Journal of Scientific Computing

, Volume 57, Issue 3, pp 439–463 | Cite as

An Efficient Approximate Residual Evaluation in the Adaptive Tensor Product Wavelet Method

  • Sebastian Kestler
  • Rob Stevenson


A wide class of well-posed operator equations can be solved in optimal computational complexity by adaptive wavelet methods. A quantitative bottleneck is the approximate evaluation of the arising residuals that steer the adaptive refinements. In this paper, we consider multi-tree approximations from tensor product wavelet bases for solving linear PDE’s. In this setting, we develop a new efficient approximate residual evaluation. Other than the commonly applied method, that uses the so-called APPLY routine, our approximate residual depends affinely on the current approximation of the solution. Our findings are illustrated by numerical results that show a considerable speed-up.


Adaptive wavelet method Multi-tree tensor product approximation  Optimal computational complexity 

Mathematics Subject Classification (2000)

41A30 41A63 65N30 65Y20 



The authors are grateful to Andreas Rupp from the Institute for Numerical Mathematics at the University of Ulm for providing implementations of \(L_2\)-orthonormal multi-wavelets on the interval.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Numerical MathematicsUniversity of UlmUlmGermany
  2. 2.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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