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dc.contributor.authorLitvinenko, Alexander
dc.contributor.authorKeyes, David E.
dc.contributor.authorKhoromskaia, Venera
dc.contributor.authorKhoromskij, Boris N.
dc.contributor.authorMatthies, Hermann G.
dc.identifier.citationLitvinenko A, Keyes D, Khoromskaia V, Khoromskij BN, Matthies HG (2019) Tucker Tensor Analysis of Matérn Functions in Spatial Statistics. Computational Methods in Applied Mathematics 19: 101–122. Available:
dc.description.abstractIn this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format.We apply the Tucker and canonical tensor decompositions to a family of Matérn- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence.We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential O(n)to a linear scaling O(drn), where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, ∥x−y∥.
dc.description.sponsorshipThe research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST).
dc.publisherWalter de Gruyter GmbH
dc.rightsArchived with thanks to Computational Methods in Applied Mathematics
dc.subjectBayesian Update
dc.subjectFourier Transform
dc.subjectGeostatistical Optimal Design
dc.subjectKalman Filter
dc.subjectLoglikelihood Surrogate
dc.subjectLow-Rank Tensor Approximation
dc.subjectMatérn Covariance Hilbert Tensor
dc.titleTucker Tensor Analysis of Matérn Functions in Spatial Statistics
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentExtreme Computing Research Center
dc.identifier.journalComputational Methods in Applied Mathematics
dc.eprint.versionPublisher's Version/PDF
dc.contributor.institutionMax-Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, 39106, , Germany
dc.contributor.institutionMax-Planck Institute for Mathematics in the Sciences, Leipzig, 04103, , Germany
dc.contributor.institutionInstitute of Scientific Computing, TU Braunschweig, Braunschweig, 38106, , Germany
kaust.personLitvinenko, Alexander
kaust.personKeyes, David E.

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