Abstract
In Cohen et al. (Math Comput 70:27–75, 2001), a new paradigm for the adaptive solution of linear elliptic partial differential equations (PDEs) was proposed, based on wavelet discretizations. Starting from a well-conditioned representation of the linear operator equation in infinite wavelet coordinates, one performs perturbed gradient iterations involving approximate matrix–vector multiplications of finite portions of the operator. In a bootstrap-type fashion, increasingly smaller tolerances guarantee convergence of the adaptive method. In addition, coarsening performed on the iterates allow one to prove asymptotically optimal complexity results when compared to the wavelet best N-term approximation. In the present paper, we study adaptive wavelet schemes for symmetric operators employing inexact conjugate gradient routines. Inspired by fast schemes on uniform grids, we incorporate coarsening and the adaptive application of the elliptic operator into a nested iteration algorithm. Our numerical results demonstrate that the runtime of the algorithm is linear in the number of unknowns and substantial savings in memory can be achieved in two and three space dimensions.
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Barsch, T.: Adaptive Multiskalenverfahren für elliptische partielle Differentialgleichungen—Realisierung, Umsetzung und numerische Ergebnisse. Shaker (2001)
Barinka, A., Barsch, T., Charton, P., Cohen, A., Dahlke, S., Dahmen, W., Urban, K.: Adaptive wavelet schemes for elliptic problems—implementation and numerical experiments. SIAM J. Sci. Comput. 23, 910–939 (2001)
Bramble, J., Cohen, A., Dahmen, W.: Multiscale problems and methods in numerical simulations. Lecture Notes in Mathematics. Springer (2003)
Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44, 141–183 (1991)
Bouras, A., Fraysse, V.: A relaxation strategy for inexact matrix–vector products for Krylov methods. Technical Report TR/PA/00/15, CERFACS, France (2000)
Berrone, S., Kozubek, T.: An adaptive wavelet algorithm for solving elliptic boundary value problems in fairly general domains. SIAM J. Sci. Comput. 28(6), 2114–2138 (2006)
Burstedde, C., Kunoth, A.: Fast iterative solution of elliptic control problems in wavelet discretization. J. Comput. Appl. Math. 196(1), 299–319 (2006)
Bramble, J., Pasciak, J., Xu, J.: Parallel multilevel preconditioners. Math. Comput. 37, 1–22 (1981)
Bramble, J.: Multigrid Methods. Pitman (1993)
Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 2nd edn. Cambridge University Press (2001)
Burstedde, C.: Fast optimised wavelet methods for control problems constrained by elliptic PDEs. Ph.D. Dissertation, Institut für Angewandte Mathematik, Universität Bonn (2005) http://hss.ulb.uni-bonn.de/diss_online/math_nat_fak/2005/burstedde_carsten
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations—convergence rates. Math. Comput. 70, 27–75 (2001)
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet techniques in numerical simulation. In: Stein, E., de Borst, R., Hughes, T. (eds.) Encyclopedia of Computational Mechanics, vol. 1. John Wiley & Sons (2004)
Cohen, A., Daubechies, I., Feauveau, J.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)
Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier (2003)
Canuto, C., Tabacco, A., Urban, K.: The wavelet element method, part i: construction and analysis. Appl. Comput. Harmon. Anal. 6, 1–52 (1999)
Dahmen, W.: Stability of multiscale transformations. J. Fourier Anal. Appl. 4, 341–362 (1996)
Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)
Dahmen, W.: Wavelet methods for pdes—some recent developments. J. Comput. Appl. Math. 128, 133–185 (2001)
DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)
Dahmen, W., Kunoth, A.: Multilevel preconditioning. Numer. Math. 63, 315–344 (1992)
Dahmen, W., Kunoth, A., Schneider, R.: Wavelet least square methods for boundary value problems. SIAM J. Numer. Anal. 39, 1985–2013 (2002)
Dahmen, W., Kunoth, A., Urban, K.: Biorthogonal spline-wavelets on the interval—stability and moment conditions. Appl. Comput. Harmon. Anal. 6, 132–196 (1999)
Dahmen, W., Prößdorf, S., Schneider, R.: Multiscale methods for pseudo-differential equations on smooth manifolds. In: Chui, C.K., Montefusco, L., Puccio, L. (eds.) Proceedings of the International Conference on Wavelets: Theory, Algorithms, and Applications, pp. 385–424. Academic Press (1994)
Dahmen, W., Schneider, R.: Composite wavelet bases for operator equations. Math. Comput. 68, 1533–1567 (1999)
Dahmen, W., Schneider, R.: Wavelets on manifolds I: construction and domain decomposition. SIAM J. Math. Anal. 31, 184–230 (1999)
DeVore, R., Temlyakov, V.: Some remarks on greedy algorithms. Adv. Comput. Math. 5, 173–1124 (1996)
Gantumur, T., Harbrecht, H., Stevenson, R.: An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp. 77, 615–629 (2007)
Golub, G., Ye, Q.: Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM J. Sci. Comput. 21, 1305–1320 (2000)
Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment. Springer (1992)
Kunoth, A., Sahner, J.: Wavelets on manifolds: an optimized construction. Math. Comput. 75, 1319–1349 (2006)
Kunoth, A.: Wavelet-based multiresolution methods for stationary PDEs and PDE-constrained control problems. In: Blowey, J., Craig, A. (eds.) Frontiers in Numerical Analysis (Durham 2004), Universitext, pp. 1–63. Springer (2005)
Oswald, P.: On discrete norm estimates related to multilevel preconditioners in the finite element method. In: Ivanov, K.G., Petrushev, P., Sendov, B. (eds.) Constructive Theory of Functions, Proc. Int. Conf. Varna, pp. 203–214 (1991)
Schneider, R.: Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme. Advances in Numerical Mathematics. Teubner Stuttgart (1998)
Stevenson, R.: On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35(5), 1110–1132 (2004)
Tchamitchian, P.: Wavelets, functions, and operators. In: Erlebacher, G., Hussaini, M.Y., Jameson, L. (eds.) Wavelets: Theory and Applications, Series in Computational Science and Engineering, pp. 83–181. Oxford University Press (1996)
van den Eshof, J., Sleijpen, G.: Inexact Krylov subspace methods for linear systems. SIAM J. Matrix Anal. A. 26(1), 125–153 (2004)
von Petersdorff, T., Schwab, C.: Fully discrete multiscale Galerkin BEM. In: Dahmen, W., Kurdila, A., Oswald, P. (eds.) Multiscale Wavelet Methods for PDEs, pp. 287–346. Academic Press (1997)
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This work was supported by the Deutsche Forschungsgemeinschaft (SFB 611) at Universität Bonn.
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Burstedde, C., Kunoth, A. A wavelet-based nested iteration–inexact conjugate gradient algorithm for adaptively solving elliptic PDEs. Numer Algor 48, 161–188 (2008). https://doi.org/10.1007/s11075-008-9164-0
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DOI: https://doi.org/10.1007/s11075-008-9164-0
Keywords
- Elliptic PDEs
- Biorthogonal spline-wavelets
- Optimal preconditioning
- Adaptive method
- Nested iteration
- Inexact conjugate gradient (CG) method