Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften v Acknowledgements I would like to sincerely thank my research advisor, Professor Stephan Dahlke, whose guidance, patience and constant willingness for discussions about wavelet and frame analysis substantially helped me to finish this thesis. I am also indebted to Professor Hans–Jürgen Reinhardt for being my second referee and in particular for guiding my first scientific steps related to the interesting field of inverse problems when I belonged to his research group. At the same time, I am also deeply grateful to Rob Stevenson for his readiness to write the third referee report and for inspiring discussions on the numerical analysis of adaptive wavelet and frame methods. Furthermore, I thank all the members of the AG Numerik/Wavelet–Analysis in Marburg for creating such a friendly atmosphere to work in. Special thanks go to Karsten Koch who has been much more than a colleague for me in the last years. I would like to thank also Manuel Werner for many fruitful discussions and for his assistance in some of the numerical experiments. I also have to express my gratitude to the current and former members of Hans– Jürgen Reinhardt's research group who made my stay in Siegen a valuable time to remember. Thanks also go to my former teacher, Heinrich Meier, for initially drawing my attention to the fascinating world of mathematics by his encouraging way of holding classes. Finally, I would like to express my sincere gratitude to my parents whose support I could always rely on, regardless of the amount of time I spent at the computer when visiting them. I also feel grateful to the Deutsche Forschungsgemeinschaft which financially supported the final phase of my stay in Marburg under Grant Da 360/7–1.

[1]  L. Nikolova,et al.  On ψ- interpolation spaces , 2009 .

[2]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[3]  Stephan Dahlke,et al.  Besov regularity for elliptic boundary value problems in polygonal domains , 1999 .

[4]  M. Roche,et al.  Rosenbrock methods for Differential Algebraic Equations , 1987 .

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

[6]  Alexander Ostermann,et al.  Runge-Kutta methods for partial differential equations and fractional orders of convergence , 1992 .

[7]  Tsogtgerel Gantumur,et al.  An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems , 2008 .

[8]  Jens Lang,et al.  Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems - Theory, Algorithm, and Applications , 2001, Lecture Notes in Computational Science and Engineering.

[9]  Kjell Gustafsson,et al.  Control of Error and Convergence in ODE Solvers , 1992 .

[10]  D. Hardin,et al.  Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets , 1999 .

[11]  J. M. Sanz-Serna,et al.  Stability and convergence at the PDE/stiff ODE interface , 1989 .

[12]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[13]  Daisuke Fujiwara,et al.  Concrete Characterization of the Domains of Fractional Powers of Some Elliptic Differential Operators of the Second Order , 1967 .

[14]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[15]  Gerd Steinebach,et al.  Order-reduction of ROW-methods for DAEs and method of lines applications , 1995 .

[16]  Rob P. Stevenson,et al.  Wavelets with patchwise cancellation properties , 2006, Math. Comput..

[17]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[18]  Rob P. Stevenson,et al.  Composite Wavelet Bases with Extended Stability and Cancellation Properties , 2007, SIAM J. Numer. Anal..

[19]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .

[20]  G. Beylkin.  Wavelets and Fast Numerical Algorithms , 1993, comp-gas/9304004.

[21]  Rong-Qing Jia,et al.  Wavelet bases of Hermite cubic splines on the interval , 2006, Adv. Comput. Math..

[22]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[23]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[24]  Angela Kunoth,et al.  Wavelets on manifolds: An optimized construction , 2006, Math. Comput..

[25]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[26]  R. DeVore,et al.  Besov regularity for elliptic boundary value problems , 1997 .

[27]  George C. Donovan,et al.  Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets , 1996 .

[28]  S. Lang.  Introduction to Complex Hyperbolic Spaces , 1987 .

[29]  W. Dahmen,et al.  Wavelets with Complementary Boundary Conditions — Function Spaces on the Cube , 1998 .

[30]  Karlheinz Gröchenig,et al.  Localized Frames Are Finite Unions of Riesz Sequences , 2003, Adv. Comput. Math..

[31]  K. Strehmel,et al.  Linear-implizite Runge-Kutta-Methoden und ihre Anwendung , 1992 .

[32]  A stable and accurate explicit scheme forparabolic evolution equationsAmir , 2007 .

[33]  H. Johnen,et al.  On the equivalence of the K-functional and moduli of continuity and some applications , 1976, Constructive Theory of Functions of Several Variables.

[34]  Dongwoo Sheen,et al.  A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature , 2000, Math. Comput..

[35]  Lloyd N. Trefethen,et al.  Reviving the Method of Particular Solutions , 2005, SIAM Rev..

[36]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[37]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Nonlinear Variational Problems , 2003, SIAM J. Numer. Anal..

[38]  Wolfgang Dahmen,et al.  Fast computation tools for adaptive wavelet schemes , 2005 .

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

[40]  C. Moler,et al.  APPROXIMATIONS AND BOUNDS FOR EIGENVALUES OF ELLIPTIC OPERATORS , 1967 .

[41]  J. Kadlec,et al.  On the regularity of the solution of the Poisson problem on a domain with boundary locally similar t , 1964 .

[42]  W. Dahmen,et al.  Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines , 2000 .

[43]  Folkmar Bornemann,et al.  An adaptive multilevel approach to parabolic equations I. General theory and 1D implementation , 1991, IMPACT Comput. Sci. Eng..

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

[45]  Peter Oswald.  Frames and Space Splittings in Hilbert Spaces , 2004 .

[46]  F. Smithies.  Linear Operators , 2019, Nature.

[47]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[48]  田辺 広城,et al.  Equations of evolution , 1979 .

