Skip to main content
SpringerLink
Log in

Adaptive wavelet methods for solving operator equations: An overview

  • Conference paper
  • First Online:
Multiscale, Nonlinear and Adaptive Approximation

Abstract

In [Math. Comp, 70 (2001), 27–75] and [Found. Comput. Math., 2(3) (2002), 203–245], Cohen, Dahmen and DeVore introduced adaptive wavelet methods for solving operator equations. These papers meant a break-through in the field, because their adaptive methods were not only proven to converge, but also with a rate better than that of their non-adaptive counterparts in cases where the latter methods converge with a reduced rate due a lacking regularity of the solution. Until then, adaptive methods were usually assumed to converge via a saturation assumption. An exception was given by the work of Dörfler in [SIAM J. Numer. Anal., 33 (1996), 1106–1124], where an adaptive finite element method was proven to converge, with no rate though.

This work contains a complete analysis of the methods from the aforementioned two papers of Cohen, Dahmen and DeVore. Furthermore, we give an overview over the subsequent developments in the field of adaptive wavelet methods. This includes a precise analysis of the near-sparsity of an operator in wavelet coordinates needed to obtain optimal computational complexity; the avoidance of coarsening; quantitative improvements of the algorithms; their generalization to frames; and their application with tensor product wavelet bases which give dimension independent rates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Allaire and M. Briane. Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh Sect. A, 126(2):297–342, 1996.

    MATH  MathSciNet  Google Scholar 

  2. T. Apel. Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics. B.G. Teubner, Stuttgart, 1999.

    Google Scholar 

  3. T. Barsch. Adaptive Multiskalenverfahren für elliptische partielle Differentialgleigungen – Realisierung, Umsetzung and numerische Ergebnisse. PhD thesis, RTWH Aachen, 2001.

    Google Scholar 

  4. A. Barinka. Fast Evaluation Tools for Adaptive Wavelet Schemes. PhD thesis, RTWH Aachen, March 2005.

    Google Scholar 

  5. A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen, and K. Urban. Adaptive wavelet schemes for elliptic problems - Implementation and numerical experiments. SISC, 23(3):910–939, 2001.

    MATH  MathSciNet  Google Scholar 

  6. A. Barinka, T. Barsch, S. Dahlke, M. Mommer, and M. Konik. Quadrature formulas for refinable functions and wavelets. II. Error analysis. J. Comput. Anal. Appl., 4(4):339–361, 2002.

    MATH  MathSciNet  Google Scholar 

  7. P. Binev and R. DeVore. Fast computation in adaptive tree approximation. Numer. Math., 97(2):193–217, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Barinka, W. Dahmen, and R. Schneider. Fast computation of adaptive wavelet expansions. Numer. Math., 105(4):549–589, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  9. H.J. Bungartz and M. Griebel. Sparse grids. Acta Numer., 13:147–269, 2004.

    Article  MathSciNet  Google Scholar 

  10. S. Berrone and T. Kozubek. An adaptive WEM algorithm for solving elliptic boundary value problems in fairly general domains. SIAM J. Sci. Comput., 28(6):2114–2138 (electronic), 2006.

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Burstedde and A. Kunoth. A wavelet-based nested iteration-inexact conjugate gradient algorithm for adaptively solving elliptic PDEs. Numer. Algorithms, 48(1-3):161–188, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Beylkin and M.J. Mohlenkamp. Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA, 99(16):10246–10251 (electronic), 2002.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.H. Bramble and J.E. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp., 50(181):1–17, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Bittner and K. Urban. Adaptive wavelet methods using semiorthogonal spline wavelets: sparse evaluation of nonlinear functions. Appl. Comput. Harmon. Anal., 24(1):94–119, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Candès and D. Donoho. New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Comm. Pure Appl. Math., 57(2):219–266, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic operator equations – Convergence rates. Math. Comp, 70:27–75, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math., 2(3):203–245, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal., 41:1785–1823, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Cohen, W. Dahmen, and R. DeVore. Sparse evaluation of compositions of functions using multiscale expansions. SIAM J. Math. Anal., 35(2):279–303 (electronic), 2003.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore. Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal., 11(2):192–226, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam, 2003.

    MATH  Google Scholar 

  22. C. Canuto, A. Tabacco, and K. Urban. The wavelet element method part I: Construction and analysis. Appl. Comput. Harmon. Anal., 6:1–52, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  23. C. Canuto and K. Urban. Adaptive optimization of convex functionals in Banach spaces. SIAM J. Numer. Anal., 42(5):2043–2075, 2005 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  24. W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numer., 6:55–228, 1997.

