Adaptive Quarklet Methods for the numerical Solution of Elliptic Partial Differential Equations with exponential Convergence DFG project number: 528343051 PI: Dr. Marc Hovemann Duration: 2023-2025
This project is concerned with the development of efficient next generation adaptive numerical methods for the solution of partial differential equations using the recently introduced Quarklets with provable exponential convergence. For that purpose we work with biorthogonal compactly sup- ported Cohen-Daubechies-Feauveau spline wavelets, that are enriched with polynomials of certain degree, namely the so-called Quarklets. They allow for the construction of wavelet counterparts of adaptive hp−finite element methods. To develop effective Quarklet methods with exponential convergence in a first step we have to assemble multivariate Quarklet systems with high smooth- ness, that can be used to characterize advanced function spaces such as Besov- or Triebel-Lizorkin spaces. Those multivariate Quarklets will be used to design adaptive near-best Quarklet tree ap- proximation techniques for given multivariate functions. For such methods we want to identify associated approximation classes, namely a large class of multivariate functions, for that our adap- tive near-best Quarklet tree approximation procedure converges with exponential order. In a next step an efficient next generation adaptive Quarklet PDE solver will be developed, that is based on a damped Richardson iteration, and uses our adaptive near-best Quarklet tree approximation technique as an important building block. For our Quarklet PDE solver then situations (in terms of conditions on the PDE, the underlying domain and the right-hand side) should be identified, such that exponential convergence can be proved. For that purpose several techniques out of regularity theory for PDEs will be used. Along with the project goes the practical implementation of all numerical Quarklet methods we designed.
LOEWE-Center 'Natur 4.0: Sensing Biodiversity' Subproject UM3: Transformation, Regularisation and Classification PI: Professor Dr. Stephan Dahlke Duration: 2019-2023 Team: Sven Heuer
Adaptive Quarklet Frame Methods of high order for elliptic operator equations PI: Professor Dr. Stephan Dahlke, Professor Dr. Thorsten Raasch DFG project number 451355735 Duration: 2021-2023 Team Marburg: Dr. Marc Hovemann
Kernel based multi level methods for approximation problems of high dimensions on sparse grids - Derivation, Analysis and Application in the Uncertainty Quantification PIs: Professor Dr. Christian Rieger; Professor Dr. Holger Wendland DFG projekt number 452806809 Duration: 2021-2024 Team Marburg: Federico Lot
Former projects
New Smoothness Spaces on Domains and their Discrete Characterizations within the D-A-CH framework (DA 360/22-1) (2018 - 2021)
New Smoothness Spaces on Domains and their Discrete Characterizations within the D-A-CH framework (DA 360/22-1) (2018 - 2021)
Regularity Theory of Stochastic Partial Differential Equations in (Quasi-)Banach Spaces (2014 - 2015)
Optimal Adaptive Finite Element and Wavelet Methods for p-Poisson Equations (2013 - 2016)
Adaptive Wavelet and Frame Techniques for Acoustic BEM within the D-A-CH framework (2013 - 2016)
Adaptive Wavelet Methods for SPDEs, Part II within DFG-SPP 1324 "Extraction of quantifiable information from complex systems" (2012 - 2015)
Coordination of SPP 1324 "Mathematical Methods for Extracting Quantifiable Information from Complex Systems", Phase II (2011-2014)
Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems, Part II within DFG-SPP 1324 "Extraction of quantifiable information from complex systems" (2011 - 2013)
Sensitivitätsanalyse, Parameterbestimmung und Modellvalidierung für komplexe biologische Prozesse (2010 - 2014)
Dynamik regulatorischer Netzwerke für Zellpolarität (2010 - 2014)
Uncertainty Principles versus Localization Properties, Function Systems for Efficient Coding Schemes - UNLocX (2010 - 2013)
Adaptive Wavelet Methods for SPDEs, Part I within DFG-SPP 1324 "Extraction of quantifiable information from complex systems" (2009 - 2012)
Coordination of SPP 1324 "Mathematical Methods for Extracting Quantifiable Information from Complex Systems", Phase I (2008-2011)
Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems, Part I within DFG-SPP 1324 "Extraction of quantifiable information from complex systems" (2008 - 2011)
Adaptive Wavelet Methods for Inverse Problems and Inverse Parabolic Equations (2006 - 2009)
Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part III within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung" (2005 - 2007)
Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part II within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung" (2003 - 2005)
Multivariate Wavelet Analysis: Constructions, Specific Applications, and Data Structures, Part I within DFG-SPP 1114 "Mathematische Methoden der Zeitreihenanalyse und digitalen Bildverarbeitung" (2001 - 2003)