Karhunen-Loéve Decomposition of Isotropic Gaussian Random Fields Using a Tensor Approximation of Autocovariance Kernel
Applications of random fields typically require a generation of random samples or their decomposition. In this contribution, we focus on the decomposition of the isotropic Gaussian random fields on a two or three-dimensional domain. The preferred tool for the decomposition of the random field is the Karhunen-Loéve expansion. The Karhunen-Loéve expansion can be approximated using the Galerkin method, where we encounter two main problems. First, the calculation of each element of the Galerkin matrix is expensive because it requires an accurate evaluation of multi-dimensional integral. The second problem consists of the memory requirements, originating from the density of the matrix. We propose a method that overcomes both problems. We use a tensor-structured approximation of the autocovariance kernel, which allows its separable representation. This leads to the representation of the matrix as a sum of Kronecker products of matrices related to the one-dimensional problem, which significantly reduces the storage requirements. Moreover, this representation dramatically reduces the computation cost, as we only calculate two-dimensional integrals.
KeywordsRandom fields sampling Karhunen-Loève decomposition Tensor approximation Numerical integration
This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science - LQ1602”. The work was also partially supported by the project LD15105 “Ultrascale computing in geo-sciences”. The authors were also supported by Grant of SGS No. SP2017/56, VŠB - Technical University of Ostrava, Czech Republic.
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