Preprint 36/2008

Fast and Accurate Tensor Approximation of Multivariate Convolution with Linear Scaling in Dimension

Boris N. Khoromskij

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Submission date: 22. Apr. 2008 (revised version: May 2009)
Pages: 28
published in: Journal of computational and applied mathematics, 234 (2010) 11, Special Issue, p. 3122-3139 
DOI number (of the published article): 10.1016/j.cam.2010.02.004
MSC-Numbers: 65F30, 65F50, 65F35
Keywords and phrases: multi-dimensional convolution, Tucker decomposition, composite grids, Richardson extrapolation
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Abstract:
In the present paper we present the tensor-product approximation of multi-dimensional convolution transform discretized via collocation-projection scheme on the uniform or composite refined grids. Examples of convolving kernels are given by the classical Newton, Slater (exponential) and Yukawa potentials, formula15, formula17 and formula19 with formula21. For piecewise constant elements on the uniform grid of size formula23, we prove the quadratic convergence formula25 in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to formula29. The fast algorithm of complexity formula31 is described for tensor-product convolution on the uniform/composite grids of size formula23, where formula35 are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker-canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order formula25 and formula29; (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on formula43 grid in the range formula45.

19.04.2013, 01:42