Tensor numerical method for optimal control problems constrained by an elliptic operator with general rank-structured coefficients

TL;DR

This paper develops tensor numerical methods for efficiently solving high-dimensional optimal control problems constrained by elliptic operators with variable coefficients, achieving linear-quadratic complexity and demonstrating effectiveness through numerical tests.

Contribution

Introduction of tensor-based numerical techniques with adaptive rank truncation for solving control problems involving elliptic operators with low-rank separable coefficients.

Findings

Method scales linearly with grid size n.

Preconditioned iterative schemes are effective for large problems.

Cascadic multigrid reduces PCG iterations significantly.

Abstract

We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a preconditioned iterative method with an adaptive rank truncation for solving the equation for the control function, governed by a sum of the elliptic operator and its inverse $M=A + A^{-1}$, both discretized over large $n^{\otimes d}$, $d=2,3$, spatial grids. Two basic solution schemes are proposed and analyzed. In the first approach, one solves iteratively the initial linear system of equations with the matrix $M$ such that the matrix vector multiplication with the elliptic operator inverse, $y=A^{-1} u,$ is performed as an embedded iteration by using a rank-structured solver for the equation of the form $A y=u$. The second numerical scheme avoids the embedded iteration by reducing the initial equation to an equivalent one with the polynomial system matrix of the form $A^2 +I$. For both schemes, a low Kronecker rank spectrally equivalent preconditioner is constructed by using the corresponding matrix valued function of the anisotropic Laplacian diagonalized in the Fourier basis. Numerical tests for control problems in 2D setting confirm the linear-quadratic complexity scaling of the proposed method in the univariate grid size $n$. Further, we numerically demonstrate that for our low rank solution method, a cascadic multigrid approach reduces the number of PCG iterations considerably, however the total CPU time remains merely the same as for the unigrid iteration.