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Explicit solution to the stochastic system of linear algebraic equations \((\alpha _{1}\boldsymbol{A}_{1} + \alpha _{2}\boldsymbol{A}_{2} +\cdots+ \alpha _{m}\boldsymbol{A}_{m})\boldsymbol{x} = \boldsymbol{b}\). (English) Zbl 1143.74051

Summary: This paper presents a novel solution strategy to the stochastic system of linear algebraic equations \((\alpha _{1}\boldsymbol{A}_{1} + \alpha _{2}\boldsymbol{A}_{2} +\cdots+ \alpha _{m}\boldsymbol{A}_{m})\boldsymbol{x} = \boldsymbol{b}\) arising from stochastic finite element modelling in computational mechanics, in which \(\alpha _{i}\) \((i = 1, \dots , m)\) denote random variables, \(\boldsymbol{A}_{i}\) \((i = 1,\dots,m)\) real symmetric deterministic matrices, \(\boldsymbol{b}\) is a deterministic/random vector and \(\boldsymbol{x}\) the unknown random vector to be solved. The system is first decoupled by simultaneously diagonalizing all the matrices \(\boldsymbol{A}_{i}\) via a similarity transformation, and then it is trivial to invert the sum of diagonalized stochastic matrices to obtain the explicit solution of the stochastic equation system. Unless all the matrices \(\boldsymbol{A}_{i}\) share exactly the same eigen-structure, the joint diagonalization can be only 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.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74E35 Random structure in solid mechanics
65F05 Direct numerical methods for linear systems and matrix inversion
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