Vorträge

In this talk, I would like to introduce myself and the topic of my master thesis "Malliavin calculus and Malliavin-Stein method".
As indicated by its name, this talk provides a basic overview of the Malliavin calculus and its operators in the case where the underlying process is an isonormal Gaussian process. After this introduction, it is shown how the Malliavin calculus can be combined with Stein's method for the purpose of one-dimensional normal approximation and, particularly, for the derivation of generalized central limit theorems.

This presentation introduces the semi-implicit Smoothed Particle Hydrodynamics (SPH)
scheme [1] for the shallow water equations following the semi-implicit finite volume and finite
difference approach of Casulli [2]. In standard explicit numerical methods, there is often a severe
limitation on the time step due to the stability restriction imposed by the CFL condition. To this
effect, a semi-implicit SPH scheme is derived, which leads to an unconditionally stable method.
The discrete momentum equation is substituted into the discrete continuity equation to obtain
a symmetric positive definite linear system for the free surface elevation. The resulting system
can be easily solved by a matrix-free conjugate gradient method. Once the new free surface
location is known, the velocity at the new time level can be directly computed and the particle
positions can subsequently be updated. We further discuss a nonlinear algorithm for treating
wetting/drying problems. We derive a mildly nonlinear system for the discrete free surface
elevation from the shallow water equations by taking into consideration a correct mass balance
in wet regions and in transition regions, i.e. the regions from wet particles to dry particles
and those from dry particles to wet particles. The scheme is validated on a two dimensional
inviscid hydrostatic free surface flows for the two dimensional shallow water equations and
wetting/drying test problem.

References
[1] A.O. Bankole, A. Iske, T. Rung, M. Dumbser, A meshfree semi-implicit Smoothed Particle
Hydrodynamics method for free surface flow. Meshfree Methods for Partial Differential
Equations VIII, M. Griebel and M.A. Schweitzer (eds.), Springer LNCSE, Vol. 115, pp.
35-52 (2017).
[2] V. Casulli, Semi-Implicit Finite Difference Methods for the Two-Dimensional Shallow
Water Equations. Jour. of Comp. Phys., Vol 86. pp. 56-74 (1990).

Meeting-ID: 820 3979 6993
Passwort: 694649

Solving differential equations rigorously is a main and vigorous topic in the
field of verified computation. Here, solving rigorously means that a computer
program supplies an approximate solution along with error bounds that respect
all numerical as well as all rounding errors that occurred during the computation.
An exact solution is proved to be enclosed within these rigorous bounds.
In this context so-called Taylor models have been used successfully for solving
ordinary differential equations (ODEs) rigorously. Implementations are COSY INFINITY [1], FLOW [2], ODEIntegretor [3], and RIOT [4]. Here, COSY INFINITY
developed by Berz and Makino and their group is the most advanced
implementation. Recently, we implemented the Taylor model approach in MATLAB/
INTLAB [5].

We give a short introduction to Taylor models, their rigorous arithmetic,
and the Taylor model method for enclosing solutions of ordinary differential
equations in a verified manner. We only treat initial value problems
$y_0 = f(t,y)$, $y(t_0) = y_0$
where the initial value $y_0$ may be an interval vector. For specific ODEs we demonstrate
how to use and call our verified ODE solver. This is designed to be very
similar to calling MATLAB's non-verified ODE solvers like ode45. Finally, results
and run times are compared to those of COSY INFINITY, RIOT and Lohner's
classical AWA.

[1] M. Berz, K. Makino, COSY INFINITY: www.bt.pa.msu.edu/index_cosy.htm
[2] X. Chen, Reachability analysis of non-linear hybrid systems using Taylor models,
Dissertation RWTH Aachen, 2015. FLOW: https://flowstar.org/dowloads/
[3] T. Dzetkulic, Rigorous integration of non-linear ordinary differential equations in
Chebyshev basis, Numer. Algor. 69, 183-205, 2015.
ODEintegrator: https://sourceforge.net/projects/odeintegrator
[4] I. Eble, Über Taylor-Modelle, Dissertation at Karlsruhe Inst. of Technology, 2007.
RIOT: www.math.kit.edu/ianm1/~ingo.eble/de
[5] S.M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing
(ed. by Tibor Csendes), Kluwer Academic Publishers, 77-104, 1999.
INTLAB: http://www.ti3.tu-harburg.de/intlab/

Vortrag (PDF, 100KB)