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.
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References
G. Allaire and M. Briane. Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh Sect. A, 126(2):297–342, 1996.
T. Apel. Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics. B.G. Teubner, Stuttgart, 1999.
T. Barsch. Adaptive Multiskalenverfahren für elliptische partielle Differentialgleigungen – Realisierung, Umsetzung and numerische Ergebnisse. PhD thesis, RTWH Aachen, 2001.
A. Barinka. Fast Evaluation Tools for Adaptive Wavelet Schemes. PhD thesis, RTWH Aachen, March 2005.
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.
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.
P. Binev and R. DeVore. Fast computation in adaptive tree approximation. Numer. Math., 97(2):193–217, 2004.
A. Barinka, W. Dahmen, and R. Schneider. Fast computation of adaptive wavelet expansions. Numer. Math., 105(4):549–589, 2007.
H.J. Bungartz and M. Griebel. Sparse grids. Acta Numer., 13:147–269, 2004.
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.
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.
G. Beylkin and M.J. Mohlenkamp. Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA, 99(16):10246–10251 (electronic), 2002.
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.
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.
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.
A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic operator equations – Convergence rates. Math. Comp, 70:27–75, 2001.
A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math., 2(3):203–245, 2002.
A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet schemes for nonlinear variational problems. SIAM J. Numer. Anal., 41:1785–1823, 2003.
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.
A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore. Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal., 11(2):192–226, 2001.
A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam, 2003.
C. Canuto, A. Tabacco, and K. Urban. The wavelet element method part I: Construction and analysis. Appl. Comput. Harmon. Anal., 6:1–52, 1999.
C. Canuto and K. Urban. Adaptive optimization of convex functionals in Banach spaces. SIAM J. Numer. Anal., 42(5):2043–2075, 2005 (electronic).
W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numer., 6:55–228, 1997.
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.
S. Dahlke and R. DeVore. Besov regularity for elliptic boundary value problems. Comm. Partial Differential Equations, 22(12):1–16, 1997. &
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.
R. DeVore. Nonlinear approximation. Acta Numer., 7:51–150, 1998.
S. Dahlke, M. Fornasier, and T. Raasch. Adaptive frame methods for elliptic operator equations. Adv. Comput. Math., 27(1):27–63, 2007.
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.
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.
W. Dahmen, H. Harbrecht, and R. Schneider. Adaptive methods for boundary integral equations - complexity and convergence estimates. Math. Comp., 76:1243–1274, 2007.
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).
R.A. DeVore, S.V. Konyagin, and V.N. Temlyakov. Hyperbolic wavelet approximation. Constr. Approx., 14(1):1–26, 1998.
R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 5. Springer-Verlag, Berlin, 1992. Evolution problems I.
W. Dahmen and R. Schneider. Wavelets with complementary boundary conditions—function spaces on the cube. Results Math., 34(34):255–293, 1998.-
W. Dahmen and R. Schneider. Composite wavelet bases for operator equations. Math. Comp., 68:1533–1567, 1999.
W. Dahmen and R. Schneider. Wavelets on manifolds I: Construction and domain decomposition. SIAM J. Math. Anal., 31:184–230, 1999.
M. Dauge and R.P. Stevenson. Sparse tensor product wavelet approximation of singular functions. Technical report, 2009. Submitted.
T.J. Dijkema and R.P. Stevenson. A sparse Laplacian in tensor product wavelet coordinates. Technical report, 2009. Submitted.
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.
W. Dahmen, R. Schneider, and Y. Xu. Nonlinear functionals of wavelet expansions—adaptive reconstruction and fast evaluation. Numer. Math., 86(1):49–101, 2000.
W. Dahmen, R. Schneider, and Y. Xu. Nonlinear functionals of wavelet expansions—adaptive reconstruction and fast evaluation. Numer. Math., 86(1):49–101, 2000.
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.
T. Gantumur. Adaptive Wavelet Algorithms for solving operator equations. PhD thesis, Department of Mathematics, Utrecht University, 2006.
T. Gantumur, H. Harbrecht, and R.P. Stevenson. An optimal adaptive wavelet method without coarsening of the iterands. Math. Comp., 76:615–629, 2007.
M. Griebel and S. Knapek. Optimized tensor-product approximation spaces. Constr. Approx., 16(4):525–540, 2000.
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.–
L. Greengard and V. Rokhlin. A fast algorithm for particle simulation. J. Comput. Phys., 73:325–348, 1987.
L. Grasedyck. Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing, 72(34):247–265, 2004.-
T. Gantumur and R.P. Stevenson. Computation of differential operators in wavelet coordinates. Math. Comp., 75:697–709, 2006.
T. Gantumur and R.P. Stevenson. Computation of singular integral operators in wavelet coordinates. Computing, 76:77–107, 2006.
W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment. Springer-Verlag, Berlin, 1992.
W. Hackbusch and B.N. Khoromskij. Tensor-product approximation to operators and functions in high dimensions. J. Complexity, 23(46):697–714, 2007.-
W. Hackbusch and Z.P. Nowak. On the fast matrix multiplication in the boundary element by panel clustering. Numer. Math., 54:463–491, 1989.
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.
H. Harbrecht and R. Stevenson. Wavelets with patchwise cancellation properties. Math. Comp., 75(256):1871–1889, 2006.
H. Harbrecht, R. Schneider, and Ch. Schwab. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math., 109(3):385–414, 2008.
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.
A. Metselaar. Handling Wavelet Expansions in Numerical Methods. PhD thesis, University of Twente, 2002.
P.-A. Nitsche. Sparse approximation of singularity functions. Constr. Approx., 21(1):63–81, 2005.
P.-A. Nitsche. Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx., 24(1):49–70, 2006.
E. Novak and H. Woźniakovski. Approximation of infinitely differentiable multivariate functions is intractable. Technical report, 2008. To appear in Journal of Complexity.
N.C. Reich. Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. PhD thesis, ETH, Zürich, 2008.
Y. Saad. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.
H.A. Schwarz. Gesammelte Mathematische Abhandlungen. volume 2, pages 133–143, Berlin, 1890. Springer.
Ch. Schwab and R.P. Stevenson. Adaptive wavelet algorithms for elliptic PDEs on product domains. Math. Comp., 77:71–92, 2008.
Ch. Schwab and R.P. Stevenson. A space-time adaptive wavelet method for parabolic evolution problems. Technical report, 2009. To appear in Math. Comp.
Ch. Schwab and R.A. Todor. Sparse finite elements for stochastic elliptic problems—higher order moments. Computing, 71(1):43–63, 2003.
R.P. Stevenson. Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal., 41(3):1074–1100, 2003.
R.P. Stevenson. On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal., 35(5):1110–1132, 2004.
R.P. Stevenson. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7(2):245–269, 2007.
R.P. Stevenson and M. Werner. Computation of differential operators in aggregated wavelet frame coordinates. IMA J. Numer. Anal., 28(2):354–381, 2008.
R.P. Stevenson and M. Werner. A multiplicative schwarz adaptive wavelet method for elliptic boundary value problems. Math. Comp., 78:619–644, 2009.
K. Urban. Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, 2009.
T. von Petersdorff and Ch. Schwab. Sparse finite element methods for operator equations with stochastic data. Appl. Math., 51(2):145–180, 2006.
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.
J. Wloka. Partielle Differentialgleichungen. B.G. Teubner, Stuttgart, 1982. Sobolevräume und Randwertaufgaben.
J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34:581–613, 1992.
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.
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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
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