Program
The program is the following. It includes large breaks to stimulate discussions.
Notice that the posters will remain visible during the whole duration of the workshop.
Place: University of Geneva, Uni Bastions in lecture hall B106 (first floor).
Université de Genève, Uni Bastions. Rue De-Candolle 5, 1205 Genève, Switzerland.
Wednesday 29th January 2020
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Thursday 30th January 2020
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Titles and abstracts of the presentations
- Assyr Abdulle (EPF Lausanne)
Stabilized explicit multirate methods for ordinary and stochastic differential equations with multiple scales
We present a new class of explicit multirate stabilized (Chebyshev) methods for ordinary and stochastic differential equations with multiple scales. The new method takes advantage of the decomposition of the right-hand side into two parts: one associated with relatively slower time-scales, and a cheap but severely stiff part associated with fast time-scales. The scheme has two components: an outer Chebyshev method that integrates a modified damped equation and a stabilisation procedure that requires the solution of an auxiliary system which depends only on the fast components of the original dynamics and is integrated with an “inner” Chebyshev method.
Unlike classical Chebyshev methods whose computational cost is governed by the fastest scale in the system of differential equations, in our multirate scheme only the inner Chebyshev method needs to resolve the cheap but severely stiff part associated with fast time-scales. In addition, no assumptions on scale separation is needed in contrast to other recently proposed multiscale methods.
The works presented are joint works with M. Grote and G. Rosilho de Souza [1],[2].
References:
[1] A. Abdulle, M. Grote and G. Rosilho de Souza, Stabilized explicit multirate methods for stiff differential equations, Preprint.
[2] A. Abdulle and G. Rosilho de Souza, Stabilized explicit multirate methods for stiff stochastic differential equations, Preprint. - Anaïs Crestetto (University of Nantes)
Micro-macro discretizations for collisional kinetic equations of Boltzmann-BGK type in the diffusive scaling
This talk aims to present asymptotic preserving (AP) schemes, based on micro-macro decomposition and particle method, for kinetic equations of Boltzmann-BGK type in the diffusive scaling.
The objective is to obtain an efficient numerical method that (i) verifies the AP property, (ii) has a computational cost similar to a fluid scheme when the limit is approached, (iii) is free from too restrictive stability condition, (iv) can be extended and used in three dimensional in space and in velocity (3Dx-3Dv) framework.
First, a one-order in time discretization will be presented for the 1Dx-1Dv radiative transport equation. It is based on a suitable formulation of the micro-macro model and the use of weighted particle method for the microscopic (kinetic) part. After that, we will discuss some extensions: a second-order in time discretization, its application to the 1Dx-1Dv Vlasov-BGK equation, the use of Monte Carlo method (instead of weighted particle method) in the radiative transport equation case and its application to the 2Dx-2Dv and 3Dx-3Dv frameworks.
Numerical results will illustrate the efficiency of this approach.
This work is a collaboration with Nicolas Crouseilles (University of Rennes and Inria Rennes - Bretagne Atlantique), Giacomo Dimarco (University of Ferrara) and Mohammed Lemou (University of Rennes and CNRS). - Jason Frank (Utrecht University)
A detectability criterion for sequential data assimilation
In [J. Frank & S. Zhuk, Nonlinearity, 2018] we propose a new continuous-time sequential data assimilation method having the general form of a Kalman-Bucy filter. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. - Laurence Halpern (University Paris 13)
Multiscale analysis of a transmission problem
Domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods. We are interested here in the case where there is an accurate but expensive model one should use in the entire domain, but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected. For the model problem of a time dependent advection reaction diffusion equation in one spatial dimension, we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain. Using a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain, we propose a heterogeneous domain decomposition method based on factorization of the underlying differential operator and show that it has better approximation properties than more classical variational or non-variational heterogeneous domain decomposition methods. - Carsten Hartmann (BTU Cottbus-Senftenberg)
Rare event simulation of slow-fast systems
Importance sampling is a variance reduction method that minimises the variance of a Monte Carlo estimator by appropriately changing the underying probability measure. For diffusion processes, the optimal change of measure can be realised by adding a control term to the equations, which then leads to a linear-quadratic stochastic regulator problem (with nonlinear state dependence). We study the situation, in which the dynamics has multiscale features, and the quantity of interest is a rare event associated with the slowest degrees of freedom. Based on a dynamic programming formulation of the control problem, we discuss conditions under which the optimal control (or: change of measure) can be expressed solely in terms of the slow variables, and we give quantitative estimates for the relative error of the corresponding rare event estimator. - Lise-Marie Imbert-Gérard (University of Maryland)
Wave propagation in inhomogeneous media: an introduction to Generalized Plane Waves
Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions of the Partial Differential Equation (PDE), making explicit use of information about the ambient medium. But wave propagation problems in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available. Generalized Plane Waves (GPWs) are functions that have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they are not exact solutions to the PDE but are instead constructed locally as high order approximate solutions. We will discuss the origin, the construction, and the properties of GPWs. The construction process introduces a consistency error, requiring a specific analysis. - Annika Lang (Chalmers/University of Gothenburg)
Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs
Consider the computation of quadratic functionals of the solution to a linear stochastic partial differential equation with multiplicative noise. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. Weak error rates are derived for a fully discrete approximation of the stochastic partial differential equation using the results obtained from the approximation of the Lyapunov equation. Numerical simulations illustrate the theoretical results.
