PRES-863065