ERROR ESTIMATES OF STOCHASTIC OPTIMAL NEUMANN BOUNDARY CONTROL PROBLEMS

• Published in: SIAM journal on numerical analysis Vol. 49; no. 3/4; pp. 1532 - 1552
• Main Authors: GUNZBURGER, MAX D., LEE, HYUNG-CHUN, LEE, JANGWOON
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Applied mathematics; Approximation; Boundary conditions; Calculus of variations and optimal control; Control theory; Covariance; Determinism; Error rates; Estimates; Exact sciences and technology; Finite element method; Lagrange multiplier; Lagrange multipliers; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Optimal control; Optimal solutions; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Random variables; Sciences and techniques of general use; 65N15; Error estimation; Differential equation; Existence of solution; Boundary condition; Optimal estimation; Partial differential equation; Non linear equation; Finite element method; Lagrange multiplier; 65K10; Integral equation; 65N30; Stochastic equation; stochastic optimal control; Transcendental equation; Neumann problem; Initial value problem; Discretization method; Equation system; 65C20; Numerical analysis; Boundary value problem; Optimal control; Algebraic equation; Karhunen-Loeve expansion
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/100801731

We study mathematically and computationally optimal control problems for stochastic partial differential equations with Neumann boundary conditions. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary-value type. Mathematically, we prove the existence of an optimal solution and of a Lagrange multiplier; we represent the input data in terms of their Karhunen—Loève expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces.