The DNA Seminar
The Differential Equations and Numerical Analysis Seminar
This seminar is a continuation of two seminar series: One on numerical analysis and one on differential equations (DIFTA).
See also Seminar on Geometric Mechanics.
Fall 2011
| Date | Speaker and title |
|---|---|
| 14.12.2011 room 1329 14:15-15:00 Wednesday | Andreas Asheim (Leuven) Regarding the absolute stability of Störmer-Cowell methods The field of astronomy has throughout history been one of the great suppliers of state-of-the-art numerical methods for ODEs. Among the earliest higher-order methods were the Størmer methods, designed by the C. F. Størmer in the early 1900s for computations related to Birkeland's aurora theory. Early numerical analysts pointed out that high order Størmer-Cowell methods, a class of methods containing Størmer's method, suffer from orbital instabilities, i.e. Kepler orbits would spiral either inwards or outwards. However, looking closely it appears that the exact mechanism behind the orbital instabilities were never investigated; apparently, some of the most cited papers in the field even got it wrong. But then recently, while cleaning his office, Syvert Nørsett found a forgotten note. Syvert and Even Thorbergsen, a master student of Syvert, investigated the stability of Størmer-Cowell methods in 1976. The work was however not entirely complete, and has not been published in any journal. In this talk I will give an outline of these results, the completion of the analysis, and place the work in an historical context. |
| 01.12.2011 room 734 14:15-15:00 Thursday | Christian Rohde (Stuttgart) Examples for Multi-Scale Modelling with Conservation Laws In the talk we will discuss several numerical multiscale methods which fit into the framework of heterogeneous multiscale methods in the sense of E&Enquist. The first part of the talk is devoted to problems where the micro-scale information will only be effective at localized subsets of the domain under consideration. As the macro-scale model we consider hyperbolic systems of conservation laws while the micro-scale model is a (usually complex) regularization of the macro-scale system. The new approach will be demonstrated on the dynamics of overshoot waves in porous media and the dynamics of phase boundaries in two-phase elastic bars. The major issue is the efficient computation of the micro-scale model which takes most of the overall computational time. In second part of the talk we will consider cell dynamics in tissues which can be described on the "micro-scale" level by a kinetic approach for the cell density. Following an analyis of Hillen we will introduce a macro-scale moment system which takes a form similar to those of the gas dynamics equations. However, it contains a pressure-like term that still depends on solution of the kinetic system. We present a numerical method that takes into account this cell level information and display several numerical simulations. Note that in contrast to the problems described above the micro-scale information is needed in the complete domain. This is joint work with A. Corli, F. Kissling, P. Engel and Ch. Surulecu. |
| 01.12.2011 room 734 15:15-16:00 Thursday | Jerome Droniou (Montpellier 2) Analysis of a fractal conservation law arising from a model of detonations in gases We will first recall some results, established in the last ten years or so, on Fractal conservation laws of the kind \[\partial_t u + \partial_x (f(u)) + (-\Delta)^{\lambda/2}u=0.\] These results concern the well-posedness of the equation and some qualitative properties on the solution (smoothness, presence of shocks, etc.). The fractional power of the Laplace operator \((-\Delta)^{\lambda/2}\) appearing in the equation can be define by Fourier transform as the operator with symbol \(|\xi|^{\lambda}\). We will then consider a more general version of the equation, appearing in a model of detonation in gases and in which \((-\Delta)^{\lambda/2}\) is replaced by a pseudo-differential operator whose symbol only asymptotically behaves as \(|\xi|^\lambda\). We will study this new equation, showing how some above-mentioned results can be extended to this model. |
| 24.11.2011 room 1329 13:15 Thursday | Begona Cano Projected explicit Lawson methods for the integration of Schrödinger equation Abstract: Some second-order explicit projected Lawson methods are introduced as an efficient tool to integrate solitary waves of cubic Schrödinger equation.The fact that projecting orthogonally onto the mass implies projecting orthogonally onto the momentum is seen to happen for any explicit RK Lawson methodas well as for some other non-solitary wave solutions. An analytical explanation will be given. |
| 24.11.2011 room 1329 14:15 Thursday | Jason Frank Statistical measure sampling and corrections for discretized geophysical fluids Abstract: Time series data from long numerical simulations of partial differential equations are frequently subjected to statistical analysis (e.g. weather/climate). However the choice of numerical method typically implies some bias on the statistical distributions. In fluid dynamics, for example, only a few specialized numerical discretizations are able to approximate the equilibrium statistical mechanics distributions predicted by recent theories. We discuss the ingredients for accurate statistical sampling. In molecular dynamics, thermostats have been developed to perturb the trajectories of a simulation such that on a long time scale, they sample a given equilibrium measure, for example, the Gibbs distribution. We discuss an approach to model reduction using thermostats for the case of a point vortices. |
| 24.11.2011 room 1329 15:15 Thursday | Serhiy Zhuk Uncertainty propagation for dynamical systems: a minimax approach Abstract: It is common in applications that a mathematical model describing dynamics of some process contains uncertain parameters reflecting the incomplete knowledge of initial data or imperfect physical parametrisation. In this context, robust control strategy of forecast based on the uncertain model should take into account available description of uncertain parameters. The probabilistic uncertainty description relies on statistics available for uncertain parameters and uses empirical probability distributions so corresponding control strategies are formulated in terms of some operations on probability distributions. However, in practice, it is not easy to guess the correct probability distribution for uncertain parameters: only bounding sets describing all possible values of these parameters are available. In this case, the minimax uncertainty description may be used. The talk describes main geometric notions of the minimax approach (admissible tube, reachability set, worst-case error) and presents minimax state estimation and control algorithms for dynamical systems described by non-linear evolution equations. We discuss main advantages of the minimax approach compared to statistical methods. We illustrate the presented approach on data assimilation problems in air quality and dynamical optical flow estimation in image processing. |
| 06.11.2011 Monday! room 1329 | Sebastian Reich (University of Potsdam, Germany) Ensemble transform filters for geophysical data assimilation Abstract: Data assimilation is the task to combine model based simulations with measurements to provide optimal state and/or parameter estimates. Data assimilation requires that one can quantify the uncertainty in the mathematical model forecasts. Bayes' theorem is used to assimilate observations into the model forecast and to reduce uncertainties. I will discuss the data assimilation filtering problem in the context of a McKean-Vlasov system for the time evolution of the probability density function characterizing model uncertainty. Specific algorithms are obtained by using ensemble/particle methods for the time evolution of the probability density functions under the model dynamics and by fitting statistical models to the ensemble of particles such as Gaussians and Gaussian mixture models prior to a data assimilation step. Contrary to particle filters (or sequential Monte Carlo methods, the data assimilation step itself is implemented as a transport of the ensemble/particles under continuous transformations. The resulting filters can be viewed as generalizations of the popular ensemble Kalman filter to non-Gaussian probability density functions. |
| 06.11.2011 Monday! room 734 | Mechthild Thalhamer (University of Innsbruck) Adaptive space and time discretisations for nonlinear Schrödinger equations Abstract: In this talk, I shall address the issue of efficient numerical methods for the space and time discretisation of nonlinear Schrödinger equations such as systems of coupled time-dependent Gross–Pitaevskii equations arising in quantum physics for the description of multi-component Bose-Einstein condensates. For the considered class of problems, a variety of contributions confirms the favourable behaviour of pseudo-spectral and exponential operator splitting methods regarding efficiency and accuracy. However, due to the fact that in the absence of an adaptive local error control in space and time, the reliability of the numerical solution and the performance of the space and time discretisation strongly depends on the experienced scientist selecting the space and time grid in advance, I will exemplify different approaches for the reliable time integration of Gross–Pitaevskii systems on the basis of a local error control for splitting methods. Joint work with Leopold–Franzens Universität Innsbruck, Austria. |
| 02.11.2011 14:15-15:00 | Kurush Ebraimi-Fard (ICMAT Madrid) Pre-Lie algebras in numerical analysis. Abstract: The notion of pre-Lie algebras can be traced back to the works of Vinberg and Gerstenhaber. Recently, pre-Lie algebras appeared in different fields. To name a few, we mention theoretical physics, where pre-Lie algebras play an important role in the context of perturbative renormalization theory; control theory, where they are known as Agrachev-Gamkrelidze's chronological algebras; theory of numerical integration methods, where they naturally appear in the context of the theory of Butcher and Lie-Butcher series. In this talk we try to give a panoramic overview of recent developments around the theory of pre-Lie algebras, emphasizing its appearance in numerical analysis. |
| 28.09.2011 14:15-16:00 room 1329 | Erik Lindgren (NTNU) Regularity for the obstacle problem Abstract: The aim of this talk is to describe parts of the regularity theory for the obstacle problem for the Laplacian, almost from scratch. The focus will be on 1) the optimal regularity of the solution and 2) some partial regularity of the free boundary. |
| 05.10.2011 15:15-16:00 room 1329 Wednesday | Hermano Frid (IMPA, Brazil) Spatially periodic solutions for a polytropic gas flow with varying specific heat Abstract:In this talk, we discuss the global existence of spatially periodic solutions for a model of polytropic gas flow with varying specific heat, where the latter is assumed to satisfy suitable smoothness and decay properties. We show that the initial total variation over one period may be taken as large as we wish as long as the mean specific heat is sufficiently large. The case of \(L^\infty\) solution, with vacuum, is also reviewed. Finally, a non-homogeneous entropy inequality implies the decay of the solution to the mean value as \(t\to\infty\). This talk reviews joint works joint with N.~Risebro and H.~Sande, and with H.~Holden and K.~Karlsen. |
| 14.10.2011 13:15-14:00 room 734 Friday | Mats Ehrnström (Hannover) A variational approach to a class of nonlocal evolution equations and existence of solitary waves of the Whitham equation Abstract: We prove the existence of solitary-wave solutions for a class of nonlocal evolution equations of the norm \[u_t + [ n(u) + Lu ]_x = 0.\] The linear operator \(L\) is nonlocal of negative order, whereas the nonlinearity \(n\) is local and of superlinear growth near the origin. Using the methods of minimisation–penalisation and concentration–compactness we find periodic solutions converging to conditionally stable, smooth minimisers of small amplitude. Our analysis includes the case of the Whitham equation, the linear terms of which match the dispersion relation for gravity water waves on finite depth. The Whitham equation has a global bifurcation branch of \(2\pi\)-periodic, smooth, traveling-wave solutions and is conjectured to admit a `highest', cusped, wave. For the solitary case, we show that the Whitham minimisers approximate the KdV-solitons in the small-amplitude limit. The talk is based on ongoing work with Mark Groves, Saarbrücken, and Erik Wahlén, Lund, as well as joint work with Henrik Kalisch, Bergen. |