    Article  MathSciNet  Google Scholar 

  25. S. Dahlke. Besov regularity for the Stokes problem. In W. Haussmann, K. Jetter, and M. Reimer, editors, Advances in Multivariate Approximation, Math. Res. 107, pages 129–138, Berlin, 1999. Wiley-VCH.

    Google Scholar 

  26. S. Dahlke and R. DeVore. Besov regularity for elliptic boundary value problems. Comm. Partial Differential Equations, 22(12):1–16, 1997. &

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Dahlke, W. Dahmen, and K. Urban. Adaptive wavelet methods for saddle point problems - Optimal convergence rates. SIAM J. Numer. Anal., 40:1230–1262, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  28. R. DeVore. Nonlinear approximation. Acta Numer., 7:51–150, 1998.

    Article  MathSciNet  Google Scholar 

  29. S. Dahlke, M. Fornasier, and T. Raasch. Adaptive frame methods for elliptic operator equations. Adv. Comput. Math., 27(1):27–63, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Dahlke, M. Fornasier, T. Raasch, R.P. Stevenson, and M. Werner. Adaptive frame methods for elliptic operator equations: The steepest descent approach. IMA J. Numer. Math., 27(4):717–740, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  31. G.C. Donovan, J.S. Geronimo, and D.P. Hardin. Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets. SIAM J. Math. Anal., 27(6):1791–1815, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  32. W. Dahmen, H. Harbrecht, and R. Schneider. Adaptive methods for boundary integral equations - complexity and convergence estimates. Math. Comp., 76:1243–1274, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  33. W. Dahmen and A. Kunoth. Adaptive wavelet methods for linear-quadratic elliptic control problems: convergence rates. SIAM J. Control Optim., 43(5):1640–1675, 2005 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  34. R.A. DeVore, S.V. Konyagin, and V.N. Temlyakov. Hyperbolic wavelet approximation. Constr. Approx., 14(1):1–26, 1998.

    Article  MathSciNet  Google Scholar 

  35. R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5. Springer-Verlag, Berlin, 1992. Evolution problems I.

    MATH  Google Scholar 

  36. W. Dahmen and R. Schneider. Wavelets with complementary boundary conditions—function spaces on the cube. Results Math., 34(34):255–293, 1998.-

    MATH  MathSciNet  Google Scholar 

  37. W. Dahmen and R. Schneider. Composite wavelet bases for operator equations. Math. Comp., 68:1533–1567, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  38. W. Dahmen and R. Schneider. Wavelets on manifolds I: Construction and domain decomposition. SIAM J. Math. Anal., 31:184–230, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  39. M. Dauge and R.P. Stevenson. Sparse tensor product wavelet approximation of singular functions. Technical report, 2009. Submitted.

    Google Scholar 

  40. T.J. Dijkema and R.P. Stevenson. A sparse Laplacian in tensor product wavelet coordinates. Technical report, 2009. Submitted.

    Google Scholar 

  41. T.J. Dijkema, Ch. Schwab, and R.P. Stevenson. An adaptive wavelet method for solving high-dimensional elliptic PDEs. Technical report, 2008. To appear in Constr. Approx.

    Google Scholar 

  42. W. Dahmen, R. Schneider, and Y. Xu. Nonlinear functionals of wavelet expansions—adaptive reconstruction and fast evaluation. Numer. Math., 86(1):49–101, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  43. W. Dahmen, R. Schneider, and Y. Xu. Nonlinear functionals of wavelet expansions—adaptive reconstruction and fast evaluation. Numer. Math., 86(1):49–101, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  44. W. Dahmen, K. Urban, and J. Vorloeper. Adaptive wavelet methods - Basic concepts and applications to the Stokes problem. In D.-X. Zhou, editor, Wavelet Analysis, New Jersey, 2002. World Scientific.

    Google Scholar 

  45. T. Gantumur. Adaptive Wavelet Algorithms for solving operator equations. PhD thesis, Department of Mathematics, Utrecht University, 2006.

    Google Scholar 

  46. T. Gantumur, H. Harbrecht, and R.P. Stevenson. An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp., 76:615–629, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  47. M. Griebel and S. Knapek. Optimized tensor-product approximation spaces. Constr. Approx., 16(4):525–540, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  48. M. Griebel and P. Oswald. Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math., 4(12):171–206, 1995.–

    Article  MATH  MathSciNet  Google Scholar 

  49. L. Greengard and V. Rokhlin. A fast algorithm for particle simulation. J. Comput. Phys., 73:325–348, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  50. L. Grasedyck. Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing, 72(34):247–265, 2004.-

    MATH  MathSciNet  Google Scholar 

  51. T. Gantumur and R.P. Stevenson. Computation of differential operators in wavelet coordinates. Math. Comp., 75:697–709, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  52. T. Gantumur and R.P. Stevenson. Computation of singular integral operators in wavelet coordinates. Computing, 76:77–107, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  53. W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment. Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  54. W. Hackbusch and B.N. Khoromskij. Tensor-product approximation to operators and functions in high dimensions. J. Complexity, 23(46):697–714, 2007.-

