Kernel quadrature (KQ) is a kernel-based approach to numerical integration, closely related to Bayesian quadrature (BQ) and probabilistic integration [38, 39, 10]. For sufficiently regular integrands, KQ can exploit spectral structure in a reproducing kernel Hilbert space (RKHS) that is invisible to plain Monte Carlo and thereby converge faster than the usual O(N 1/2) rate in the number of points [3, 28]. Unconstrained kernel-based rules, however, may produce numerically unstable weights, motivating longstanding interest in positively weighted rules [13, 21, 29, 46]. In this paper, positive weights mean nonnegative weights that sum to one, i.e., simplex or convex-combination weights. Whether positive-weight KQ can systematically improve over Monte Carlo is a subtle question. Kernel herding and related constructions suggested fast rates under favorable assumptions [13], but the conditional-gradient viewpoint of Bach et al. [4] clarified that the strongest such assumptions are not generally available in infinite-dimensional RKHSs. Subsequent herding-type analyses in broad RKHS settings have therefore mostly remained at the Monte-Carlo scale, except under additional structure or modified algorithms such as sparse herding variants [31, 44, 43]. Beyond herding, subsampling-based positive KQ methods such as thinning [16, 15] and recombination [21, 24] have obtained rates beyond Monte Carlo, but a general mechanism for such improvement in the simple i.i.d.

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d.

This report examines numerical aspects of constructing Karhunen-Loève expansions (KLEs) for second-order stochastic processes. The KLE relies on the spectral decomposition of the covariance operator via the Fredholm integral equation of the second kind, which is then discretized on a computational grid, leading to an eigendecomposition task. We derive the algebraic equivalence between this Fredholm-based eigensolution and the singular value decomposition of the weight-scaled sample matrix, yielding consistent solutions for both model-based and data-driven KLE construction. Analytical eigensolutions for exponential and squared-exponential covariance kernels serve as reference benchmarks to assess numerical consistency and accuracy in 1D settings. The convergence of SVD-based eigenvalue estimates and of the empirical distributions of the KL coefficients to their theoretical $\mathcal{N}(0,1)$ target are characterized as a function of sample count. Higher-dimensional configurations include a two-dimensional irregular domain discretized by unstructured triangular meshes with two refinement levels, and a three-dimensional toroidal domain whose non-simply-connected topology motivates a comparison between Euclidean and shortest interior path distances between the grid points. The numerical results highlight the interplay between the discretization strategy, quadrature rule, and sample count, and their impact on the KLE results.

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

This paper presents new quadrature rules for functions in a reproducing kernel Hilbert space using nodes drawn by a sampling algorithm known as randomly pivoted Cholesky. The resulting computational procedure compares favorably to previous kernel quadrature methods, which either achieve low accuracy or require solving a computationally challenging sampling problem.

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Thislinkbetween the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error,that depends onthespectrum oftheRKHS kernel.

Integrals appear in manyscientific fields as quantities of interest per se.