Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs - Analytic Regularity and Generalized Polynomial Chaos Approximation
Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs - Analytic Regularity and Generalized Polynomial Chaos Approximation
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参数和随机抛物型偏微分方程的稀疏张量伽辽金离散 - 解析正则性和广义多项式混沌逼近
DOI：
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发表时间：
2013
影响因子：
2
通讯作者：
C. Schwab
中科院分区：
文献类型：
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作者：
V. H. Hoang;C. Schwab
对于随机系数线性抛物型偏微分方程的初边值问题，证明了其解关于参数的解析性，并给出了Bochner空间尺度上的$N$-项广义多项式混沌逼近的先验误差分析.通过Galerkin投影到参数空间的双支撑多项式组上，将问题化为无穷维参数空间上的一类参数族确定性初边值问题.一致稳定性的支持所产生的耦合抛物型系统的建立。解析的解决方案相对于可数多个参数的建立，和正则性结果的参数的解决方案被证明为兼容以及不兼容的初始数据和源项。目前的结果意味着收敛速度和稳定性稀疏，自适应时空张量积Galerkin离散这些无限维…
For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an a priori error analysis for $N$-term generalized polynomial chaos approximations in a scale of Bochner spaces. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space by Galerkin projection onto finitely supported polynomial systems in the parameter space. Uniform stability with respect to the support of the resulting coupled parabolic systems is established. Analyticity of the solution with respect to the countably many parameters is established, and a regularity result of the parametric solution is proved for both compatible as well as incompatible initial data and source terms. The present results imply convergence rates and stability of sparse, adaptive space-time tensor product Galerkin discretizations of these infinite dimensional...