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The stochastic partial differential equation approach to Gaussian processes (GPs) represents Matern GP priors in terms of $n$ finite element basis functions and Gaussian coefficients with sparse precision matrix. Such representations enhance the scalability of GP regression and classification to datasets of large size $N$ by setting $napprox N$ and exploiting sparsity. In this paper we reconsider the standard choice $n approx N$ through an analysis of the estimation performance. Our theory implies that, under certain smoothness assumptions, one can reduce the computation and memory cost without hindering the estimation accuracy by setting $n ll N$ in the large $N$ asymptotics. Numerical experiments illustrate the applicability of our theory and the effect of the prior lengthscale in the pre-asymptotic regime.
In this work we introduce a reduced-rank algorithm for Gaussian process regression. Our numerical scheme converts a Gaussian process on a user-specified interval to its Karhunen-Lo`eve expansion, the $L^2$-optimal reduced-rank representation. Numeric
We present a performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation. This addresses a problem arising when one seeks solutions to semi-definite progra
The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physi
Series expansions of isotropic Gaussian random fields on $mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Lo`eve expansions of iso
In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as forest-of-trees. Th