On the approximation of functions with line singularities by ridgelets

In [GO15], the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal - by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate. Structurally, the proof resembles [Can01], where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes [CDD01] we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

[1]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[2]  S. Dahlke.  Extraction of quantifiable information from complex systems , 2014 .

[3]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[4]  Konstantin Grella,et al.  Sparse Discrete Ordinates Method in Radiative Transfer , 2011, Comput. Methods Appl. Math..

[5]  Wolfgang Dahmen,et al.  Adaptive Petrov-Galerkin Methods for First Order Transport Equations , 2011, SIAM J. Numer. Anal..

[6]  M. Nielsen,et al.  Frame Decomposition of Decomposition Spaces , 2007 .

[7]  Arnulf Jentzen,et al.  Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise , 2015, Applied Mathematics & Optimization.

[8]  Rob P. Stevenson,et al.  Adaptive Solution of Operator Equations Using Wavelet Frames , 2003, SIAM J. Numer. Anal..

[9]  Wolfgang Dahmen,et al.  Efficient Resolution of Anisotropic Structures , 2014 .

[10]  Albino Perego,et al.  AN ADVANCED LEAKAGE SCHEME FOR NEUTRINO TREATMENT IN ASTROPHYSICAL SIMULATIONS , 2015, 1511.08519.

[11]  Philipp Grohs,et al.  FFRT: A Fast Finite Ridgelet Transform for Radiative Transport , 2014, Multiscale Model. Simul..

[12]  Philipp Grohs,et al.  Optimal adaptive ridgelet schemes for linear advection equations , 2016 .

[13]  Marsha J. Berger,et al.  An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes , 2015, J. Sci. Comput..

[14]  Gitta Kutyniok,et al.  Shearlets: Multiscale Analysis for Multivariate Data , 2012 .

[15]  Demetrio Labate,et al.  Harmonic and Applied Analysis - From Groups to Signals , 2015, Harmonic and Applied Analysis.

[16]  Wang-Q Lim,et al.  Sparse multidimensional representation using shearlets , 2005, SPIE Optics + Photonics.

[17]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[18]  Lena Schwartz,et al.  Theory Of Function Spaces Ii , 2016 .

[19]  Minh N. Do,et al.  Ieee Transactions on Image Processing the Contourlet Transform: an Efficient Directional Multiresolution Image Representation , 2022 .

[20]  H. Triebel.  Theory of Function Spaces III , 2008 .

[21]  Emmanuel J. Candès.  Ridgelets and the Representation of Mutilated Sobolev Functions , 2001, SIAM J. Math. Anal..

[22]  Wang-Q Lim,et al.  Shearlets and Optimally Sparse Approximations , 2011, ArXiv.

[23]  Gitta Kutyniok,et al.  Parabolic Molecules , 2012, Found. Comput. Math..

[24]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[25]  Arnulf Jentzen,et al.  On stochastic differential equations with arbitrary slow convergence rates for strong approximation , 2015, 1506.02828.

[26]  Axel Clemens Obermeier,et al.  Ridgelets — An Optimally Adapted Representation System For Solving Advection Equations , 2015 .

[27]  Arnulf Jentzen,et al.  Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients , 2015, Found. Comput. Math..

[28]  Rolf Rannacher,et al.  Numerical methods in multidimensional radiative transfer , 2009 .

[29]  D. Donoho.  Sparse Components of Images and Optimal Atomic Decompositions , 2001 .

[30]  E. Candès,et al.  Continuous curvelet transform , 2003 .

[31]  Massimo Fornasier,et al.  Adaptive frame methods for elliptic operator equations , 2007, Adv. Comput. Math..

[32]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[33]  M. Fornasier,et al.  Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach , 2007 .

[34]  M. Modest.  Radiative heat transfer , 1993 .

[35]  C. Schwab,et al.  Numerical analysis of lognormal diffusions on the sphere , 2016, Stochastics and Partial Differential Equations: Analysis and Computations.

[36]  Arnulf Jentzen,et al.  Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values , 2014, Journal of Mathematical Analysis and Applications.

[37]  P. Grohs.  Ridgelet-type Frame Decompositions for Sobolev Spaces related to Linear Transport , 2012 .

[38]  Herbert S. Wilf,et al.  Generating functionology , 1990 .

[39]  E. Candès,et al.  Continuous curvelet transform: II. Discretization and frames , 2005 .

[40]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[41]  K. Tamvakis.  Molécules , 2020, Merveilleuses structures.

[42]  Loukas Grafakos,et al.  Modern Fourier Analysis , 2008 .

[43]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[44]  Gitta Kutyniok,et al.  Anisotropic multiscale systems on bounded domains , 2015, Adv. Comput. Math..

[45]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[46]  Rob Stevenson,et al.  Fractional Space-Time Variational Formulations of (Navier-) Stokes Equations , 2017, SIAM Journal on Mathematical Analysis.

[47]  Rob P. Stevenson,et al.  Computation of differential operators in wavelet coordinates , 2005, Math. Comput..

[48]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[49]  C. Bennett,et al.  Interpolation of operators , 1987 .

[50]  Martin Frank,et al.  APPROXIMATE MODELS FOR RADIATIVE TRANSFER , 2007 .

[51]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .