Singular operators in multiwavelet bases

Authors:
G. Fann;G. Beylkin;R. J. Harrison;K. E. Jordan
Affiliations:
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831;Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831;Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831;Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Venue:
IBM Journal of Research and Development
Year:
2004

Citing 6
Cited 1

Ten lectures on wavelets

Ten lectures on wavelets
On the representation of operators in bases of compactly supported wavelets

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A class of bases in L2 for the sparse representations of integral operators

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A direct adaptive Poisson solver of arbitrary order accuracy

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A Multiresolution Approach to Regularization of Singular Operators and Fast Summation

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Adaptive solution of partial differential equations in multiwavelet bases

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Fast transform from an adaptive multi-wavelet representation to a partial Fourier representation

Journal of Computational Physics

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Abstract

We review some recent results on multiwavelet methods for solving integral and partial differential equations and present an efficient representation of operators using discontinuous multiwavelet bases, including the case for singular integral operators. Numerical calculus using these representations produces fast O(N) methods for multiscale solution of integral equations when combined with low separation rank methods. Using this formulation, we compute the Hilbert transform and solve the Poisson and Schr脙露dinger equations. For a fixed order of multiwavelets and for arbitrary but finite- precision computations, the computational complexity is O(N). The computational structures are similar to fast multipole methods but are more generic in yielding fast O(N) algorithm development.