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

• Published in: Journal of computational physics Vol. 424; p. 109865
• Main Authors: Heidel, Gennadij, Khoromskaia, Venera, Khoromskij, Boris N., Schulz, Volker
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.01.2021; Elsevier Science Ltd
• Subjects: Approximation; Cartesian coordinates; Complexity; Computation; Computational physics; Control systems; Eigenvalues; Fast Fourier transformations; Finite difference method; Format; Fourier transforms; Fractional multidimensional Laplacian; Iterative methods; Low Kronecker rank preconditioner; Mathematical analysis; Matrix algebra; Matrix methods; Numerical methods; Operators (mathematics); Optimal control; Optimal control problems; Quadratures; Reduced higher order SVD; Tensor numerical methods; Tensors; Tucker and canonical tensor decompositions; Reduced higher order SVD; Low Kronecker rank preconditioner; Tensor numerical methods; Optimal control problems; Fractional multidimensional Laplacian; Tucker and canonical tensor decompositions
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2020.109865

We introduce the tensor numerical method for solution of the 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 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 spectral 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 pre-computed. 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(nlog⁡n), outperforming the standard FFT based methods of complexity O(n3log⁡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. •Low rank tensor approximation of the discrete fractional multidimensional Laplacian.•Rank-structured preconditioner for iterative solving the equation for control function.•Proof of the existence of low rank tensor decomposition for functions of Laplacian.•The O(nlog⁡n)-solvers for 3D control problems constrained by fractional Laplacian.