    Article  MATH  MathSciNet  Google Scholar 

  55. W. Hackbusch and Z.P. Nowak. On the fast matrix multiplication in the boundary element by panel clustering. Numer. Math., 54:463–491, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  56. V.H. Hoang and Ch. Schwab. High-dimensional finite elements for elliptic problems with multiple scales. SIAM J. Multiscale Model. Simul., 3(1):168–194, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  57. H. Harbrecht and R. Stevenson. Wavelets with patchwise cancellation properties. Math. Comp., 75(256):1871–1889, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  58. H. Harbrecht, R. Schneider, and Ch. Schwab. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math., 109(3):385–414, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  59. D. Labate, W-Q. Lim, G. Kutyniok, and G. Weiss. Sparse multidimensional representation using shearlets. In M. Papadakis, A. Laine, and M. Unser, editors, Wavelets XI, SPIE, Proc. 5914, pages 254–262, 2005.

    Google Scholar 

  60. A. Metselaar. Handling Wavelet Expansions in Numerical Methods. PhD thesis, University of Twente, 2002.

    Google Scholar 

  61. P.-A. Nitsche. Sparse approximation of singularity functions. Constr. Approx., 21(1):63–81, 2005.

    MATH  MathSciNet  Google Scholar 

  62. P.-A. Nitsche. Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx., 24(1):49–70, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  63. E. Novak and H. Woźniakovski. Approximation of infinitely differentiable multivariate functions is intractable. Technical report, 2008. To appear in Journal of Complexity.

    Google Scholar 

  64. N.C. Reich. Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. PhD thesis, ETH, Zürich, 2008.

    Google Scholar 

  65. Y. Saad. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.

    MATH  Google Scholar 

  66. H.A. Schwarz. Gesammelte Mathematische Abhandlungen. volume 2, pages 133–143, Berlin, 1890. Springer.

    MATH  Google Scholar 

  67. Ch. Schwab and R.P. Stevenson. Adaptive wavelet algorithms for elliptic PDEs on product domains. Math. Comp., 77:71–92, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  68. Ch. Schwab and R.P. Stevenson. A space-time adaptive wavelet method for parabolic evolution problems. Technical report, 2009. To appear in Math. Comp.

    Google Scholar 

  69. Ch. Schwab and R.A. Todor. Sparse finite elements for stochastic elliptic problems—higher order moments. Computing, 71(1):43–63, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  70. R.P. Stevenson. Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal., 41(3):1074–1100, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  71. R.P. Stevenson. On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal., 35(5):1110–1132, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  72. R.P. Stevenson. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7(2):245–269, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  73. R.P. Stevenson and M. Werner. Computation of differential operators in aggregated wavelet frame coordinates. IMA J. Numer. Anal., 28(2):354–381, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  74. R.P. Stevenson and M. Werner. A multiplicative schwarz adaptive wavelet method for elliptic boundary value problems. Math. Comp., 78:619–644, 2009.

    Article  MathSciNet  Google Scholar 

  75. K. Urban. Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, 2009.

    Google Scholar 

  76. T. von Petersdorff and Ch. Schwab. Sparse finite element methods for operator equations with stochastic data. Appl. Math., 51(2):145–180, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  77. J. van den Eshof and G.L.G. Sleijpen. Inexact Krylov subspace methods for linear systems. SIAM J. Matrix Anal. Appl., 26(1):125–153, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  78. J. Wloka. Partielle Differentialgleichungen. B.G. Teubner, Stuttgart, 1982. Sobolevräume und Randwertaufgaben.

    Google Scholar 

  79. J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34:581–613, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  80. Ch. Zenger. Sparse grids. In Parallel algorithms for partial differential equations (Kiel, 1990), volume 31of Notes Numer. Fluid Mech., pages 241–251. Vieweg, Braunschweig, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Amsterdam, Korteweg - de Vries (KdV) Institute for Mathematics, P.O. Box 94248, 1090 GE, Amsterdam, The Netherlands

    Rob Stevenson

Authors
  1. Rob Stevenson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rob Stevenson .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Texas A & M University, Blocker Hall 608K, College Station, 77843, U.S.A.

    Ronald DeVore

  2. Institut für Mathematik, Universität Paderborn, Warburger Str. 100, Paderborn, 33098, Germany

    Angela Kunoth

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Stevenson, R. (2009). Adaptive wavelet methods for solving operator equations: An overview. In: DeVore, R., Kunoth, A. (eds) Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03413-8_13

Download citation

Publish with us

Policies and ethics

Search

Navigation