Multilevel quasiseparable matrices in PDE-constrained optimization

TL;DR

This paper introduces a novel approach using multilevel quasiseparable matrices to efficiently solve large-scale PDE-constrained optimization problems, achieving nearly linear computational complexity.

Contribution

It extends quasiseparable matrix techniques to higher-dimensional PDE problems and develops linear-cost preconditioners for saddle-point systems.

Findings

Preconditioners enable nearly linear solution times.

Quasiseparable matrices effectively approximate Green's functions in higher dimensions.

Algorithms outperform traditional methods in large-scale PDE-constrained optimization.

Abstract

Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems of saddle-point type. In this paper we propose and develop a novel approach to solving such systems by exploiting so-called quasiseparable matrices. One may think of a usual quasiseparable matrix as of a discrete analog of the Green's function of a one-dimensional differential operator. Nice feature of such matrices is that almost every algorithm which employs them has linear complexity. We extend the application of quasiseparable matrices to problems in higher dimensions. Namely, we construct a class of preconditioners which can be computed and applied at a linear computational cost. Their use with appropriate Krylov methods leads to algorithms of nearly linear complexity.