Article2003

SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATA

Ivo M. Babuska · Kang-Man Liu · Raul F. Tempone

70 citations0/0 files storedMathematical Models and Methods in Applied Sciences

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Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Model Reduction and Neural Networks

Abstract

We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, ω) is expressed by a finite number NKL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable NKL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, ω) is completely known. When a random function a(x, ω) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

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References

1991Stochastic Finite Elements: A Spectral Approach 2006Stochastic Differential Equations—An Introduction With Applications 2001Solution of stochastic partial differential equations using Galerkin finite element techniques 1994Numerical solution of stochastic differential equations 1996Numerical solution of spectral stochastic finite element systems 1999Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach 1995Probabilistic Structural Mechanics Handbook 1999Statistical analyses of scatterplots to identify important factors in large-scale simulations, 1: Review and comparison of techniques

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