Sparse Multi-Level Approximations for PDE-Constrained Optimization Under Uncertainty

Explore advanced computational methods for solving PDE-constrained optimization problems under uncertainty in this 58-minute conference talk. Learn about the challenges of minimizing expected values of functionals constrained by random partial differential equations, which traditionally require extremely high computational costs. Discover a novel framework that uses multilevel and sparse quadrature formulae while preserving the properties of the original optimization problem, unlike existing approaches that may destroy convexity through negative quadrature weights. Examine the innovative approach of solving sequences of optimization problems with different discretization levels for both physical and probability spaces, followed by a postprocessing step that combines adjoint variables from multiple levels to obtain the final control approximation. Gain insights into the complete convergence analysis for multilevel quadrature formulae and review numerical experiments that demonstrate improved computational complexity beyond theoretical assumptions.