| Abstract: |
| Inversion and control of systems under uncertainty pose significant computational challenges, particularly in high-dimensional stochastic environments. In this talk, I will present a rigorous, end-to-end mathematical and computational framework for solving constrained stochastic optimization problems governed by linear operator equations with uncertain coefficients. We will begin by establishing the problem within the setting of Bochner spaces. By adopting a finite-dimensional noise (FDN) assumption, I will demonstrate how to transform the primary optimization problem into a regularized saddle-point framework via a continuous augmented Lagrangian. I will also share our convergence analysis for the discretized system using a finite-dimensional Galerkin approximation, proving that our discrete solutions reliably converge to the continuous minimizer.
To address practical implementation, the second half of the talk will focus on a newly developed Uzawa-type iterative algorithm with guaranteed strong convergence. Finally, I will introduce a fully discrete computational framework utilizing tensor products of spatial and stochastic basis functions. By detailing the explicit assembly of parameter-dependent global block matrices and the derivation of closed-form block gradients and Hessians, this talk will highlight how we can successfully bridge the gap between high-level functional analysis and efficient, robust numerical implementation. |
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