Abstract
This paper deals with the numerical modeling of differential equations with coefficients in the form of random fields. Using the Karhunen-Lo´eve expansion, we approximate these coefficients as a sum of independent random variables and real functions. This allows us to use the computational probabilistic analysis. In particular, we apply the technique of probabilistic extensions to construct the probability density functions of the processes under study. As a result, we present a comparison of our approach with Monte Carlo method in terms of the number of operations and demonstrate the results of numerical experiments for boundary value problems for differential equations of the elliptic type.
Highlights
Numerical modeling is a tool widely used to predict the behavior of complex systems, as well as to assess risks and make decisions
We apply the technique of probabilistic extensions to construct the probability density functions of the processes under study
We present a comparison of our approach with Monte Carlo method in terms of the number of operations and demonstrate the results of numerical experiments for boundary value problems for differential equations of the elliptic type
Summary
Numerical modeling is a tool widely used to predict the behavior of complex systems, as well as to assess risks and make decisions. Such predictions are obtained by using mathematical models whose solutions describe a phenomenon of interest. Input uncertainties of the model manifest in coefficients, forcing terms, boundary and initial condition data, geometry, etc. The solution of such problems is discussed in Stochastic Finite element methods [2]. We focus on problems with random coefficients For these purposes, we use Computational Probabilistic Analysis (CPA) [3]. When sufficient data are available, we can aggregate them into probability distributions
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