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Calculation of the expectation of the solution of a one-dimensional stochastic PDE using a reduced base. (Calcul de l’espérance de la solution d’une EDP stochastique unidimensionnelle à l’aide d’une base réduite.) (French. Abridged English version) Zbl 1225.60102

Summary: We present an efficient method to approximate the expectation of the response of a one-dimensional elliptic problem with stochastic inputs. In conventional methods, the computational effort and cost of the approximation of the response can be dramatic. Our method presented here is based on the Karhunen-Loève (K-L) expansion of the inverse of the diffusion parameter, allowing us to build a base of random variables in reduced numbers, from which we construct a projected solution. We show that the expectation of this projected solution is a good approximation, and give an a priori error estimate. A numerical example is presented to show the efficiency of this approach.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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