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

We introduce the tensor numerical method for solution of d-dimensional optimal control problems (d=2,3) with spectral fractional Laplacian type operators in constraints discretized on large n?d tensor-product Cartesian grids. The approach is based rank-structured approximation matrix valued functions corresponding finite difference Laplacian. solve equation function, where system includes sum and its inverse. discrete Laplace operator a grid are diagonalized by using fast Fourier transform (FFT). Then low rank tensors obtained folding diagonal matrices eigenvalues computed, which allows to governing function tensor-structured format. existence canonical class involved justified sinc quadrature applied generating function. linear equations solved PCG iterative truncation at each iteration step, Kronecker preconditioner pre-computed. right-hand side, vector, maintained format beneficially reduces cost O(nlog?n), outperforming standard FFT methods complexity O(n3log?n) 3D case. Numerical tests 2D confirm scaling univariate size n.