The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
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DOI：
10.1016/j.jcp.2008.07.009
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发表时间：
2008-11
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通讯作者：
J. Foo;X. Wan;G. Karniadakis
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作者：
J. Foo;X. Wan;G. Karniadakis
随机谱方法是一种求解随机参数偏微分方程的数值方法。在这项工作中，我们提出并研究了多元概率配置法（ME-PCM），这是一种广义形式的概率配置法。在ME-PCM中，参数空间被离散化，并在每个单元上指定配点/体积网格。充分和稀疏张量积网格的基础上高斯和Clenshaw-Curtis求积规则被认为是。我们证明解析和观察在数值试验中，作为参数空间网格细化，解决方案的收敛速度取决于每个元素的求积规则，只有通过其精确度。此外，还研究了张量积插值的L2误差，并给出了一种自适应算法。数值例子表明自适应ME-PCM，包括低正则性问题和长时间的集成。我们测试的ME-PCM二维Navier-Stokes的例子和随机扩散问题与各种随机输入分布和多达50个维度。虽然ME-PCM的收敛速率在50维中恶化，但平均值和方差的误差比仅使用少量样本的蒙特卡罗方法获得的误差低两个数量级（例如，100）。ME-PCM的计算成本相比，其他方法，包括随机伽辽金，蒙特卡罗和准随机序列方法的成本是有利的。
Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multi-element probabilistic collocation method (ME-PCM), which is a generalized form of the probabilistic collocation method. In the ME-PCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and Clenshaw-Curtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L2error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive ME-PCM are shown, including low-regularity problems and long-time integration. We test the ME-PCM on two-dimensional Navier-Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of ME-PCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasi-random sequence methods.