Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints

TL;DR

This paper presents a tensor numerical method that efficiently solves high-dimensional optimal control problems involving fractional Laplacian operators by leveraging low-rank tensor approximations and FFT, significantly reducing computational complexity.

Contribution

The paper introduces a novel tensor-structured approach for solving fractional Laplacian constrained control problems, achieving linear complexity in grid size through low-rank approximations and FFT.

Findings

Achieves $O(n \, log \, n)$ complexity for 3D problems.

Numerical tests confirm linear scaling with grid size.

Outperforms standard FFT-based methods in high dimensions.

Abstract

We introduce the tensor numerical method for solution of the $d$-dimensional optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids. The approach is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional finite difference Laplacian. We solve the equation for the control function, where the system matrix includes the sum of the fractional $d$-dimensional Laplacian and its inverse. The matrix valued functions of discrete Laplace operator on a tensor grid are diagonalized by using the fast Fourier transform (FFT). Then the low rank approximation of the $d$-dimensional tensors obtained by folding of the corresponding large diagonal matrices of eigenvalues are computed, which allows to solve the governing equation for the control function in a tensor-structured format. The existence of low rank canonical approximation to the class of matrix valued functions involved is justified by using the sinc quadrature approximation method applied to the Laplace transform of the generating function. The linear system of equations for the control function is solved by the PCG iterative method with the rank truncation at each iteration step, where the low Kronecker rank preconditioner is precomputed. The right-hand side, the solution vector, and the governing system matrix are maintained in the rank-structured tensor format which beneficially reduces the numerical cost to $O(n\log n)$, outperforming the standard FFT based methods of complexity $O(n^3\log n)$ for 3D case. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size $n$.