Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

Kanagawa, Motonobu; Sriperumbudur, Bharath K.; Fukumizu, Kenji

doi:10.1007/s10208-018-09407-7

Cited by 43 publications

(62 citation statements)

References 47 publications

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“…See [63,Corollary 11.33] or [64,Proposition 3.6] for earlier and slightly more restricted results that require r > d/2 and [29, Proposition 4] for a version specifically for numerical integration. Some assumptions, satisfied by all domains of interest to us, are needed; see for instance [29,Section 3] for precise definitions. This assumption essentially says that the boundary of Ω is sufficiently regular (Lipschitz boundary) and that there is no "pinch point" on the boundary of Ω (interior cone condition).…”

Section: Kernels Inducing Sobolev-equivalent Rkhssmentioning

confidence: 99%

“…One of our principal aims is to stimulate new research on quadrature weights in the context of probabilistic numerics. While convergence rates of Bayesian quadrature rules have been studied extensively in recent years [10,28,29], analysis of the weights themselves has not attracted much attention. On the other hand, the earliest work [36,55,3] (see [44] for a recent review) done in the 1970s on kernel-based quadrature already revealed certain interesting properties of the Bayesian quadrature weights.…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, we discuss the behaviour of the sum of absolute weights, n i=1 |w BQ X,i |, that strongly affects stability and robustness of Bayesian quadrature. If this quantity is small, the quadrature rule is robust against misspecification of the Gaussian process prior [29] and errors in integrand evaluations [20] and kernel means [60, pp. 298-300].…”

Section: Introductionmentioning

confidence: 99%

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On the positivity and magnitudes of Bayesian quadrature weights

2019

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This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g., Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

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Section: Kernels Inducing Sobolev-equivalent Rkhssmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the positivity and magnitudes of Bayesian quadrature weights

2019

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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].…”

Section: Introductionmentioning

confidence: 99%

“…The mean (green; Equation(29)) and maximal (red; Equation (30)) relative integration errors obtained when simultaneously approximating D global illumination integrals (28) using Bayesian cubature (BC) with random points and fully symmetric multi-output Bayesian cubature (MOBC). Here J = 3 and J = 6 random generator vectors were used to produce a fully symmetric point set of size N = 144 (for J = 3) and N = 288 (for J = 6).…”

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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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Multi‐fidelity data fusion through parameter space reduction with applications to automotive engineering

Romor,

Tezzele,

Mrosek

et al. 2023

Numerical Meth Engineering

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Multi‐fidelity models are of great importance due to their capability of fusing information coming from different numerical simulations, surrogates, and sensors. We focus on the approximation of high‐dimensional scalar functions with low intrinsic dimensionality. By introducing a low dimensional bias we can fight the curse of dimensionality affecting these quantities of interest, especially for many‐query applications. We seek a gradient‐based reduction of the parameter space through linear active subspaces or a nonlinear transformation of the input space. Then we build a low‐fidelity response surface based on such reduction, thus enabling nonlinear autoregressive multi‐fidelity Gaussian process regression without the need of running new simulations with simplified physical models. This has a great potential in the data scarcity regime affecting many engineering applications. In this work we present a new multi‐fidelity approach that involves active subspaces and the nonlinear level‐set learning method, starting from the preliminary analysis previously conducted (Romor F, Tezzele M, Rozza G. Proceedings in Applied Mathematics & Mechanics. Wiley Online Library; 2021). The proposed framework is tested on two high‐dimensional benchmark functions, and on a more complex car aerodynamics problem. We show how a low intrinsic dimensionality bias can increase the accuracy of Gaussian process response surfaces.

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Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

Cited by 43 publications

References 47 publications

On the positivity and magnitudes of Bayesian quadrature weights

On the positivity and magnitudes of Bayesian quadrature weights

Symmetry exploits for Bayesian cubature methods

Multi‐fidelity data fusion through parameter space reduction with applications to automotive engineering

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