A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty

Abstract

We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are non-convex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments.

MG/OPT and MLMC for Robust Optimization of PDEs

“…The first model problem consists of the Laplace equation where the right hand side (the source term) can be controlled at any point in the domain. This is the ubiquitous model problem analyzed in many papers, e.g., [1,10,14,18,29,28,32,35]. In the second problem, the flux at the boundary of the Laplace PDE is to be controlled by the Dirichlet condition at that boundary.…”

“…for a given correction term τ k , whose function it is to remedy the discrepancies between the approximate optimization problems (18). The V-cycle employs some convergent optimization algorithm for problem (19) at level k, one iteration of which is denoted by S k .…”

“…A Proper orthogonal decomposition method in this context has been investigated in [8], and reduced basis methods in, e.g., [10]. Recently, quasi-Monte Carlo methods have been investigated in this context [18]. Many techniques apply some existing (multigrid) solver to a sampled version of the problem.…”

Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements

“…Hence for a fixed parameter ω ∈ Ω, the optimal control problem inherits directly the properties from its deterministic infinite dimensional counterpart. Due to the convexity of the functional and the linearity of the state equation, the problem (2.1)-(2.3) is amenable to a well established existence and uniqueness theory [2,30,46,62]. Its solution is uniquely characterized by the following first-order necessary and sufficient optimality conditions for a fixed outcome ω ∈ Ω.…”

“…This procedure describes the standard Monte Carlo (MC) simulation regularly employed in such problems, [25,49,64]. In the present work, motivated by sensitivity analysis applied to optimal control problems on random geometries and the large amount of computer resources we need, we improve upon the slow convergence of MC simulation considering a Quasi-Monte Carlo (QMC) method, which is a more effective quadrature method for tackling high dimensional stochastic integrals, [2,23,30,35,43].…”