arXiv subjects

Gennadij Heidel

Publications and source records attributed to Gennadij Heidel.

3 recordsLinked to original sources

arXiv2020

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 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 $n$.

math.NA↗

arXiv2017

A Riemannian trust-region method for low-rank tensor completion

The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low-rank constraint. This may be formulated as a least-squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure, thus methods of Riemannian optimization are applicable. In our work, we derive the Riemannian Hessian of an objective function on the low-rank tensor manifolds using the Weingarten map, a concept from differential geometry. We discuss the convergence properties of Riemannian trust-region methods based on the exact Hessian and standard approximations, both theoretically and numerically. We compare our approach to Riemannian tensor completion methods from recent literature, both in terms of convergence behaviour and computational complexity. Our examples include the completion of randomly generated data with and without noise and recovery of multilinear data from survey statistics.

math.NA↗

arXiv2015

Preconditioning for boundary control problems in incompressible fluid dynamics

PDE-constrained optimization is a field of numerical analysis that combines the theory of PDEs, nonlinear optimization and numerical linear algebra. Optimization problems of this kind arise in many physical applications, prominently in incompressible fluid dynamics. In recent research, efficient solvers for optimization problems governed by the Stokes and Navier--Stokes equations have been developed which are mostly designed for distributed control. Our work closes a gap by showing the effectiveness of an appropriately modified preconditioner to the case of Stokes boundary control. We also discuss the applicability of an analogous preconditioner for Navier--Stokes boundary control and provide some numerical results.

math.NA↗