A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus

Abstract

In many stochastic partial differential equations (SPDEs) involving random coefficients, modeling the randomness by spatial white noise may lead to ill-posed problems. Here we consider an elliptic problem with spatial Gaussian coefficients and present a methodology that resolves this issue. It is based on stochastic convolution implemented via generalized Malliavin operators in conjunction with weighted Wiener spaces that ensure the ellipticity condition. We present theoretical and numerical results that demonstrate the fast convergence of the method in the proper norm. Our approach is general and can be extended to other SPDEs and other types of multiplicative noise. where ω indicates randomness. Recently, this model has been actively investigated in the context of uncertainty quantification (UQ) for mathematical and computational models, see, e.g., refs. 3 and 4 and the references therein. The diffusion coefficient a(x, ω) is typically modeled by the Karhunen-Loève type expansion a(x, ω) =ā(x) + ε(x). Here,ā(x) is the mean and ε(x) = k≥1 σ k (x)ξ k is the noise term, where σ k (x) are deterministic matrices and ξ := {ξ i } i≥1 is a set of uncorrelated random variables with zero mean and unit variance. For the problem of Eq. 1 to be well posed, the ellipticity condition is required. However, if the noise ε(x) is Gaussian, (or any other type with infinite negative part), the standard ellipticity condition would not hold, no matter how small the variance of ε(x, ω) is. Various modifications of the aforementioned setting have been used to mitigate this problem. For example, Gaussian models were replaced by distributions with finite range (e.g., uniform distribution). We note that from the statistical and, by implication, UQ perspectives, Gaussian perturbations are of paramount importance and should be modeled judiciously.In this article, we propose a well-posed modification of the model of Eq. 1 with Gaussian noise ε = ε(x). More specifically, we replace Eq. 1 by the following SPDE:where "δ ε(x) " denotes the Malliavin divergence operator (MDO) with respect to Gaussian noise ε(x) (see, e.g., refs. 5 and 6 and the next section). The MDO δ ε (ζ) is a stochastic integral and specifically, in the present setting, a convolution of the integrand ζ with the driving Gaussian noise ε. Although replacing the ordinary product by stochastic convolution may be surprising at first, it should be appreciated that physical systems rarely produce an instant response to sharp inputs such as multiplicative forcing.Therefore, in modeling systems subject to sharp perturbations, convolutions often become a model of choice. Historically, stochastic convolution-based models were used to bypass the exceeding singularity of models with (literally understood) multiplicative noise. In fact, the idea of reduction to MDO could be traced back to the pioneering work of K. Itô on stochastic calculus and stochastic differential equations. Specifically, in his seminal work (7), Itô has replaced the "product" modelby the "stochastic convolution" mode...

A note on stochastic elliptic models

“…Such a difference between the Wick product and the regular product stems from the fact that the Wick product should be interpreted from the stochastic integral point of view. The connection of Wick product with Itô-Skorohod integral can be found in [7,11,9,18].…”

“…(27b) and (27c), the coefficients u II, a and u III, a can be solved one-by-one, which can be very efficient. We refer to [1,5,17,16,18,10] for more details about the accuracy and efficiency of the stochastic finite element methods for models (I) and (II). We subsequently summarize some theoretical results for models (I) and (II) with a(x, x) being a log-normal random process.…”

Elliptic equations of higher stochastic order

“…More specifically, using the lower triangular structure and the linearity of the uncertainty propagator and incorporating the estimates of the operator norms, we provide an a priori error estimate for the convergence of spectral/hp finite element method. For equations of stochastic order one, this analysis has been carried out in [22]. The same strategy can be replicated to obtain new a priori error estimates for other numerical methods for solving the propagator.…”

Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

“…In fact, as we will see, the Wick propagator, i.e. the system of equations for the deterministic coefficients of the polynomial chaos expansion of the solution, is quasi-linear and sparse for Wick-type equations, including equations with polynomial Wick nonlinearities [6,9], Wick-multiplicative Gaussian noise [4,8,11], as well as noises represented by nonlinear transformations of Gaussian processes. By 'quasi-linear' here, we mean that only the equation for the mean field is nonlinear, whereas the rest of the system is linear, forming a hierarchy that can be readily solved.…”

“…This issue has been answered only partially in the past, for example, by Wan et al [8], where it was shown numerically that for elliptic SPDEs, the variance of the Wick solution was close to the stochastic solution for small perturbations but deviated greatly for large noise levels. Here, we revisit this issue and also study higher-order Wick-Malliavin approximations to the stochastic Burgers and NS equations.…”