A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry

We present a novel computational framework for shape optimization problems arising in spectral geometry. The goal in such problems is to identify domains in Rd which are the global optima of certain functions of the spectrum of elliptic operators on the domains. We propose the use of a combined finite element and Bayesian optimization (FEM–BO) framework in this context, and demonstrate the key ideas on two concrete examples. We study the Pólya–Szegö conjecture on polygons, and demonstrate that our proposed framework yields the theoretically proven result for triangles and quadrilaterals, and also provides compelling numerical evidence for the case of pentagons. We next study a variant of this conjecture for the Steklov eigenvalue problem.