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Explore a scalable computational method for solving large-scale PDE- and bound-constrained optimization problems in this 30-minute conference talk from the MFEM Workshop 2025. Discover how to address model uncertainty through unknown spatially distributed parameter fields ρ(x) while incorporating bound-constraints like ρ(x)≥ρℓ(x) to introduce additional knowledge such as nonnegativity of diffusivity parameters. Learn about the computational challenges posed by bound-constraints that introduce complementarity conditions into nonlinear optimality systems and understand how a robust, full-space, interior-point method provides an effective solution approach. Examine the implementation of a Gauss-Newton search direction to avoid computational costs associated with regularizing the inertia of linearized optimality system matrices. Analyze two related preconditioned Krylov-subspace solution strategies for linear systems and see how the number of preconditioned Krylov-subspace iterations remains independent of both discretization and the ill-conditioning that typically affects interior-point linear systems. Review parallel scaling results demonstrated through nonlinear elliptic and linear parabolic PDE- and bound-constrained optimization example problems, all implemented using a native computational framework that extensively utilizes MFEM, a scalable C++ finite element library for high-order mathematical calculations in large-scale scientific simulations.
