Order reduction of semilinear differential matrix and tensor equations

In this thesis, we are interested in approximating, by model order reduction, the solution to large-scale matrix- or tensor-valued semilinear Ordinary Differential Equations (ODEs). Under specific hypotheses on the linear operators and the considered domain, these ODEs often stem from the space discretization on a tensor basis of semilinear Partial Differential Equations (PDEs) with a dimension greater than or equal to two. The bulk of this thesis is devoted to the case where the discrete system is a matrix equation. We consider separately the cases of general Lipschitz continuous nonlinear functions and the Differential Riccati Equation (DRE) with a quadratic nonlinear term. In both settings, we construct a pair of left-right approximation spaces that leads to a reduced semilinear matrix differential equation with the same structure as the original problem, which can be more rapidly integrated with matrix-oriented integrators. For the DRE, under certain assumptions on the data, we show that a reduction process onto rational Krylov subspaces obtains significant computational and memory savings as opposed to current approaches. In the more general setting, a challenging difference lies in selecting and constructing the two approximation bases to handle the nonlinear term effectively. In addition, the nonlinear term also needs to be approximated for efficiency. To this end, in the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to a practical, structure-aware low order approximation of the original problem. In the final part of the thesis, we consider the multidimensional setting. Here we extend the matrix-oriented POD-DEIM algorithm to the tensor setting and illustrate how we can apply it to systems of such equations. Moreover, we discuss how to integrate the reduced-order model and, in particular, how to solve the resulting tensor-valued linear systems.