7 research outputs found
A variational approach to modeling slow processes in stochastic dynamical systems
The slow processes of metastable stochastic dynamical systems are difficult
to access by direct numerical simulation due the sampling problem. Here, we
suggest an approach for modeling the slow parts of Markov processes by
approximating the dominant eigenfunctions and eigenvalues of the propagator. To
this end, a variational principle is derived that is based on the maximization
of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be
estimated from statistical observables that can be obtained from short
distributed simulations starting from different parts of state space. The
approach forms a basis for the development of adaptive and efficient
computational algorithms for simulating and analyzing metastable Markov
processes while avoiding the sampling problem. Since any stochastic process
with finite memory can be transformed into a Markov process, the approach is
applicable to a wide range of processes relevant for modeling complex
real-world phenomena
Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
The global behavior of dynamical systems can be studied by analyzing the
eigenvalues and corresponding eigenfunctions of linear operators associated
with the system. Two important operators which are frequently used to gain
insight into the system's behavior are the Perron-Frobenius operator and the
Koopman operator. Due to the curse of dimensionality, computing the
eigenfunctions of high-dimensional systems is in general infeasible. We will
propose a tensor-based reformulation of two numerical methods for computing
finite-dimensional approximations of the aforementioned infinite-dimensional
operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD).
The aim of the tensor formulation is to approximate the eigenfunctions by
low-rank tensors, potentially resulting in a significant reduction of the time
and memory required to solve the resulting eigenvalue problems, provided that
such a low-rank tensor decomposition exists. Typically, not all variables of a
high-dimensional dynamical system contribute equally to the system's behavior,
often the dynamics can be decomposed into slow and fast processes, which is
also reflected in the eigenfunctions. Thus, the weak coupling between different
variables might be approximated by low-rank tensor cores. We will illustrate
the efficiency of the tensor-based formulation of Ulam's method and EDMD using
simple stochastic differential equations
Effective dynamics for a kinetic Monte-Carlo model with slow and fast time scales
We consider several multiscale-in-time kinetic Monte Carlo models, in which
some variables evolve on a fast time scale, while the others evolve on a slow
time scale. In the first two models we consider, a particle evolves in a
one-dimensional potential energy landscape which has some small and some large
barriers, the latter dividing the state space into metastable regions. In the
limit of infinitely large barriers, we identify the effective dynamics between
these macro-states, and prove the convergence of the process towards a kinetic
Monte Carlo model. We next consider a third model, which consists of a system
of two particles. The state of each particle evolves on a fast time-scale while
conserving their respective energy. In addition, the particles can exchange
energy on a slow time scale. Considering the energy of the first particle, we
identify its effective dynamics in the limit of asymptotically small ratio
between the characteristic times of the fast and the slow dynamics. For all
models, our results are illustrated by representative numerical simulations
Pseudo generators of spatial transfer operators
Metastable behavior in dynamical systems may be a significant challenge for a
simulation based analysis. In recent years, transfer operator based approaches
to problems exhibiting metastability have matured. In order to make these
approaches computationally feasible for larger systems, various reduction
techniques have been proposed: For example, Sch\"utte introduced a spatial
transfer operator which acts on densities on configuration space, while Weber
proposed to avoid trajectory simulation (like Froyland et al.) by considering a
discrete generator.
In this manuscript, we show that even though the family of spatial transfer
operators is not a semigroup, it possesses a well defined generating structure.
What is more, the pseudo generators up to order 4 in the Taylor expansion of
this family have particularly simple, explicit expressions involving no
momentum averaging. This makes collocation methods particularly easy to
implement and computationally efficient, which in turn may open the door for
further efficiency improvements in, e.g., the computational treatment of
conformation dynamics. We experimentally verify the predicted properties of
these pseudo generators by means of two academic examples
A direction preserving discretization for computing phase-space densities
Ray flow methods are an efficient tool to estimate vibro-acoustic or electromagnetic energy transport in complex domains at high-frequencies. Here, a Petrov--Galerkin discretization of a phase-space boundary integral equation for transporting wave energy densities on two-dimensional surfaces is proposed. The directional dependence of the energy density is approximated at each point on the boundary in terms of a finite local set of directions propagating into the domain. The direction of propagation can be preserved for transport across multicomponent domains when the directions within the local set are inherited from a global direction set. The range of applicability and computational cost of the method will be explored through a series of numerical experiments, including wave problems from both acoustics and elasticity in both single and multicomponent domains. The domain geometries considered range from both regular and irregular polygons to curved surfaces, including a cast aluminium shock tower from a Range Rover car
Etude mathématique de modèles quantiques et classiques pour les matériaux aléatoires à l'échelle atomique
Les contributions de cette thèse portent sur deux sujets.La première partie est dédiée à l'étude de modèles de champ moyen pour la structure électronique de matériaux avec des défauts.Dans le chapitre~ref{chap:ergodic_crystals}, nous introduisons et étudions le modèle de Hartree-Fock réduit (rHF) pour des cristaux désordonnés. Nous prouvons l'existence d'un état fondamental et établissons, pour les interactions de Yukawa (à courte portée), certaines propriétés de cet état. Dans le chapitre~ref{chap:défauts_étendus}, nous considérons des matériaux avec des défauts étendus. Dans le cas des interactions de Yukawa, nous prouvons l'existence d'un état fondamental, solution de l'équation auto-cohérente. Nous étudions également le cas de cristaux avec une faible concentration de défauts aléatoires. Dans le chapitre~ref{chap:numerical_simuation}, nous présentons des résultats de simulations numériques de systèmes aléatoires en dimension un.Dans la deuxième partie, nous étudions des modèles Monte-Carlo cinétique multi-échelles en temps. Nous prouvons, pour les trois modèles présentés au chapitre~ref{chap:kMC}, que les variables lentes convergent, dans la limite de la grande séparation des échelles de temps, vers une dynamique effective. Nos résultats sont illustrés par des simulations numériques.The contributions of this thesis concern two topics.The first part is dedicated to the study of mean-field models for the electronic structure of materials with defects. In Chapter~ref{chap:ergodic_crystals}, we introduce and study the reduced Hartree-Fock (rHF) model for disordered crystals. We prove the existence of a ground state and establish, for (short-range)Yukawa interactions, some properties of this ground state. In Chapter~ref{chap:défauts_étendus}, we consider crystals with extended defects. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. We also investigate the case of crystals with low concentration of random defects. In Chapter~ref{chap:numerical_simuation}, we present some numerical results obtained from the simulation of one-dimensional random systems.In the second part, we consider multiscale-in-time kinetic Monte Carlo models. We prove, for the three models presented in Chapter~ref{chap:kMC}, that in the limit of large time-scale separation, the slow variables converge to an effective dynamics. Our results are illustrated by numerical simulations.CERGY PONTOISE-Bib. electronique (951279901) / SudocSudocFranceF