[49]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[50]  Peter Oswald.  Multilevel Frames and Riesz Bases in Sobolev Spaces , 2004 .

[51]  Peter Deuflhard,et al.  Numerische Mathematik II , 1994 .

[52]  Kai Bittner,et al.  Biorthogonal Spline Wavelets on the Interval , 2005 .

[53]  Jens Lang,et al.  ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems , 2000 .

[54]  C. Lubich,et al.  Linearly implicit time discretization of non-linear parabolic equations , 1995 .

[55]  R. Schneider,et al.  Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme , 1995 .

[56]  Folkmar A. Bornemann,et al.  An adaptive multilevel approach to parabolic equations : II. Variable-order time discretization based on a multiplicative error correction , 1991, IMPACT Comput. Sci. Eng..

[57]  Qingtang Jiang,et al.  Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets , 2002, SIAM J. Matrix Anal. Appl..

[58]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[59]  C. Chui,et al.  Wavelets on a Bounded Interval , 1992 .

[60]  Reinhold Schneider,et al.  Multiskalen- und Wavelet-Matrixkompression , 1998 .

[61]  Folkmar Bornemann,et al.  Adaptive multilevel discretization in time and space for parabolic partial differential equations. , 1989 .

[62]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

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

[64]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[65]  Peter Deuflhard,et al.  Numerische Mathematik. I , 2002 .

[66]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[67]  R. Gribonval,et al.  Bi-framelet systems with few vanishing moments characterize Besov spaces , 2004 .

[68]  Jens Lang,et al.  On Global Error Estimation and Control for Initial Value Problems , 2007, SIAM J. Sci. Comput..

[69]  GermanyNumerische Mathematik,et al.  Multilevel Preconditioning , 1992 .

[70]  Alexander Ostermann,et al.  Rosenbrock methods for partial differential equations and fractional orders of convergence , 1993 .

[71]  R. Gribonval,et al.  Nonlinear approximation with bi-framelets , 2005 .

[72]  H. Triebel.  Theory Of Function Spaces , 1983 .

[73]  Willem Hundsdorfer,et al.  A Second-Order Rosenbrock Method Applied to Photochemical Dispersion Problems , 1999, SIAM J. Sci. Comput..

[74]  Wolfgang Dahmen,et al.  A semigroup approach to the numerical solution of parabolic differential equations , 2005 .

[75]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[76]  K. Gröchenig.  Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator , 2004 .

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

[78]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods : Basic Concepts and Applications to the Stokes Problem , 2002 .

[79]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods for Saddle Point Problems - Optimal Convergence Rates , 2002, SIAM J. Numer. Anal..

[80]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[81]  Albert Cohen,et al.  Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition , 2000, Numerische Mathematik.

[82]  P. Lemarié,et al.  Base d'ondelettes sur les groupes de Lie stratifiés , 1989 .

[83]  J. M. Sanz-Serna,et al.  Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations , 1987 .

[84]  Wolfgang Dahmen,et al.  Local Decomposition of Refinable Spaces and Wavelets , 1996 .

[85]  Wolfgang Dahmen,et al.  Stable multiscale bases and local error estimation for elliptic problems , 1997 .

[86]  Claudio Canuto,et al.  The wavelet element method. Part II: Realization and additional features in 2D and 3D , 1997 .

[87]  Vom Fachbereich Mathematik,et al.  Stabile biorthogonale Spline-Waveletbasen auf dem Intervall , 2006 .

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

[89]  Rob Stevenson,et al.  On the Compressibility of Operators in Wavelet Coordinates , 2004, SIAM J. Math. Anal..

[90]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[91]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .

[92]  Wolfgang Dahmen,et al.  Adaptive Wavelet Schemes for Elliptic Problems - Implementation and Numerical Experiments , 2001, SIAM J. Sci. Comput..

[93]  Wolfgang Dahmen,et al.  Sparse Evaluation of Compositions of Functions Using Multiscale Expansions , 2003, SIAM J. Math. Anal..

[94]  Martin – Luther – Universität Halle – Wittenberg Institut für Mathematik High-order finite element – linearly implicit two-step peer methods for time-dependent PDEs , 2006 .

[95]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[96]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[97]  W. Dahmen.  Stability of Multiscale Transformations. , 1995 .

[98]  Joachim Rang,et al.  New Rosenbrock W-Methods of Order 3 for Partial Differential Algebraic Equations of Index 1 , 2005 .

[99]  Massimo Fornasier,et al.  Intrinsic Localization of Frames , 2005 .

[100]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[101]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[102]  K. Gröchenig.  Describing functions: Atomic decompositions versus frames , 1991 .

[103]  Rob P. Stevenson,et al.  An optimal adaptive wavelet method without coarsening of the iterands , 2006, Math. Comput..

[104]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[105]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[106]  C. Canuto,et al.  Numerical solution of elliptic problems by the Wavelet Element Method , 1998 .

[107]  Function Theory , 1951, Nature.

[108]  Albert Cohen,et al.  Wavelet Methods for Second-Order Elliptic Problems, Preconditioning, and Adaptivity , 1999, SIAM J. Sci. Comput..

[109]  Wolfgang Dahmen,et al.  Composite wavelet bases for operator equations , 1999, Math. Comput..

[110]  I. Lasiecka.  Unified theory for abstract parabolic boundary problems—a semigroup approach , 1980 .

[111]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[112]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[113]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[114]  Albert Cohen,et al.  Wavelet methods in numerical analysis , 2000 .

[115]  Karlheinz Gröchenig,et al.  Localization of frames II , 2004 .

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

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

[118]  Dongwoo Sheen,et al.  A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature , 2003 .

[119]  Wolfgang Dahmen,et al.  Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations , 1997 .