A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients. (English) Zbl 1307.49026
Summary: In this work, we apply the MG/OPT framework to a multilevel-in-sample-space discretization of optimization problems governed by PDEs with uncertain coefficients. The MG/OPT algorithm is a template for the application of multigrid to deterministic PDE optimization problems. We employ MG/OPT to exploit the hierarchical structure of sparse grids in order to formulate a multilevel stochastic collocation algorithm. The algorithm is provably first-order convergent under standard assumptions on the hierarchy of discretized objective functions as well as on the optimization routines used as pre- and postsmoothers. We present explicit bounds on the total number of PDE solves and an upper bound on the error for one V-cycle of the MG/OPT algorithm applied to a linear quadratic control problem. We provide numerical results that confirm the theoretical bound on the number of PDE solves and show a dramatic reduction in the total number of PDE solves required to solve these optimization problems when compared with standard optimization routines applied to a fixed sparse-grid discretization of the same problem.
MSC:
49M25 | Discrete approximations in optimal control |
49M15 | Newton-type methods |
65K05 | Numerical mathematical programming methods |
65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
90C15 | Stochastic programming |
49N10 | Linear-quadratic optimal control problems |