This is joint work with Adam Andersson (Smartr) and Andreas Petersson (Chalmers/University of Gothenburg). - Frédéric Legoll (ENPC, ParisTech)
Multiscale stochastic dynamics: effective dynamics and parareal computations
The question of coarse-graining is ubiquitous in many applied sciences. In a multiscale context, it is typically characterized by a mapping which projects the full state of the system (containing slow and fast variables) to a smaller class of variables (where there is no fast modes). While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. Starting with a non-reversible dynamics, we study an effective dynamics which approximates the (non-closed) projected dynamics. Under fairly weak conditions on the system (encoding in particular the time-scale separation between slow and fast modes), we prove error bounds on the trajectorial error between the projected and the effective dynamics.
We next discuss how to adapt the parareal algorithm (which concurrently use the reference and the effective dynamics) to such contexts. In particular, we numerically investigate several strategies for going from the macroscopic to the microscopic representation of the system, and for the iterative updating of the solution.
Joint work with T. Lelievre and U. Sharma (ENPC), and with K. Myerscough and G. Samaey (KU Leuven). - Gabriel Lord (Radboud University)
Numerics and a model for the stochastically forced vorticity equation
This talk will introduce the stochastically forced vorticity equation and a reduced stochastic differential equation model. Through numerical simulation we illustrate that the SDE model captures the infinite dimensional behaviour. The talk will discuss some of the issues in performing numerical simulations for these systems.
This work is joint with : Margaret Beck, Eric Cooper and Konstantinos Spiliopoulos. - Greg Pavliotis (Imperial College London)
Mean field limits for weakly interacting diffusions: phase transitions, multiscale analysis and fluctuations
In this talk we will present some recent results on the mean field limit of weakly interacting periodic oscillators in the presence of thermal noise, the so-called noisy Kuramoto model. We show that in the mean field limit, described by the McKean-Vlasov equation, a nonlinear, nonlocal PDE of Fokker-Planck type, the model exhibits phase transitions, in the sense that as sufficiently low temperatures there are multiple steady states. Furthermore, we study the joint mean field-diffusive limits and we show that, when the phase transition has occurred, the two limits do not commute. Furthermore, we discuss about Gaussian and non-Gaussian fluctuations around the mean field (law of large numbers) limit. - Caroline Wormell (University of Sydney)
Linear response in high-dimensional chaotic systems
The response of long-term averages of observables of chaotic systems to small, time-invariant dynamical perturbations can often be predicted to first order using linear response theory (LRT), but some very basic chaotic systems are known to have a non-differentiable response. However, complex dissipative chaotic systems' macroscopic observables are widely assumed to have a linear response, but the mechanism for this is not well-understood.
We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional weakly coupled deterministic dynamical systems, where the weak coupling is via a mean field and the microscopic subsystems may or may not obey LRT. Through a stochastic reduction to mean-field dynamics, we provide conditions for linear response theory to hold both in large, finite-dimensional systems and in the thermodynamic limit. In particular, we demonstrate that in large systems of finite size, linear response is induced by self-generated noise.
Conversely, we will present an example in the thermodynamic limit where the macroscopic observable does not satisfy LRT, despite all microscopic subsystems satisfying LRT when uncoupled. This latter example is associated with emergent non-trivial dynamics of the macroscopic observable.
This is joint work with Georg Gottwald. - Kostas Zygalakis (University of Edinburgh)
Hybrid modeling for the stochastic simulation of multi-scale chemical kinetics
It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be adequately modeled by Markov processes and, for such systems, methods such as Gillespie’s algorithm are typically employed. While such schemes are easy to implement and are exact, the computational cost of simulating such systems can become prohibitive as the frequency of the reaction events increases. This has motivated numerous coarse-grained schemes, where the “fast” reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation for systems where all reactants are present in large concentrations, the approximation breaks down when the fast chemical species exist in small concentrations, giving rise to significant errors in the simulation. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as computing observables of cell cycle models.
