Seminari del Dipartimento di Matematica

Seminario del 2022

Valeria Simoncini

Matrix equations: from theory to (computational) practice

nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022

analisi numerica

The numerical solution of possibly large dimensional algebraic linear systems permeates scientific modelling. Often systems with multiple right-hand sides arise, whose efficient numerical solution usually requires ad-hoc procedures. In the past decades a new class of linear equations has shown to be the natural algebraic framework in the discretization of mathematical models in a variety of scientific applications. These problems are given by multiterm linear matrix equations of the form A_1 X B_1 + A_2 X B_2 + ... + A_k X B_k = C where all appearing terms are matrices of conforming dimensions, and X is an unknown matrix. The case k=2 is called the Sylvester equation, and computational methods for its solution are well established, especially for small dimensions. The general multiterm case turns out to be a key ingredient in problems such as time-space, stochastic and parametric partial differential equations. Its numerical solution is the current challenge, though little is known also about its algebraic properties. In this lecture we give a gentle introduction to the problem, and discuss various attempts to numerically solve it.