Colloquium with Dr. Sven Leyffer, of Argonne National Laboratory.

This Colloquium featured Dr. Sven Leyffer from Argonne National Laboratory. Dr. Leyffer discussed topological design problems, which arise in critical engineering and scientific fields such as additive manufacturing and the design of cloaking devices. He explained how these problems are formulated as massive mixed-integer PDE-constrained optimization (MIPDECO) problems. Despite their apparent complexity, Dr. Leyffer demonstrated that MIPDECOs can be solved efficiently, with a computational cost comparable to solving a single continuous PDE-constrained optimization problem. He highlighted two key solution methods: rounding techniques, which are asymptotically optimal, and trust-region techniques, which converge as mesh refinement occurs. The talk included examples from topology optimization to illustrate these methods in action.