Tensor approach to optimal control problems with fractional d-dimensional elliptic operator in constraints

Gennadij Heidel, Venera Khoromskaia, Boris N. Khoromskij, Volker Schulz

Published 2018-09-06Version 1

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 spacial grids. It is based on the rank-structured approximation of the matrix valued functions of the corresponding fractional elliptic operator. The functions of finite element (finite difference) Laplacian on a tensor grid are diagonalized by using the fast Fourier transform (FFT) matrix and then the low rank tensor approximation to the multi-dimensional core diagonal tensor is computed. The existence of low rank canonical approximation to the class of matrix valued functions of the fractional Laplacian is proved based on the sinc quadrature approximation method applied to the integral transform of the generating function. The equation for the control function is solved by the PCG method with the rank truncation at each iteration step where the low Kronecker rank preconditioner is precomputed by using the canonical decomposition of the core tensor for the inverse of system matrix. The right-hand side, the solution, and the governing operator are maintained in the rank-structured tensor format. Numerical tests for the 2D and 3D control problems confirm the linear complexity scaling of the method in the univariate grid size.