Parametric PDEs: sparse or low-rank approximations?
Parametric PDEs: sparse or low-rank approximations?
复制标题
参数偏微分方程：稀疏或低秩近似？
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
W. Dahmen
中科院分区：
文献类型：
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作者：
M. Bachmayr;A. Cohen;W. Dahmen
我们考虑一类参数算子方程，其中所涉及的参数可以是确定性的或随机性的。在这两种情况下，我们都关注涉及大量参数的场景。解决高维问题的典型策略是使用基于空间变量和参数变量分离的低秩近似解。一个这样的策略是基于执行稀疏的最佳n项近似的解决方案映射在一个先验选择的系统中的张量积形式的参数变量。这种方法已被广泛分析的情况下，张量积勒让德多项式基地，近似率已建立。本文的目的是调查可以得到什么进一步利用低秩结构，特别是使用优化系统的奇异值分解技术获得的基函数。在理论方面，我们表明，优化的低秩展开可以带来显着或没有改善稀疏多项式展开，这取决于参数问题的类型。在计算方面，我们分析了自适应求解器，在接近最佳的计算成本，这种类型的近似，利用低秩结构以及稀疏的基础扩展。
We consider a class of parametric operator equations where the involved parameters could either be of deterministic or stochastic nature. In both cases we focus on scenarios involving a large number of parameters. Typical strategies for addressing the challenges posed by high dimensionality use low-rank approximations of solutions based on a separation of spatial and parametric variables. One such strategy is based on performing sparse best n-term approximations of the solution map in an a priori chosen system of tensor product form in the parametric variables. This approach has been extensively analyzed in the case of tensor product Legendre polynomial bases, for which approximation rates have been established. The objective of this paper is to investigate what can be gained by exploiting further low rank structures, in particular using optimized systems of basis functions obtained by singular value decomposition techniques. On the theoretical side, we show that optimized low-rank expansions can either bring significant or no improvement over sparse polynomial expansions, depending on the type of parametric problem. On the computational side, we analyze an adaptive solver which, at near-optimal computational cost for this type of approximation, exploits low-rank structure as well as sparsity of basis expansions.
影响因子：
2.9
作者：
Charrier, J.;Scheichl, R.;Teckentrup, A. L.
通讯作者：
Teckentrup, A. L.