Bilateral German-Hungarian Project

CSITI - Coupled Systems and Innovative Time Integrators

financed by DAAD and Magyar Ösztöndíj Bizottság (MÖB).

(01/2019 - 12/2020)

Scientific goals

Coupled systems and the underlying equations provide the mathematically tractable model for many phenomena that occur in physics, biology, chemistry, social sciences or economics. Hence the efficient numerical treatment of such problems is at the basis of our civilization and therefore has been attracting an increasing amount of attention parallel to the technological development. We carry out the mathematical study of three mathematical procedures for the numerical solution of such coupled problems:

Operator splitting
Magnus integrators
Exponential Runge-Kutta methods

We further develop these, and provide the theoretical basis for their applicability. For testing the developed methods we propose three problems to be solved numerically, each of which addresses real-life questions of vital importance and is mathematically modelled by a coupled system:

(plastic) contamination of Arctic
urban smog
early phase of planet formation.

The coupling highly depends on the actual problem. Due to the complicated structure of each problem, we need to apply an innovative time integrator (or even their combination) to obtain an efficient numerical time integration method.

Main objective:

Since coupled systems of our interest contain various types of equations (linear, semilinear, non-autonomous linear, nonlinear), in the present project we study first each innovative integrator's (splitting, exponential Runge-Kutta, and Magnus) numerical features (such as convergence, stability, preservation of qualitative properties, etc.) separately for problems formulated as abstract Cauchy problems. Then we focus on the three real-life problems outlined above. Since each of these coupled systems has a rather complicated structure, we need to combine the innovative time integrators to derive effective numerical methods for solving them. Thus, in the computer implementation phase we will study the theoretical and numerical properties of the combined methods.

German team:

Christian Budde
Anna Clevenhaus
Matthias Ehrhardt
Bálint Farkas (Leader)
Lorenc Kapllani
Tatiana Kossaczká
Michelle Muniz

Hungarian team:

Ágnes Bodó
Szilvia Császár
Petra Csomós (Leader)
Bálint Máté Takács

German institutions:

Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Hungarian institutions:

Eötvös Loránd University, Budapest, Department of Applied Analysis and Computational Mathematics

Publications related to the Project

2019

Lorenc Kapllani, Multistep Schemes for Solving Backward Stochastic Differential Equations on GPU, Master Thesis, University of Wuppertal, March 2019, (Supervisors: Long Teng, Matthias Ehrhardt).
L. Kapllani, L. Teng and M. Ehrhardt, A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs, Preprint 19/20, June 2019.
P. Csomós, B. Takács, Operator splitting for space-dependent epidemic model, submitted to Applied Numerical Mathematics
S. Kovács, Sz. György, On the qualitative behaviour of parameter dependent systems, Miskolc Mathematical Notes, to appear.
S. Treibert, H. Brunner and M. Ehrhardt, Compartment models for vaccine effectiveness and non-specific effects for Tuberculosis, Mathematical Biosciences and Engineering, 2019, 16(6): 7250-7298. DOI:10.3934/mbe.2019364 (open access)
B. Érdi, E. Kövári, The kite central configurations of four bodies with three equal masses, Symmetry, submitted.
C. Budde, Positive Miyadera-Voigt perturbations of bi-continuous semigroups, arXiv:1912.07334, submitted, Dec. 2019
C. Budde and R. Heymann, Extrapolation of operator-valued multiplication operators, arXiv:1911.08175, submitted, Nov. 2019.
C. Budde and M. Kramar Fijavž, Bi-continuous semigroups for flows in infinite networks, arXiv:1901.10292, submitted, Jan. 2019.
S. Kovács, Sz. György, N. Gyúró, On an invasive species model with harvesting, BIOMAT series, submitted, 2019.
C. Budde, General extrapolation spaces and perturbations of bi-continuous semigroups, Ph.D. thesis, University of Wuppertal, 2019, (Supervisor: Bálint Farkas).
E. Kövári, Linearized stability of the central four-body problem with axial symmetry (in Hungarian), Master's thesis, Loránd Eötvös University, Budapest, 2019, (Supervisor: Bálint Érdi).

2020

L. Teng, X. Wu, M. Günther and M. Ehrhardt, A new methodology to create valid time-dependent correlation matrices via isospectral flows, ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), Volume 54, Number 2, (March-April 2020), 361-371. DOI:10.1051/m2an/2019064
M. Muniz, M. Ehrhardt and M. Günther, Approximating correlation matrices using stochastic Lie group methods, Preprint 20/05, March 2020.
A. Clevenhaus, M. Ehrhardt, M. Günther and D. Sevcovic, Pricing American Options with a Non-constant Penalty Parameter, J. Risk Financial Manag. 2020, 13(6), 124. DOI 10.3390/jrfm13060124
P. Csomós, M. Ehrhardt and B. Farkas, Operator splitting for abstract Cauchy problems with dynamical boundary condition, accepted: Operators and Matrices (2020).

Talks related to the Project

2019

Bálint Farkas, Analytic Semigroups, Budapest, April 15, 2019.
Christian Budde, An Introduction to non-autonomous problems, Budapest, April 15, 2019.
Szilvia György, Exponential dichotomy and stability of delay differential equations, Budapest, April 15, 2019.
Michelle Muniz, Lie group methods for matrix ODEs and SDEs, Budapest, April 15, 2019.
Matthias Ehrhardt, Operator Splitting Schemes, Budapest, April 16, 2019.
Petra Csomós, Exponential and Magnus integrators, Budapest, April 16, 2019.
Anna Clevenhaus, ADI Schemes for option pricing in the Heston model with stochastic correlation, Budapest, April 17, 2019.
Bálint Takács, Numerical solution of space-dependent SIR models, Budapest, April 17, 2019.
Emese Kövári, Central configurations of four bodies with an axis of symmetry, Wuppertal, December 12, 2019.
Michelle Muniz, Using stochastic Lie group methods to approximate correlation matrices in financial markets, Wuppertal, December 12, 2019.
Anna Clevenhaus, Pricing American Options with an adaptive Penalization Strategy and the Combination Technique, Wuppertal, December 12, 2019.
Lorenc Kapllani, A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs, Wuppertal, December 12, 2019.

2020

Oct. 2020
Matthias Ehrhardt, Miklós Farkas Seminar on Applied Analysis, Budapest University of Technology and Economics, Budapest, April 9, 2020. (shifted to October 2020)