Approximate multiplication in adaptive wavelet methods

Dana Černá 1  and Václav Finěk 1
  • 1 Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17, Liberec, Czech Republic

Abstract

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2-problem, finding the convergent iteration process for the l 2-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Černá D., Finěk V., Construction of optimally conditioned cubic spline wavelets on the interval, Adv. Comput. Math., 2011, 34(2), 219–25 http://dx.doi.org/10.1007/s10444-010-9152-5

  • [2] Cohen A., Dahmen W., DeVore R., Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75 http://dx.doi.org/10.1090/S0025-5718-00-01252-7

  • [3] Cohen A., Dahmen W., DeVore R., Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245 http://dx.doi.org/10.1007/s102080010027

  • [4] Dahmen W., Wavelet and multiscale methods for operator equations, In: Acta Numer., 6, Cambridge University Press, Cambridge, 1997, 55–228

  • [5] DeVore R.A., Nonlinear approximation, In: Acta Numer., 7, Cambridge University Press, Cambridge, 1998, 51–150

  • [6] Dijkema T.J., Schwab Ch., Stevenson R., An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constr. Approx., 2009, 30(3), 423–455 http://dx.doi.org/10.1007/s00365-009-9064-0

  • [7] Stevenson R., Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal., 2003, 41(3), 1074–1100 http://dx.doi.org/10.1137/S0036142902407988

  • [8] Stevenson R., On the compressibility operators in wavelet coordinates, SIAM J. Math. Anal., 2004, 35(5), 1110–1132 http://dx.doi.org/10.1137/S0036141002411520

OPEN ACCESS

Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.

Search