A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations

2017/04/20 by Juan Galvis, Galvis, Juan, O. Andrés Cuervo +1 · 1 citation
Decision Sciences · Engineering · Environmental Science · #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Seismic Performance and Analysis #Wind and Air Flow Studies

paper · pdf · doi:10.48550/arxiv.1704.06339

openalex publication_date 2017/04/20 · openalex created_date 2017/05/05 · openalex updated_date 2026/07/28

Abstract

We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.

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