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

Lang, Annika; Pereira, Mike

doi:10.48550/arxiv.2107.02667

Cited by 3 publications

(3 citation statements)

References 40 publications

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“…The latter may be particularly suitable to mitigate the effects of boundary conditions (see [15]). Second, the techniques in this paper may be extended to generalized Matérn fields on compact Riemannian fields, see [24,23,28,22]. Finally, in Bayesian inverse problems, it is straightforward to extend the techniques to nonlinear forward problems within a Newton-based solver for the MAP estimate [31].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The paper in [18] also works in a similar setting as ours in bounded domains but solves the linear systems in parallel using multilevel preconditioned iterative solvers with linear complexity in the degrees of freedom; additionally, their approach is also applicable to compact metric spaces. Also relevant to this discussion are [22,20] which deal with sampling from random fields on compact Riemannian manifolds. Our efficient approach for representing covariance matrices and sampling from the random field may be applicable to that setting as well.…”

Section: Related Workmentioning

confidence: 99%

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Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Antil¹,

Saibaba²

2022

Preprint

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This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle-Matérn Gaussian random fields. The Whittle-Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including non-stationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen-Loève expansion. We show how to incorporate this prior representation into the infinitedimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the techniques developed here are applicable to solving systems with fractional Laplacians and Gaussian random fields. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Antil¹,

Saibaba²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The extension of the Matérn field based on SPDEs to space-time is provided by Cameletti et al [39] and subsequently by Bakka et al [15], Clarotto et al [42], while the multivariate Matérn case has been explored in Bolin and Wallin [33]. Alternative approximations based on Galerkin methods on manifolds have been provided by Lang and Pereira [92]. An interesting approach that allows working on manifolds with huge datasets is proposed by Pereira et al [121].…”

Section: Implementation As An Spdementioning

confidence: 99%

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

Porcu¹,

Bevilacqua²,

Schaback³

et al. 2023

Preprint

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The Matérn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Matérn model has been central to disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Matérn-based journey across these disciplines. First, we reflect on the importance of the Matérn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Matérn model has been influential. Then, we position the Matérn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Matérn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.

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Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields

Jansson¹,

Kovács²,

Lang³

2021

Preprint

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Spherical Matérn-Whittle Gaussian random fields are considered as solutions to fractional elliptic stochastic partial differential equations on the sphere. Approximation is done with surface finite elements. While the non-fractional part of the operator is solved by a recursive scheme, a quadrature of the Dunford-Taylor integral representation is employed for the fractional part. Strong error analysis is performed, obtaining polynomial convergence in the white noise approximation, exponential convergence in the quadrature, and quadratic convergence in the mesh width of the discretization of the sphere. Numerical experiments for different choices of parameters confirm the theoretical findings.

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

Cited by 3 publications

References 40 publications

Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

Efficient algorithms for Bayesian Inverse Problems with Whittle--Matérn Priors

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields

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