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首页> 外文期刊>SIAM Journal on Numerical Analysis >AN ANISOTROPIC SPARSE GRID STOCHASTIC COLLOCATIONMETHOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITHRANDOM INPUT DATA
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AN ANISOTROPIC SPARSE GRID STOCHASTIC COLLOCATIONMETHOD FOR PARTIAL DIFFERENTIAL EQUATIONS WITHRANDOM INPUT DATA

机译：具有随机输入数据的偏微分方程的各向异性稀疏网格随机匹配方法

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摘要

This work proposes and analyzes an anisotropic sparse grid stochastic collocationmethod for solving partial differential equations with random coefficients and forcing terms (inputdata of the model). The method consists of a Galerkin approximation in the space variables and acollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis orGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solutionof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a prioriand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.These procedures seem to be very effective for the problems under study. The proposed methodcombines the advantages of isotropic sparse collocation with those of anisotropic full tensor productcollocation: the first approach is effective for problems depending on random variables which weighapproximately equally in the solution, while the benefits of the latter approach become apparent whensolving highly anisotropic problems depending on a relatively small number of random variables, asin the case where input random variables are Karhunen–Loeve truncations of "smooth" randomfields. This work also provides a rigorous convergence analysis of the fully discrete problem anddemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence inthe preasymptotic regime, with respect to the total number of collocation points. It also showsthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.Numerical examples illustrate the theoretical results and are used to compare this approach withseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensionalproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient andsuperior to all examined methods.
机译：这项工作提出并分析了一种各向异性的稀疏网格随机配置方法，用于求解带有随机系数和强迫项的偏微分方程（模型的输入数据）。该方法包括利用Clenshaw-Curtis或高斯结在稀疏张量积网格上的空间变量中的Galerkin近似和概率空间中的并置。甚至在存在非线性的情况下，搭配方法也可以解决未耦合的确定性问题，就像蒙特卡洛方法一样。这项工作包括先验后验程序，以使稀疏网格的各向异性适应每个给定问题。这些程序对于研究中的问题似乎非常有效。所提出的方法结合了各向同性稀疏搭配和各向异性全张量积搭配的优点：第一种方法对于依赖于在解决方案中权重近似相等的随机变量的问题是有效的，而后一种方法的优点在解决根据相对少量的随机变量，例如输入随机变量是“平滑”随机域的Karhunen-Loeve截断的情况。这项工作还对完全离散的问题进行了严格的收敛性分析，并证明了在渐近状态下（子）指数收敛和在渐近状态下的代数收敛（相对于搭配点总数）。这也表明各向异性近似打破了一系列问题的维数诅咒。数值示例说明了理论结果，并用于将该方法与其他方法进行比较，包括标准的蒙特卡洛方法。特别是，对于中等大小的问题，具有适当选择的各向异性的稀疏网格方法似乎非常有效，并且优于所有检查的方法。

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