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

Understand

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).