Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

People Research Publications Teaching

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In this research seminar we discuss modern multirate time integration methods for the solution of systems of ordinary differential equations with significant different time scales in the components. The goal is to develop new methods by means of operator-splitting approaches, the error due to the communication between fine and coarse grid solution and to examine the stability of the overall process.

Literature:

• A. Bátkai, P. Csomós, K.J. Engel, B. Farkas, Stability and convergence of product formulas for operator matrices, Int. Eq. Op. Th. 74(2) (2012), 281-299.
• B. Kehlet, A. Logg, A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations, Numerical Algorithms 76(1) (2017), 191-210.
• J.M. Sexton, D.R. Reynolds, Relaxed Multirate Infinitesimal Step Methods for Initial-Value Problems, arXiv:1808.03718 [math.NA] (March 2019).
• J.H. Chaudhry, D. Estep, V. Ginting, S. Tavener, A posteriori analysis of an iterative multi-discretization method for reaction-diffusion systems, Computer Methods in Applied Mechanics and Engineering 267, (2013), 1-22.
• J.H. Chaudhry, D. Estep, S. Tavener, V. Carey, J. Sandelin, A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm, SIAM J. Numer. Anal. 54(5) (2016), 2974-3002.
• J.B. Collins, D. Estep, S. Tavener, A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques, BIT Numerical Mathematics 55(4) (2015), 1017-1042.
• D. Estep, V. Ginting, S. Tavener, A Posteriori analysis of a multirate numerical method for ordinary differential equations, Comput. Methods Appl. Mech. Engrg. 223-224 (2012), 10-27.
• D. Estep, V. Ginting, D. Ropp, J. Shadid, S. Tavener, An a posteriori-a priori analysis of multiscale operator splitting, SIAM J. Numer. Anal. 46 (2008) 1116-1146.
• M. Striebel, M. Günther, A charge oriented mixed multirate method for a special class of index - 1 network equations in chip design, Appl. Numer. Math. 53 (2005), 489-507.
• V. Savcenco, W. Hundsdorfer, J.G. Verwer, A multirate time stepping strategy for stiff ordinary differential equations, BIT Numerical Mathematics 47(1) (2007), 137-155.
• M. Günther, A. Kværnø, O. Rentrop, Multirate Partitioned Runge-Kutta Methods, BIT Numerical Mathematics 41(3) (2001), 504-514.
• H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.
• S. Roberts, A. Sarshar, A. Sandu, Coupled Multirate Infinitesimal GARK Schemes for Stiff Systems with Multiple Time Scales, arXiv:1812.00808 [math.NA] (Feb 2019).
• A. Sarshar, S. Roberts, A. Sandu, Design of High-Order Decoupled Multirate GARK Schemes, arXiv:1804.07716 [cs.NA] (Nov 2018).
• L. Bonaventura, F. Casella, L. Delpopolo, A. Ranade, A self adjusting multirate algorithm based on the TR-BDF2 method, arXiv:1801.09118 [math.NA] (Jan 2018).
• K.R. Green, R.J. Spiteri, Gating-enhanced IMEX splitting methods for cardiac monodomain simulation, Numerical Algorithms, (2019), 1-15.
• J. Cervi, R.J. Spiteri, Methods and Algorithms for Scientific Computing High-Order Operator Splitting for the Bidomain and Monodomain Models, SIAM J. Sci. Comput., 40(2), A769-A786.
• J. Cervi, R.J. Spiteri, High-Order Operator-Splitting Methods for the Bidomain and Monodomain Models, in: D. Boffi, L. Pavarino, G. Rozza, S. Scacchi, C. Vergara (eds.) Mathematical and Numerical Modeling of the Cardiovascular System and Applications, SEMA SIMAI Springer Series, vol 16. Springer, 2018, pp. 23-40.

