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Abstract
Kernel quadratures and other kernelbased 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.
Original language  English 

Pages (fromto)  A697A720 
Number of pages  24 
Journal  SIAM Journal on Scientific Computing 
Volume  40 
Issue number  2 
DOIs  
Publication status  Published  2018 
MoE publication type  A1 Journal articlerefereed 
Keywords
 numerical integration
 kernel quadrature
 Bayesian quadrature
 reproducing kernel
 Hilbert spaces
 fully symmetric sets
 sparse grids
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 2 Finished

Sequential Monte Carlo Methods for State and Parameter Estimation in Stochastic Dynamic Systems
Särkkä, S., Karvonen, T. & Tronarp, F.
01/01/2016 → 31/08/2016
Project: Academy of Finland: Other research funding

Sequential Monte Carlo Methods for State and Parameter Estimation in Stochastic Dynamic Systems
01/06/2015 → 31/08/2018
Project: Academy of Finland: Other research funding