Abstract
In this paper, we study a newly developed hybrid shearlet-wavelet system on bounded domains which yields frames for Hs(Ω) for some \(s\in \mathbb {N}\), \({\Omega } \subset \mathbb {R}^{2}\). We will derive approximation rates with respect to Hs(Ω) norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-wavelet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties.
Similar content being viewed by others
References
Bittner, K.: Biorthogonal spline wavelets on the interval. In: Wavelets and Splines: Athens 2005. pp. 93–104, Nashboro Press, Brentwood (2006)
Bubba, T.A., Labate, D., Zanghirati, G., Bonettini, S., Goossens, B.: Shearlet-based regularized ROI reconstruction in fan beam computed tomography. In: Wavelets and Sparsity XVI, pp. 95970K–95970K–11. Proceedings of the SPIE, San Diego (2015)
Buckheit, J., Donoho, D.: Wavelab and reproducible research. In: Wavelets and Statistics, pp. 55–81. Springer (1995)
Candès, E., Donoho, D.: New tight frames of curvelets and optimal representations of objects with piecewise c 2 singularities. Comm. Pure Appl. Math. 57(2), 219–266 (2004)
Candès, E. J., Donoho, D.L.: Curvelets: a surprisingly effective nonadaptive representation of objects with edges. In:: Curve and surface fitting, pp. 105–120. Vanderbilt University Press, Nashville (2000)
Canuto, C., Tabacco, A., Urban, K.: The wavelet element method. I. Construction and analysis. Appl. Comput. Harmon. Anal. 6(1), 1–52 (1999)
Christensen, O.: An introduction to frames and Riesz bases. Birkhääuser Boston, Inc., Boston (2003)
Cohen, A.: Wavelet Methods in Numerical Analysis. North-Holland, Amsterdam (2000)
Cohen, A., Dahmen, W., DeVore, R.: Multiscale decompositions on bounded domains. Trans. Amer. Math Soc. 352(8), 3651–3685 (2000)
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp. 70(233), 27–75 (2001)
Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1(1), 54–81 (1993)
Dahlke, S., Fornasier, M., Primbs, M., Raasch, T., Werner, M.: Nonlinear and adaptive frame approximation schemes for elliptic PDEs: theory and numerical experiments. Numer. Methods Partial Differ. Equ. 25(6), 1366–1401 (2009)
Dahlke, S., Fornasier, M., Raasch, T.: Adaptive frame methods for elliptic operator equations. Adv. Comput. Math. 27(1), 27–63 (2007)
Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27(2), 195–214 (2009)
Dahlke, S., Raasch, T., Werner, M., Fornasier, M., Stevenson, R.: Adaptive frame methods for elliptic operator equations: the steepest descent approach. IMA J. Numer. Anal. 27(4), 717–740 (2007)
Dahlke, S., Steidl, G., Teschke, G.: Shearlet coorbit spaces: compactly supported analyzing shearlets, traces and embeddings. J. Fourier Anal. Appl. 17(6), 1232–1255 (2011)
Dahmen, W., Huang, C., Kutyniok, G., Lim, W.-Q., Schwab, C., Welper, G.: Efficient resolution of anisotropic structures. In: Extraction of quantifiable information from complex systems, volume 102 of Lect. Notes Comput. Sci. Eng., pp. 25–51. Springer, Cham (2014)
Dahmen, W., Kunoth, A., Urban, K.: A wavelet Galerkin method for the Stokes equations. Computing 56(3), 259–301 (1996)
Dahmen, W., Kutyniok, G., Lim, W., Schwab, C., Welper, G.: Adaptive anisotropic Petrov-Galerkin methods for first order transport equations. J. Comput. Appl. Math. 340, 191–220 (2018)
Dahmen, W., Schneider, R.: Wavelets with complementary boundary conditions—function spaces on the cube. Results Math. 34(3-4), 255–293 (1998)
Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)
DeVore, R.A.: Nonlinear approximation. In: Acta numerica, pp 51–150. Cambridge Univ. Press, Cambridge (1998)
Do, M.N., Vetterli, M.: Contourlets. In: Beyond wavelets, volume 10 of Stud. Comput. Math., pp 83–105. Academic Press/Elsevier, San Diego (2003)
Dobrowolski, M.: Angewandte Funktionalanalysis. funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Springer, Berlin (2010)
Donoho, D.L.: Sparse components of images and optimal atomic decompositions. Constr. Approx. 17(3), 353–382 (2001)
