Order reduction of matrix exponentials by proper orthogonal decomposition

Abstract

Many applications of the matrix exponential exp(At) of a real matrix A and a real parameter t require repeated evaluation of it for different values of t. Such evaluations are time consuming and encounter the curse of dimensionality, especially in the large scale cases. Therefore, reduced order versions of the matrix exponential will be of great use and ease in these situations. The proper orthogonal decomposition (POD) method has wide applications in the reduced order modeling of dynamical systems. The matrix exponential can be considered as the solution of an initial value problem (IVP) as a dynamical system and the reduced order approximations to it be obtained by utilizing the POD method. Following this approach, the matrix exponential exp(At) is viewed as a scalar to matrix function and the optimal subspace(s) of its range space for the order reduction is computed by the POD method. Next, the IVP defining the matrix exponential is projected on the obtained optimal subspace(s). Then the solution of projected IVP is adopted as the desired reduced order approximation of the matrix exponential. An upper estimate for the error of the reduced order approximation is obtained thorough an error analysis. Numerical experiments illustrating efficiency of the method and quality of the obtained approximations is provided.