HIM Junior Trimester Program Computational Mathematics

Numerical methods in molecular simulation (Bonn, April 7-11th 2008)

Proposed talks and lectures

 Wim Briels Coarse graining of soft matter systems

 Christophe Chipot Reaction coordinates versus order parameters in free energy calculations

 Aline Kurtzmann Self-interacting diffusions

 We will study the asymptotic behavior and ergodic (limit quotient) properties of the family of processes $X$, solution to SDEs $$dX_t = dB_t - [V'(X_t) + 1/t \int_0^t \partial_x W(X_t,X_s)ds] dt.$$ These processes have so far been studied on compact spaces by Benaïm, Ledoux and Raimond, using stochastic approximation methods. We extend these methods to $\mathbb{R}^d$, assuming a confinement potential satisfying some conditions. These hypotheses on the confinement potential are required since in general the process can be transient, and is thus very difficult to analyze.

 Cosmin Marinica Stability and mobility of small interstitials clusters in iron

 The microstructural evolution of the radiation damage in materials science needs to manage the physics of infrequent-event systems where several time-scales are typically contained: the point defects diffusion (self-interstitial atoms, vacancies), the defect clustering (growth or dissociation), the dislocations glide, the lattice vibrations etc. The energy landscape of self-interstitial atoms, the competition between the configurations with high and low mobility is an important issue in the microstructural evolution of ferritic materials (e.g. defect clustering under/or after irradiation). The interstitial defects in iron have atypical properties and theirs dynamics are not completely understood. A method for systematic search of the minima and saddle points configurations, the activation relaxation technique "nouveau" (ARTn), is implemented and applied to the physics of self-interstitial defects in iron. We propose some simple algorithms to improve the efficiency of the ARTn for this particular system. The energy landscape at zero K of small interstitial clusters (from 1 to 4 self-interstitial atoms) is presented using and recent empirical potential. The results are discussed and compared to the molecular dynamics simulations performed at higher temperatures. New self-interstitial configurations and migration paths are reported.

 Paul Fleurat-Lessard A chemist view on reaction path

 Computationnal chemistry has now become a standard tool to evaluate energy and free energy differences along a reaction path. More, new methods (like the Transition Path Theory or the Transition Path Sampling) have recently emerged that allow chemists to find the free energy path connecting the reactant to the products without any bias. However, when dealing with chemical reactions, accurate description of bond formation and breaking requires good electronic potentials. Most of the time, one relies on ab initio programs to obtain the desired accuracy. This leads in turn to simulation costs much higher than for classical force fields. As a consequence, one must look for ways to i) generate initial path as close as possible to the unknown actual path, and ii) efficient ways to optimize this path. In the molecular electronic structure world, lots of work have been devoted to the coordinate system used in geometry optimizations. In this presentation, we will test the efficiency of different coordinate systems to generate a good initial path and to optimize it. These tests will be conducted on simple reactions using cartesian, Z-Matrix and redundant coordinates.

 Arnaud Guillin Long time behavior of Markov processes: Various approaches

 We present an Adaptively Biased Molecular Dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving potential. It has several useful features, including a favorable scaling with time and a small number of control parameters. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures or collective variables. This is beneficial not only in terms of speed and accuracy of a calculation, but also in terms of the amound of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Mme peptide in a gaseous and solvated environment.

 Manuel Athènes Mapping equilibrium and non-equilibrium entropy landscapes from path-sampling

 Several recent studies have investigated the relevance of computing equilibrium thermodynamic quantities by means of non-equilibrium techniques such as path-sampling. We review these approaches in the context of free energy calculations. The advantage of path-sampling with respect to standard sampling methods for computing equilibrium probability densities in systems exhibiting rugged energy landscapes is illustrated on the 38-atom Lennard-Jones cluster. An extension of path-sampling will also be presented : it will be shown that the non-equilibrium probability densities and the non-equilibrium entropy for systems driven out-of-equilibrium can also be measured by means of a space-time thermodynamic integration.

 Pierre Del Moral Particle Rare Event Stochastic Simulation Methods

 Benjamin Jourdain Numerical analysis of the QMC method in a simple case

 In this talk, we will present the Diffusion Monte Carlo method which is widely used in chemistry to compute the electronic ground state energy of a molecule. We will explain the bias of the DMC estimator which is a a consequence of the so-called Fixed Node Approximation. Finally, on a simplified model, we will analyse the numerical error introduced when computing this estimator thanks to particles evolving according to a stochastic differential equation discretized in time and interacting through selection steps.

 Roland Assaraf The generalized correlated sampling approach: toward an exact calculation of energy derivatives in Diffusion Monte Carlo.

