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

Abstract

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form p t=1 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.

Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure

“…In the case α ∈ (0, 1], the difference is zero for the homogeneous initial value u(x, 0) = 0, see, e.g., [30]. Another interesting idea is to put the discretized equations into a compact linear system, and then formulate a gradient-descent iterative algorithm to solve the linear system; see, e.g., a treatment for the case of heat and Poisson equations in [19].…”

Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation

“…If we carefully set parameters of GI algorithm, then the generated sequence would converge to the desired solution. In the last five years, many GI algorithms have been introduced; see e.g., GI [22,23], relaxed GI [24], accelerated GI [25], accelerated Jacobi GI [26], modified Jacobi GI [27], gradient-descent algorithm [28], and global generalized Hessenberg algorithm [29]. For LS solutions of Sylvester-type matrix equations, there are iterative solvers, e.g., [30,31].…”