Scholarlink

<p>We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the co-variance kernel by a positive mixture of squared-exponentials, fitted via Newton optimization with a theoretically informed initialization; we also provide convergence estimates for this Gaussian-mixture approximation. The resulting separable kernel enables a Legendre-Galerkin discretization with a Kronecker product structure across dimensions, together with submatrices exhibiting even/odd parity. For assembly, we employ a Duffy-type transformation followed by Gaussian quadrature. These structural properties substantially reduce memory usage and arithmetic cost compared with naive formulations. All algorithms and numerical experiments are released in an open-source repository that reproduces every figure and table. For completeness, a concise derivation of the three-term recurrence for Legendre polynomials is included in appendix.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Efficient Karhunen-Loève expansions via Legendre-Galerkin discretization and tensor structure

  • 期刊论文

  • 发表日期：

    2025年12月5日

查看全文
  • Michal Béreš

摘要

We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the co-variance kernel by a positive mixture of squared-exponentials, fitted via Newton optimization with a theoretically informed initialization; we also provide convergence estimates for this Gaussian-mixture approximation. The resulting separable kernel enables a Legendre-Galerkin discretization with a Kronecker product structure across dimensions, together with submatrices exhibiting even/odd parity. For assembly, we employ a Duffy-type transformation followed by Gaussian quadrature. These structural properties substantially reduce memory usage and arithmetic cost compared with naive formulations. All algorithms and numerical experiments are released in an open-source repository that reproduces every figure and table. For completeness, a concise derivation of the three-term recurrence for Legendre polynomials is included in appendix.