In this talk, we present a hybrid scheme for simulating well-mixed stochastic kinetics, using Gillespie–type dynamics to simulate the network in regions of low reactant concentration, and chemical Langevin dynamics when the concentrations of all species are large. These two regimes are coupled via an intermediate region in which a “blended” jump-diffusion model is introduced. Examples of gene regulatory networks involving reactions occurring at multiple scales, as well as a cell-cycle model are simulated, using the exact and hybrid scheme, and compared, both in terms of weak error, as well as computational cost. If there is time, we will also discuss the extension of these methods for simulating spatial reaction kinetics models, blending together partial differential equation with compartment based approaches, as well as compartment based approaches with individual particle models.
Titles and abstracts of posters
- Ibrahim Almuslimani (University of Geneva)
Explicit stabilized integrators for stiff optimal control problems
Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this poster we present explicit stabilized integrators of orders one and two for optimal control problems. We analyze their favourable stability and symplecticity properties based on the continuous optimality system. Numerical experiments including the optimal control of Burgers equation illustrate the efficiency of the new approach. - Asma Barbata (Faculty of science of Monastir Tunisia)
Observer based control for large scale interconnected system with multiplicative noises
In this paper, we propose a new theorem for large scale interconnected system with \ell subsystem to ensure the almost sure exponential stability, this theorem is used to design an observer based control for stochatic nonlinear systems. - Guillaume Bertoli (University of Geneva)
Strang splitting method for semilinear parabolic problems with inhomogeneous boundary conditions: a correction based on the flow of the nonlinearity
The Strang splitting method, formally of order two, can suffer from order reduction when applied to semilinear parabolic problems with inhomogeneous boundary conditions. The recent work [L .Einkemmer and A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions. SIAM J. Sci. Comput., 37, 2015. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., 38, 2016] introduces a modification of the method to avoid the reduction of order based on the nonlinearity. In this paper we introduce a new correction constructed directly from the flow of the nonlinearity and which requires no evaluation of the source term or its derivatives. The goal is twofold. One, this new modification requires only one evaluation of the diffusion flow and one evaluation of the source term flow at each step of the algorithm and it reduces the computational effort to construct the correction. Second, numerical experiments suggest it is well suited in the case where the nonlinearity is stiff. We provide a convergence analysis of the method for a smooth nonlinearity and perform numerical experiments to illustrate the performances of the new approach. - Faten Ezzine (Faculty of science of Sfax)
Non-linear integral inequalities and applications to asymptotic stability of deterministic and stochastic perturbed systems
Our basic goal of this paper is to study the behavior of the solutions of the stochastic and deterministic perturbed systems with respect to the solutions of the stochastic unperturbed system. In the second part, we present sufficient conditions for global practical uniform exponential stability of stochastic perturbed systems based on Gronwall inequalities by means of generalized Gronwall inequalities. Furthermore, a numerical example is presented to illustrate the applicability of the our result. - Tahar Khalifa (University of Sousse)
Output feedback regulation of a class of stochastic nonlinear system via sampled-data control
Download abstract (PDF). - Adrien Laurent (University of Geneva)
Order conditions for sampling the invariant measure of ergodic SDEs on manifolds
We derive a new methodology for the construction of high order integrators for sampling the invariant measure of ergodic stochastic differential equations subjected to a scalar constraint. For a class of Runge-Kutta type methods, we derive the conditions for the order two of accuracy. Numerical experiments in dimension 3 on the sphere and the torus confirm the theoretical findings. We also discuss a possible extension of the exotic aromatic B-series formalism. - Atef Lechiheb (University of Sousse)
Homogenization of linear Evolution Equations with generalised broad-sense stationary random Initial Conditions
Download abstract (PDF). - Conor McCoid (University of Geneva)
Robust Algorithm for Triangle Intersections
We present an algorithm for the calculation of the intersection between two triangles in two dimensions. The algorithm is asymmetric: one triangle is treated differently than the other. This allows the algorithm to be robust, and we can prove that the algorithm is backwards stable. A sketch of the proof is also presented. - Tommaso Vanzan (University of Geneva)
Multilevel Optimized Schwarz Methods and Application to the Stokes-Darcy system
Optimized Schwarz methods are domain-decomposition methods based on enhanced transmission conditions which are optimized in order to accelerate the convergence. We introduce a multilevel optimized Schwarz method where the transmission conditions are tuned not to improve the convergence behavior but instead the smoothing property of the iterative scheme on each grid. We present convergence results both with overlap and without overlap, which also suggests how to choose the optimized parameters in a two-level settings. Numerical results show the effectiveness of our approach and we apply the method to the Stokes-Darcy coupling.