Duffin, R., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)
Easley, G.R., Labate, D., Colonna, F.: Shearlet-based total variation diffusion for denoising. Trans. Img Proc. 18(2), 260–268 (February 2009)
Etter, S., Grohs, P., Obermeier, A.: FFRT - A fast finite ridgelet transform for radiative transport. Multiscale Model Simul. 13(1), 1–42 (2014)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Grohs, P., Kutyniok, G., Ma, J., Petersen, P., Raslan, M.: Anisotropic multiscale systems on bounded domains. arXiv:1510.04538 (2015)
Grohs, P., Obermeier, A.: Optimal adaptive ridgelet schemes for linear advection equations. Appl. Comput Harmon. Anal. 41(3), 768–814 (2016)
Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. In:: Wavelets and splines: Athens 2005, pp. 189–201. Nashboro Press, Brentwood (2006)
Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 39(1), 298–318 (2007)
Guo, K., Labate, D., Lim, W.-Q., Weiss, G., Wilson, E.: The theory of wavelets with composite dilations. In: Harmonic analysis and applications, pp. 231–250. Birkhäuser Boston, Boston (2006)
Häuser, S., shearlet, G. Steidl.: Convex multiclass segmentation with regularization. Int. J. Comput. Math. 90(1), 62–81 (2013)
King, E.J, Reisenhofer, R., Kiefer, J., Lim, W.-Q., Li, Z., Heygster, G.: Shearlet-based edge detection: flame fronts and tidal flats. In: SPIE Optical Engineering+ Applications, pp. 959905–959905. International Society for Optics and Photonics (2015)
Kittipoom, P., Kutyniok, G., Lim, W.-Q.: Construction of compactly supported shearlet frames Constr. Approx 35(1), 21–72 (2012)
Kutyniok, G., Lim, W.-Q.: Compactly supported shearlets are optimally sparse. J. Approx. Theory 163(11), 1564–1589 (2011)
Kutyniok, G., Lim, W.-Q.: Shearlets on bounded domains. In: Approximation Theory XIII: San Antonio 2010, pp. 187–206. Springer (2012)
Kutyniok, G., Lim, W.-Q., Reisenhofer, R.: ShearLab 3D: Faithful digital shearlet transforms based on compactly supported shearlets. ACM T. Math. Software 42(1), 1–42 (2015)
Labate, D., Lim, W.-Q., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: Wavelets XI, pp. 254–262. Proceedings of the SPIE, San Diego (2005)
Li, Y., Nirenberg, L.: Estimates for elliptic systems from composite material. Commun Pure Appl. Math. 56(7), 892–925 (2003)
Lim, W.: The discrete shearlet transform: a new directional transform and compactly supported shearlet frames. IEEE Trans. Image Process. 19, 1166–1180 (2010)
Ma, J.: Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets. Appl. Comput. Harmon. Anal. 42(2), 294–318 (2017)
Petersen, P.: Shearlet approximation of functions with discontinuous derivatives. J. Approx. Theory 207(C), 127–138 (2016)
Petersen, P.: Shearlets on Bounded Domains and Analysis of Singularities Using Compactly Supported Shearlets. Dissertation. Technische Universität, Berlin (2016)
Primbs, M.: Stabile biorthogonale Spline-Waveletbasen auf dem Intervall. Dissertation Universität, Duisburg (2006)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970)
Stevenson, R.: Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal. 41(3), 1074–1100 (2003)
Whitney, H.: Hassler Whitney Collected Papers, Chapter Analytic Extensions of Differentiable Functions Defined in Closed Sets, pp. 228–254. Birkhäuser, Boston (1992)
Acknowledgments
P. Petersen and M. Raslan thank P. Grohs and G. Kutyniok for valuable discussions.
Funding
This work received support from the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and the Berlin Mathematical School. P. Petersen is supported by a DFG Research Fellowship “Shearlet-based energy functionals for anisotropic phase-field methods.”
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Yang Wang
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Petersen, P., Raslan, M. Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces. Adv Comput Math 45, 1581–1606 (2019). https://doi.org/10.1007/s10444-019-09679-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-019-09679-9