An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations. (English) Zbl 1388.93110
Summary: Based on the alternating direction method of multipliers (ADMM), we develop three numerical algorithms incrementally for solving the optimal control problems constrained by random Helmholtz equations. First, we apply the standard Monte Carlo technique and finite element method for the random and spatial discretization, respectively, and then ADMM is used to solve the resulting system. Next, combining the multi-modes expansion, Monte Carlo technique, finite element method, and ADMM, we propose the second algorithm. In the third algorithm, we preprocess certain quantities before the ADMM iteration, so that nearly no random variable is in the inner iteration. This algorithm is the most efficient one and is easy to implement. The error estimates of these three algorithms are established. The numerical experiments verify the efficiency of our algorithms.
MSC:
| 93E25 | Computational methods in stochastic control (MSC2010) |
| 90C15 | Stochastic programming |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65K10 | Numerical optimization and variational techniques |
| 65C05 | Monte Carlo methods |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
random optimal control problems; Helmholtz equation; alternating direction method of multipliers; multi-modes representationReferences:
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