Data Sparse Computation of the Karhunen-Loeve Expansion
Data Sparse Computation of the Karhunen-Loeve Expansion
复制标题
Karhunen-Loeve展开式的数据稀疏计算
DOI：
10.1063/1.2990920
复制
发表时间：
2008
影响因子：
7.3
通讯作者：
A. Litvinenko
中科院分区：
文献类型：
--
作者：
B. Khoromskij;A. Litvinenko
物理过程的现实数学模型包含不确定性。这些模型通常由随机微分方程（SDEs）或带有乘性噪声的随机偏微分方程（SPDE）描述，其中右侧或系数中的不确定性表示为随机场。为了解决一个给定的SPDE数值一个离散的确定性运营商以及随机领域。SPDE的总维数是确定性部分和随机部分维数的乘积。为了用尽可能少的随机变量逼近随机场，但仍然保留基本信息，Karhunen-Loeve展开（KLE）变得重要。随机场的KLE需要求解一个大的特征值问题。通常，它是解决了Krylov子空间方法与稀疏矩阵近似。我们演示了使用低秩和数据稀疏层次矩阵技术来解决这个问题。
Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise, where uncertainties in, e.g. the right‐hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. For approximation of random fields with as few random variables as possible, but still retaining the essential information, the Karhunen‐Loeve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of the low‐rank and data sparse hierarchical matrix technique for solving this problem....