Multiple point evaluation on combined tensor product supports

Authors:
R. Hiptmair;G. Phillips;G. Sinha
Affiliations:
SAM, ETH Zürich, Zürich, Switzerland 8092;Neue Kantonsschule Aarau, Aarau, Switzerland 5000;Department of Mathematics, California Institute of Technology, Pasadena, USA 91125
Venue:
Numerical Algorithms
Year:
2013

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Abstract

We consider the multiple point evaluation problem for an n-dimensional space of functions [驴驴驴1,1[ d 驴驴 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驴驴驴n point evaluations of a function in this space with computational cost O(mlog d n). We resort to nested segment tree data structures built in a preprocessing stage with an asymptotic effort of O(nlog d驴驴驴1 n).