Explicit solution to the stochastic system of linear algebraic equations (α1A1 + α2A2 + ⋯ + αmAm)x = b Original Research Article

This paper presents a novel solution strategy to the stochastic system of linear algebraic equations (α1A1 + α2A2 + ⋯ + αmAm)x = b arising from stochastic finite element modelling in computational mechanics, in which αi (i = 1, … , m) denote random variables, Ai (i = 1, … , m) real symmetric deterministic matrices, b a deterministic/random vector and x the unknown random vector to be solved. The system is first decoupled by simultaneously diagonalizing all the matrices Ai via a similarity transformation, and then it is trivial to invert the sum of the diagonalized stochastic matrices to obtain the explicit solution of the stochastic equation system. Unless all the matrices Ai share exactly the same eigen-structure, the joint diagonalization can only be approximately achieved. Hence, the solution is approximate and corresponds to a particular average eigen-structure of the matrix family. Specifically, the classical Jacobi algorithm for the computation of eigenvalues of a single matrix is modified to accommodate multiple matrices and the resulting Jacobi-like joint diagonalization algorithm preserves the fundamental properties of the original version including its convergence and an explicit solution for the optimal Givens rotation angle. Three numerical examples are provided to illustrate the performance of the method.