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Multiple point evaluation on combined tensor product supports

by R. Hiptmair and G. Phillips and G. Sinha

(Report number 2011-63)

Abstract
We consider the multiple point evaluation problem for an $n$-dimensional space of functions $[-1,1[^{d}\mapsto \bbR$ spanned by $d$-variate basis functions that are the restrictions of simple (say linear) functions to tensor product domains. For arbitrary evaluation points this task is faced in the context of (semi-)Lagrangian schemes using adaptive sparse tensor approximation spaces for boundary value problems in moderately high dimensions. We devise a fast algorithm for performing $m\geq n$ point evaluations of a function in this space with computational cost $O(m\log^{d}n)$. We resort to nested segment tree data structures built in a preprocessing stage with an asymptotic effort of $O(n\log^{d-1}n)$.

Keywords: (Multilevel) segment tree, adaptive sparse tensor product approximation

@Techreport{HPS11_113,
  author = {R. Hiptmair and G. Phillips and G. Sinha},
  title = {Multiple point evaluation on combined tensor product supports},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-63},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-63.pdf },
  year = {2011}
}

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