Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Adisorn Kittisopaporn,Pattrawut Chansangiam

doi:10.1186/s13662-020-02785-9

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Adisorn Kittisopaporn, Pattrawut Chansangiam

Open Access

https://doi.org/10.1186/s13662-020-02785-9

Copy DOI

Abstract

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form sum_{t=1}^{p}A_{t}XB_{t}=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

Highlights

Linear matrix equations have played a crucial role in control theory and differential equations; see, e.g., [1,2,3,4]
There was much attention given to the following matrix equations: the equation AXB = C, the Sylvester equation AX +XB = C, the Kalman–Yakubovich equation AXB + X = C, and, more generally, the equation AXB + CXD = F
We find that Algorithm 4.1 gives the fastest convergence

Summary

Introduction

Linear matrix equations have played a crucial role in control theory and differential equations; see, e.g., [1,2,3,4]. Proposition 1.2 ([33]) If the equation AXB = C has a unique solution X∗, the gradient-based iterative (GI) algorithm, X(k + 1) = X(k) + μAT C – AX(k)B BT ,. We introduce a gradient-descent iterative algorithm for solving the generalized Sylvester equation that takes the form p. Note that this equation includes all mentioned matrix equations as special cases. We shall propose a new iterative method to solve (2) based on gradients and the steepest descend which provides an appropriate sequence of convergent factors for minimizing an error at each iteration.

Proposed algorithm

Method

Conclusion