Low-Rank Tensor Approximation for High-Order Correlation Functions of Gaussian Random Fields

Kreßner, Daniel; Kumar, Rajesh; Nobile, Fabio; Tobler, Christine

doi:10.1137/140968938

Cited by 14 publications

(12 citation statements)

References 24 publications

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“…Once the second order statistics of the Gaussian random field solution u in (1.1) have been determined, by its gaussianity all higher moments are determined. Given the second order statistics, tensor-structured linear algebra techniques for the efficient deterministic numerical evaluation of these higher moments are available; see, for example, [30]. In the present work, we therefore focus on the computation of the second order statistics of the Gaussian random field solution u in (1.1).…”

Section: Introductionmentioning

confidence: 99%

Covariance regularity and $$\mathcal {H}$$ H -matrix approximation for rough random fields

2016

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In an open, bounded domain D ⊂ R n with smooth boundary ∂D or on a smooth, closed and compact, Riemannian n-manifold M ⊂ R n+1 , we consider the linear operator equation Au = f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order r ∈ R with analytic coefficients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, f ∈ L 2 (Ω; H t) is an H t-valued random field with finite second moments, with H t denoting the (isotropic) Sobolev space of (not necessarily integer) order t modelled on the domain D or manifold M, respectively. We prove that the random solution's covariance kernel Ku = (A −1 ⊗A −1)K f on D×D (resp. M×M) is an asymptotically smooth function provided that the covariance function K f of the random data is a Schwartz distributional kernel of an elliptic pseudodifferential operator. As a consequence, numerical H-matrix calculus allows deterministic approximation of singular covariances Ku of the random solution u = A −1 f ∈ L 2 (Ω; H t−r) in D × D with work versus accuracy essentially equal to that for the mean field approximation in D, overcoming the curse of dimensionality in this case.

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Section: Introductionmentioning

confidence: 99%

Covariance regularity and $$\mathcal {H}$$ H -matrix approximation for rough random fields

2016

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“…This can be made more efficient by similar techniques used for the covariance matrix factorization (see e.g. [36,37,50]). Alternative ideas for an efficient computation of the KL expansion are based on domain decomposition [15], randomized linear algebra [48], or domain modification [45].…”

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confidence: 99%

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Khristenko¹,

Scarabosio²,

Swierczynski³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

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We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole R d , without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

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“…Although L1MC-RF can outperform others in results presented so far, it has limitation on matrices with exponentially decaying singular values, which can be found in certain real world applications [72]. Such singular value distribution makes truncation in L1MC-RF more difficult while GeomPursuit performs much better by design.…”

Section: ) Limitation On Matrices With Exponentially Decaying Singulmentioning

confidence: 99%

Rank-One Matrix Completion With Automatic Rank Estimation via L1-Norm Regularization

Shi

Cheung

2018

IEEE Trans. Neural Netw. Learning Syst.

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Completing a matrix from a small subset of its entries, i.e., matrix completion is a challenging problem arising from many real-world applications, such as machine learning and computer vision. One popular approach to solve the matrix completion problem is based on low-rank decomposition/factorization. Low-rank matrix decomposition-based methods often require a prespecified rank, which is difficult to determine in practice. In this paper, we propose a novel low-rank decomposition-based matrix completion method with automatic rank estimation. Our method is based on rank-one approximation, where a matrix is represented as a weighted summation of a set of rank-one matrices. To automatically determine the rank of an incomplete matrix, we impose L1-norm regularization on the weight vector and simultaneously minimize the reconstruction error. After obtaining the rank, we further remove the L1-norm regularizer and refine recovery results. With a correctly estimated rank, we can obtain the optimal solution under certain conditions. Experimental results on both synthetic and real-world data demonstrate that the proposed method not only has good performance in rank estimation, but also achieves better recovery accuracy than competing methods.

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Low-Rank Tensor Approximation for High-Order Correlation Functions of Gaussian Random Fields

Cited by 14 publications

References 24 publications

Covariance regularity and $$\mathcal {H}$$ H -matrix approximation for rough random fields

Covariance regularity and $$\mathcal {H}$$ H -matrix approximation for rough random fields

Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields

Rank-One Matrix Completion With Automatic Rank Estimation via L1-Norm Regularization

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