Classical quadrature rules via Gaussian processes

Karvonen, Toni; Särkkä, Simo

doi:10.1109/mlsp.2017.8168195

Cited by 14 publications

(16 citation statements)

References 103 publications

(138 reference statements)

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“…It is shown in [53] that polynomial-based quadrature rules can be interpreted as Bayesian quadrature in a model with zero mean for a suitably chosen (polynomial) kernel; the optimal n-point set (with n = p+1 for polynomials of degree p) minimizing the posterior variance (4.6) realizes the cubature rule. One may refer to the discussion in Remark B.1 of Appendix B for models that include a linearly parameterized mean: any cubature rules can be interpreted as Bayesian integration; see [52].…”

Section: 2mentioning

confidence: 99%

“…In the same paper, these results are used to show that for any n-point cubature rule there exists n functions h i (•) such that the rule corresponds to Bayesian integration for model (B.1). One may also refer to [53] for the relation between polynomial-based quadrature rules and Bayesian quadrature (for a suitably chosen polynomial kernel) when β in (B.1) is considered as a vector of known constants (for instance, zero), so that the posterior variance is given by (2.11).…”

Section: Conclusion Optimal Designs For Bayesian Integration Of An Umentioning

confidence: 99%

See 1 more Smart Citation

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

Pronzato¹,

Zhigljavsky²

2020

SIAM/ASA J. Uncertainty Quantification

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A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-lling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-lling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive denite kernels dene strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-lling properties.

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Section: 2mentioning

confidence: 99%

Section: Conclusion Optimal Designs For Bayesian Integration Of An Umentioning

confidence: 99%

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

Pronzato¹,

Zhigljavsky²

2020

SIAM/ASA J. Uncertainty Quantification

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“…, p = d + 1 results in a fully symmetric kernel. See [58,30] for some results on how quadrature rules for such kernels are related to classical quadrature rules. This assumption holds, for example, for Ω = [−1, 1] d equipped with the uniform measure and Ω = R d equipped with the Gaussian measure as well as for many other cases of interest.…”

Section: This Class Of Kernels Includes (I) Isotropic Kernels (Ii) Pmentioning

confidence: 99%

Fully Symmetric Kernel Quadrature

Karvonen¹,

Särkkä²

2018

SIAM J. Sci. Comput.

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Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many distinct weights as there are fully symmetric sets and that these weights can be solved from a linear system of equations constructed out of row sums of certain submatrices of the full kernel matrix. We present several numerical examples that show feasibility, both for a large number of nodes and in high dimensions, of the developed fully symmetric kernel quadrature rules. Most prominent of the fully symmetric kernel quadrature rules we propose are those that use sparse grids.

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“…as N → ∞ has been studied in both the well-specified [4,60,7,19,8] and mis-specified [28,29] regimes. Some relationships between the posterior mean estimator and classical cubature methods have been documented in [16,56,30]. In [35,49,32] the Bayes-Sard framework was studied, where it was proposed to incorporate an explicit parametric component [48] into the prior model in order that contextual information, such as trends, can be properly encoded.…”

Section: Introductionmentioning

confidence: 99%

Symmetry exploits for Bayesian cubature methods

2019

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Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, compared to many classical cubature methods, comes at a computational cost which is cubic in the number of evaluations of the integrand. It has been recently observed that fully symmetric point sets can be exploited in order to reduce -in some cases substantially -the computational cost of the standard Bayesian cubature method. This work identifies several additional symmetry exploits within the Bayesian cubature framework. In particular, we go beyond earlier work in considering non-symmetric measures and, in addition to the standard Bayesian cubature method, present exploits for the Bayes-Sard cubature method and the multi-output Bayesian cubature method.

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Classical quadrature rules via Gaussian processes

Cited by 14 publications

References 103 publications

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

Fully Symmetric Kernel Quadrature

Symmetry exploits for Bayesian cubature methods

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