Numerical Optimization of Partial Differential Equations with Uncertain Coefficients
Many optimization problems in engineering and science are governed by partial differential equations (PDEs) with uncertain parameters. Although such such problems can be formulated as optimization problems in suitable Banach spaces and frameworks for derivative based optimization methods can in principle be applied, the numerical solution of these problems is more challenging than the solution of deterministic PDE constrained optimization problems. If parameters in the PDE are uncertain, the PDE solution is a random field and the numerical solution of the PFDE requires a discretization of the PDE in space/time as well as in the random variables. As a consequence, the size of discretized optimization problems are substantially larger than the already large problem sizes arising in deterministic PDE constrained optimization.
In this talk we discuss the use stochastic collocation methods for the numerical solution of such optimization problems, we explore the structure of this method in gradient and Hessian computations and we present initial approaches to cope with the size of the discretized problems, including adaptivity and model reduction.