Sparse polynomial approximation for optimal control problems constrained by elliptic PDEs with lognormal random coefficients
This work provides theoretical guarantees for efficient approximation of optimal control under uncertainty, benefiting computational scientists dealing with PDE-constrained optimization in random media.
The paper establishes dimension-independent convergence rates for sparse polynomial approximations of optimal control problems constrained by elliptic PDEs with lognormal random coefficients, and demonstrates numerically that the sparse quadrature error converges faster than Monte Carlo methods.
In this work, we consider optimal control problems constrained by elliptic partial differential equations (PDEs) with lognormal random coefficients, which are represented by a countably infinite-dimensional random parameter with i.i.d. normal distribution. We approximate the optimal solution by a suitable truncation of its Hermite polynomial chaos expansion, which is known as a sparse polynomial approximation. Based on the convergence analysis in \cite{BachmayrCohenDeVoreEtAl2017} for elliptic PDEs with lognormal random coefficients, we establish the dimension-independent convergence rate of the sparse polynomial approximation of the optimal solution. Moreover, we present a polynomial-based sparse quadrature for the approximation of the expectation of the optimal solution and prove its dimension-independent convergence rate based on the analysis in \cite{Chen2018}. Numerical experiments demonstrate that the convergence of the sparse quadrature error is independent of the active parameter dimensions and can be much faster than that of a Monte Carlo method.