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Multiplicative white noise: The Wick-Malliavin approximation

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Numerical Methods for Stochastic Partial Differential Equations with White Noise

Part of the book series: Applied Mathematical Sciences ((AMS,volume 196))

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Abstract

In this chapter, we consider Wiener chaos expansion (WCE) for elliptic equations with multiplicative noise. Unlike the stochastic collocation methods (SCM), a direct application of WCE will lead to a fully coupled linear system. To sparsify the resulting linear system, we present WCE with the use of Ito-Wick product and an approximation/reduction technique called Wick-Malliavin approximation. Specifically, we consider Wick-Malliavin approximation for elliptic equations with lognormal coefficients and use the Wick product for elliptic equations with spatial white noise as coefficients. Numerical results demonstrate that high-order Wick-Malliavin approximation is efficient even when the noise intensity is relatively large.

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Notes

  1. 1.

    The constant here is usually not the same as in the estimate \(\left \Vert \mathcal{A}^{-1}\mathcal{M}_{k}\right \Vert _{1} \leq C_{k}\left \Vert v\right \Vert _{1}\). Here we use the same constant for simplicity.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts, USA

    Zhongqiang Zhang

  2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    George Em Karniadakis

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  1. Zhongqiang Zhang
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  2. George Em Karniadakis
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Zhang, Z., Karniadakis, G.E. (2017). Multiplicative white noise: The Wick-Malliavin approximation. In: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-57511-7_11

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