Multi-level higher order QMC Galerkin discretization for affine parametric operator equations
Multi-level higher order QMC Galerkin discretization for affine parametric operator equations
复制标题
仿射参数算子方程的多级高阶QMC Galerkin离散化
DOI：
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
C. Schwab
中科院分区：
文献类型：
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作者：
J. Dick;F. Kuo;Q. Gia;C. Schwab
将高阶拟蒙特卡罗(QMC)求积与一般的Petrov-Galerkin离散相结合的多层拟蒙特卡罗(QMC)算法，推广了[emph{F.Y.~Kuo，CH.~Schwab，I.H.~Sloan，一类具有随机系数的椭圆型偏微分方程的多层拟蒙特卡罗有限元方法]中的多层一阶分析和[emph{J.~Dick，F.Y.~Kuo，Q.T.~Le~Gia，D.~Nuyens，]中的单层高阶分析.和Ch.Schwab，参数算子方程的高阶QMC Galerkin离散化}(综述)]。特别地，我们涵盖了非光滑区域上的正定和不定的强椭圆型偏微分方程组，并详细讨论了随机偏微分方程组输入的参数化中特征函数的高阶导数对收敛结果的影响。在此基础上，给出了算法参数的具体选择，以便在最小的计算量下达到预定的精度。确定了多层高阶QMC Petrov-Galerkin算法优于相应的单层算法的问题类别和数据的充分条件。数值实验验证了理论结果。
We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [emph{F.Y.~Kuo, Ch.~Schwab, and I.H.~Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient} (in review)] and the single level higher order analysis in [emph{J.~Dick, F.Y.~Kuo, Q.T.~Le~Gia, D.~Nuyens, and Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations} (in review)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of {KL} eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our emph{a-priori} error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov-Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results.
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影响因子：
2.7
作者：
Giles, Michael B.
通讯作者：
Giles, Michael B.
影响因子：
--
作者：
Cliffe, K. A.;Giles, M. B.;Teckentrup, A. L.
通讯作者：
Teckentrup, A. L.
影响因子：
2.9
作者：
Charrier, J.;Scheichl, R.;Teckentrup, A. L.
通讯作者：
Teckentrup, A. L.
影响因子：
2.1
作者：
Barth, Andrea;Schwab, Christoph;Zollinger, Nathaniel
通讯作者：
Zollinger, Nathaniel