Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs*
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs*
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DOI：
10.1017/s0962492911000055
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发表时间：
2011-04
期刊：
影响因子：
14.2
通讯作者：
C. Schwab;C. J. Gittelson
中科院分区：
文献类型：
--
作者：
C. Schwab;C. J. Gittelson
随机输入数据，如随机载荷和系数的偏微分方程（PDE），重新制定为参数，确定性偏微分方程的参数空间的高，可能是无限维。推导出其随机解的空间和时间k点相关函数的张量化算子方程。参数，确定性偏微分方程的随机解的法律。建立了无限维参数空间中随机解的“广义多项式混沌”（GPC）级数表示。介绍了有关这些参数偏微分方程解的规律性的最新结果。最好的N项近似，自适应随机Galerkin和配置离散的参数，确定性偏微分方程的收敛速度，建立。稀疏张量积的分层（多级）离散化在物理空间（和时间），GPC扩展在参数空间中，收敛速度是独立的参数空间的维度。对随机系数偏微分方程的多层蒙特卡罗（MLMC）离散化进行了收敛性分析。对于具有随机系数的线性椭圆、抛物和双曲偏微分方程，建立了稀疏张量离散优于MLMC离散的随机输入充分条件。
Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.