  Semigroups for Partial Differential Equations
• K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer, 2000.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer, 1983.
  Operator Splitting
• A. Bátkai, P. Csomós, B. Farkas, G. Nickel, Operator splitting with spatial-temporal discretization, Spectral theory, mathematical system theory, evolution equations, differential and difference equations, 161-171, Oper. Theory Adv. Appl. 221, Birkhäuser/Springer Basel AG, Basel, 2012.
• J. Geiser, Decomposition Methods for Differential Equations: Theory and Applications, Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series, 2009.
• J. Geiser, Iterative Splitting Methods for Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series, 2011.
  Fokker-Planck Equation
• B. Gaviraghi, M. Annunziato, A. Borzì,
  • Chapter 22: Splitting Methods for Fokker-Planck Equations Related to Jump-Diffusion Processes, pp. 409-422.
  • Chapter 23: A Fokker-Planck Based Approach to Control Jump Processes, pp. 423-439.
  in M. Ehrhardt, M. Günther, E.J.W. ter Maten (eds.), Novel Methods in Computational Finance, Springer, 2017.
• B. Gaviraghi, M. Annunziato, A. Borzì, Analysis of splitting methods for solving a partial-integro differential Fokker-Planck equation, Applied Mathematics and Computation 294 (2016), 1-17.
• M. Mohammadi, A. Borzì, Hermite approximation of a hyperbolic Fokker-Planck optimality system to control a piecewise-deterministic process, International Journal of Control 89 (2016), 1382-1395.
• M. Mohammadi, A. Borzì, A Hermite spectral method for a Fokker-Planck optimal control problem in an unbounded domain, International Journal for Uncertainty Quantification 5 (2015), 233-254.
• M. Mohammadi, A. Borzì, Analysis of the Chang-Cooper Discretization Scheme for a Class of Fokker-Planck Equations, Journal of Numerical Mathematics 23 (2015), 271-288.
• M. Mohammadi, Analysis of discretization schemes for Fokker-Planck equations and related optimality systems, PhD Thesis, University of Würzburg, Würzburg, Germany, Feb 2015.
• L. Pareschi, M.Zanella, Structure Preserving Schemes for Nonlinear Fokker-Planck Equations and Applications, Journal of Scientific Computing 74(3) (2018), 1575-1600.
• G. Toscani, A Rosenau-type approach to the approximation of the linear Fokker-Planck equation, arXiv:1703.10909 [math.NA] (March 2017).

  Events

• Workshop , April 2019, Budapest, Hungary.

  Projects

• bilateral German-Hungarian DAAD Project Coupled Systems and Innovative Time Integrators, (01/2019-12/2020)



Scheinkriterium:

  Präsentation

• Erläuterung des Anwendungsproblems
• Mathematische Modellierung (evtl. mehrstufig)
• Lösung / Analyse / Numerik
• Diskussion der Resultate und ihrer Relevanz für die Anwendung
• Für die Präsentation wird Prosper bzw. HA-Prosper oder ifmslide oder PPower4 oder Beamer empfohlen
  ( Verschiedene Präsentationsprogramme)
• Evtl. numerische Simulationen (möglichst in Matlab, Scilab oder Octave)
  Schriftliche Ausarbeitung (ca. 10 Seiten, inkl. Beispiele)
  Regelmässige Teilnahme am Seminar

Vorkenntnisse: Basiswissen mathematischer Grundvorlesungen wird vorausgesetzt.

Didaktische Vortragstipps:

• M. Lehn, Wie halte ich einen Seminarvortrag, (pdf-file)
• S.P. Jones, How to write a good research paper and give a good research talk
• I. Parberry, How to present a paper in theoretical computer science: A speaker's guide for students, Bulletin of the EATCS,(37), 1989.
• S.P. Jones, J. Launchbury, J. Hughes, How to give a good research talk, SIGPLAN Notices 28(11), November 1993.
• G. Aiglstorfer, A short guide for students talks and papers, TU Munich, 2004.
• H. Kraft, Das Verfassen und Präsentieren wissenschaftlicher Arbeiten, TU Munich, 2006.
Vortragsliste

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University of Wuppertal Faculty of Mathematics and Natural Sciences Department of Mathematics Applied Mathematics & Numerical Analysis Group

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