Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation

Pereira, Mike; Desassis, Nicolas

doi:10.1016/j.spasta.2019.100359

Cited by 10 publications

(8 citation statements)

References 31 publications

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“…In practice though, the order K of the polynomial approximation is set differently, which allows to work with relatively small orders. Pereira and Desassis (2019) suggest to set K by controlling the deviation in distribution between the samples obtained with and without the polynomial approximation. We propose here an approach purely based on the numerical properties of Chebyshev series, and show in the numerical experiments that it allows to limit the approximation order.…”

Section: Convergence Analysismentioning

confidence: 99%

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

Lang

Pereira

2021

Preprint

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A new numerical approximation method for a class of Gaussian random fields on compact Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace-Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin-Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin-Chebyshev approximation are shown and confirmed through numerical experiments.

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Section: Convergence Analysismentioning

confidence: 99%

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

Lang

Pereira

2021

Preprint

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“…Since f only needs to be evaluated at the points corresponding to the eigenvalues λ i,i∈ [d] of Q, it suffices to find a good approximation of B −1 on the interval [λ min , λ max ] whose extremal values can be lower and upper-bounded easily using the Gershgorin circle theorem [37, Theorem 7.2.1]. In the literature [22,44,76], the function f has been built using Chebyshev polynomials [64] which are a family (T k ) k∈N of polynomials defined over [-1,1] by…”

Section: Algorithm 24 Sampler When Q Is a Block Circulant Matrix With...mentioning

confidence: 99%

“…The steps to generate arbitrary Gaussian vectors using this polynomial approximation are listed in Algorithm 3.2. The main drawback of such an approach is the choice of the order of the Chebyshev series K which involves some hand-tuning or additional computationally intensive statistical tests [76].…”

Section: Algorithm 24 Sampler When Q Is a Block Circulant Matrix With...mentioning

confidence: 99%

High-dimensional Gaussian sampling: a review and a unifying approach based on a stochastic proximal point algorithm

Vono¹,

Dobigeon²,

Chainais³

2020

Preprint

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Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stake issue. In past years, multiple methods have been proposed from different communities to tackle this difficult sampling task ranging from iterative numerical linear algebra to Markov chain Monte Carlo (MCMC) approaches. Surprisingly, no complete review and comparison of these methods have been conducted. This paper aims at reviewing all these approaches by pointing out their differences, close relations, benefits and limitations. In addition to this state of the art, this paper proposes a unifying Gaussian simulation framework by deriving a stochastic counterpart of the celebrated proximal point algorithm in optimization. This framework offers a novel and unifying revisit of most of the existing MCMC approaches while extending them. Guidelines to choose the appropriate Gaussian simulation method for a given sampling problem are proposed and illustrated with numerical examples.

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“…Then we use a polynomial trick to efficiently compute the product between these covariance matrices and vectors. Such an approach has been proposed by Dietrich and Newsam (1995) and Pereira and Desassis (2019) to deal with the simulation of Gaussian random fields, and by for instance Higham (2008) and Hammond et al (2011) for dealing with applications involving matrix functions.…”

Section: Outputmentioning

confidence: 99%

A matrix-free approach to geostatistical filtering

Pereira¹,

Desassis²,

Magneron³

et al. 2020

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In this paper, we present a novel approach to geostatistical filtering which tackles two challenges encountered when applying this method to complex spatial datasets: modeling the non-stationarity of the data while still being able to work with large datasets. The approach is based on a finite element approximation of Gaussian random fields expressed as an expansion of the eigenfunctions of a Laplace-Beltrami operator defined to account for local anisotropies. The numerical approximation of the resulting random fields using a finite element approach is then leveraged to solve the scalability issue through a matrix-free approach. Finally, two cases of application of this approach, on simulated and real seismic data are presented.

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Efficient simulation of Gaussian Markov random fields by Chebyshev polynomial approximation

Cited by 10 publications

References 31 publications

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

High-dimensional Gaussian sampling: a review and a unifying approach based on a stochastic proximal point algorithm

A matrix-free approach to geostatistical filtering

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