<div class="csl-bib-body">
<div class="csl-entry">Heitzinger, C., Pammer, G., & Rigger, S. (2018). Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Differential Equations. <i>SIAM/ASA Journal on Uncertainty Quantification</i>, <i>6</i>(1), 213–242. https://doi.org/10.1137/17M1125418</div>
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dc.identifier.issn
2166-2525
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/139555
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dc.description.abstract
The numerical solution of stochastic partial differential equations and numerical Bayesian estimation is computationally demanding. If the coefficients in a stochastic partial differential equation exhibit symmetries, they can be exploited to reduce the computational effort. To do so, we show that permutation-invariant functions can be approximated by permutation-invariant polynomials in the space of continuous functions as well as in the space of p-integrable functions defined on r0, 1ss for 1 ď p ă 8. We proceed to develop a numerical strategy to compute cubature formulas that exploit permutation-invariance properties related to multisymmetry groups in order to reduce computational work. We show that in a certain sense there is no curse of dimensionality if we restrict ourselves to multisymmetric functions, and we provide error bounds for formulas of this type. Finally, we present numerical results, comparing the proposed formulas to other integration techniques that are frequently applied to high-dimensional problems such as quasi-Monte Carlo rules and sparse grids.
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dc.description.sponsorship
Fonds zur Förderung der wissenschaftlichen Forschung (FWF)
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dc.language.iso
en
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dc.publisher
SIAM PUBLICATIONS
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dc.relation.ispartof
SIAM/ASA Journal on Uncertainty Quantification
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dc.rights.uri
http://rightsstatements.org/vocab/InC/1.0/
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dc.subject
Cubature formulas
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dc.subject
Multisymmetric polynomials
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dc.subject
Multivariate integration
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dc.subject
Permutation-invariance
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dc.subject
Quadrature
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dc.subject
Stochastic partial differential equation
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dc.title
Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Differential Equations