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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.

Summary 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 with Riemannian tensor completion methods from recent literature, both in terms of convergence behavior and computational complexity. Our examples include the completion of randomly generated data with and without noise and the recovery of multilinear data from survey statistics.

PDE-constrained optimization problems 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.

This paper presents the development and mathematical implementation of a production scheduling model utilizing mixed-integer linear programming (MILP). A simplified model of a real-world multi-product batch plant constitutes the basis. The paper shows practical extensions to the model, resulting in a digital twin of the plant. Apart from sequential arrangement, the final model contains maintenance periods, campaign planning and storage constraints to a limited extend. To tackle weak computational performance and missing model features, a condensed mathematical formulation is introduced at first. After stating that these measures do not suffice for applicability in a restrained time period, a novel solution strategy is proposed. The overall non-iterative algorithm comprises a multi-step decomposition approach, which starts with a reduced scope and incrementally complements the schedule in multiple subproblem stages. Each of those optimizations holds less decision variables and makes use of warmstart information obtained from the predecessor model. That way, a first feasible solution accelerates the subsequent improvement process. Furthermore, the optimization focus can be shifted beneficially leveraging the Gurobi solver parameters. Findings suggest that correlation may exist between certain characteristics of the scheduling scope and ideal parameter settings, which yield potential for further investigation. Another promising area for future research addresses the concurrent multi-processing of independent MILPs on a single machine. First observations indicate that significant performance gains can be achieved in some cases, though sound dependencies were not discovered yet.