Article
| Title: | Efficient Karhunen-Loève expansions via Legendre-Galerkin discretization and tensor structure (English) |
| Author: | Béreš, Michal |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 70 |
| Issue: | 6 |
| Year: | 2025 |
| Pages: | 941-991 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We develop an efficient framework for Karhunen-Loève expansions of isotropic Gaussian random fields on hyper-rectangular domains. The approach approximates the covariance 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. (English) |
| Keyword: | Karhunen-Loève expansion |
| Keyword: | Legendre-Galerkin basis |
| Keyword: | separable covariance |
| Keyword: | Gaussian-mixture approximation |
| Keyword: | tensor structure |
| MSC: | 42C05 |
| MSC: | 60G15 |
| MSC: | 60G60 |
| MSC: | 65D30 |
| MSC: | 65F15 |
| DOI: | 10.21136/AM.2025.0163-25 |
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| Date available: | 2025-12-20T07:03:50Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153230 |
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