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International Conference Approximation Theory

AT 2016: Approximation Theory XV: San Antonio 2016 pp 375–398Cite as

Kernel-Based Approximation Methods for Partial Differential Equations: Deterministic or Stochastic Problems?

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 201))

Abstract

In this article, we present the kernel-based approximation methods to solve the partial differential equations using the Gaussian process regressions defined on the kernel-based probability spaces induced by the positive definite kernels. We focus on the kernel-based regression solutions of the multiple Poisson equations. Under the kernel-based probability measures, we show many properties of the kernel-based regression solutions including approximate formulas, convergence, acceptable errors, and optimal initialization. The numerical experiments show good results for the kernel-based regression solutions for the large-scale data.

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Acknowledgements

I would like to express my gratitude to the grant of the “Thousand Talents Program” for junior scholars of China, the grant of the Natural Science Foundation of China (11601162), and the grant of South China Normal University (671082, S80835, and S81031).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, 510631, China

    Qi Ye

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  1. Qi Ye
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Corresponding author

Correspondence to Qi Ye .

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Editors and Affiliations

  1. Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado, USA

    Gregory E. Fasshauer

  2. Department of Mathematics, Vanderbilt University , Nashville, Tennessee, USA

    Larry L. Schumaker

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Ye, Q. (2017). Kernel-Based Approximation Methods for Partial Differential Equations: Deterministic or Stochastic Problems?. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XV: San Antonio 2016. AT 2016. Springer Proceedings in Mathematics & Statistics, vol 201. Springer, Cham. https://doi.org/10.1007/978-3-319-59912-0_19

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