 The Diffusion Monte Carlo (DMC) method is a stochastic approach used to solve the Schroedinger equation. The algorithm consists in a stochastic dynamic producing a sample of the unkwnown groundstate (lowest eigenstate) of a given Hamiltonian (Schroedinger operator). The groundstate energy (lowest eigenvalue of the operator) is then readily calculated as an average of some simple local function over that sample. However, interesting physical properties can in general be obtained from derivatives of these energies with respect to some parameter of the Hamiltonian. Computing exactly these derivatives with finite an small statistical fluctuations is a challenging problem especially because the sampled distribution and hence its derivatives are not analitically known. Only the stochastic process is known. In this work we present a strategy based on the calculation of the derivative of the stochastic process, trough a generalized correlated sampling scheme. In essence, two close but different stochastic process are correlated. The stability of the method is determined by the behaviour of the mean square distance between the two processes and depends on the chaotic properties of the stochastic dynamical system. Different applications on different molecular systems are shown, enlighting different stability behaviours. We discuss some strategies to enhance the stability, in order to develop a robust method.

 Raz Kupferman Averaging and coarse-graining: the mathematics of modeling

 In many applications, the governing equations cannot be solved directly, neither analytically nor numerically, due to the complexity of the system, which may be a combination of nonlinearity and high-dimension. In this context, the term "modeling" refers to the replacement of the complex system by a simpler one, usually of lower-dimension, that is tractable, and whose solutions approximate well the solutions of the original system. Modeling approaches are ubiquitous in the modern sciences, and range from methods that are based on rigorous analysis through computational dimensional-reduction algorithms, to uncontrolled approximations based on heuristic considerations. In these lectures I will review some of the history of this field, and describe in detail a number of particular systems.

 Dimitrios Tsagkarogiannis Reconstruction schemes for coarse-grained models

 Starting from a microscopic stochastic lattice system and the corresponding coarse-grained model we suggest a mathematical strategy to recover microscopic information given the coarse-grained data. This amounts to comparing the initial microscopic measure conditioned on some coarse-grained configuration (multi-canonical constraint) with the suggested reconstructed measures. The comparison is done in the context of the relative entropy norm between the two microscopic measures. Moreover, we give a hierarchy of reconstructed measures and suggest numerical schemes of improving accuracy. This is joint work with Jose Trashorras.

 Sylvain Maire A Monte Carlo method to compute principal eigenelements of some linear operators

 We describe a Monte Carlo method to compute the principal eigenvalue of some linear operators like neutron transport operators or the Laplace operator. The principal eigenvalue is obtained by combining the spectral expansion of the solution of the Cauchy problem and its Feynman-Kac approximation. We study the accuracy of our MC estimators and propose a branching mecanism to reduce the variance which also gives an approximation of the leading eigenfunction of the adjoint operator.

 Andreas Eberle Quantitative approximations of evolving probability measures

 Under global mixing conditions, we derive non-asymptotic bounds for the approximation of evolving probability measures by an interacting particle system that is a continuous time version of a sequential Monte Carlo sampler. The results show that sometimes already a limited number of particles is enough to guarantee reasonable approximation properties even when the relative density of the final and the initial measure is extremely singular.

 Carsten Hartmann Model reduction for partially-observed stochastic differential equations

 We study model reduction for stochastic differential equations from the perspective of linear control theory. In doing so, we adopt ideas from large deviations theory and ask to which extend certain variables in the system are "controllable" by noise and give observable output. Model reduction then consists in, firstly, transforming the system in such a way that those degrees of freedom that are least sensitive to the noise also give the least output (balancing), and, secondly, neglecting the respective low energy modes (truncation). For partially-observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.

 Markos Katsoulakis Coarse-graining, reconstruction and importance sampling methods for the simulation of many-particle stochastic systems

 We discuss recent work on coarse-graining methods for microscopic stochastic lattice systems. We emphasize the numerical analysis of the schemes, focusing on error quantification as well as on the construction of improved algorithms capable of operating in wider parameter regimes. We also present adaptive coarse-graining schemes which have the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. The methods employed in the development and the analysis of the algorithms rely on a combination of statistical mechanics methods (renormalization and cluster expansions), statistical tools (reconstruction and importance sampling) and PDE-inspired analysis (a posteriori estimates). We also discuss the connections and extensions of our work on lattice systems to the coarse-graining of polymers.

 Cedric Bernardin Fourier law for Hamiltonian microscopic dynamics perturbed by a conservative noise

 We review recent results for a system of (an)harmonic oscillators perturbed by a noise conserving energy or energy and momentum. (with G. Basile and S.Olla)

 Greg Pavliotis From ballistic to diffusive motion in periodic potentials

 The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.

 Stephan De Bièvre Motion through an oscillator chain: diffusion and linear response

 (collaboration with P. Parris, A. Silvius and P. Lafitte) We study the fully Hamiltonian motion of a particle through a chain of uncoupled monochromatic harmonic oscillators in thermal equilibrium. We show that, in contrast to what is commonly believed, such a monochromatic heat bath induces normal transport properties for the particle, namely diffusive behaviour in absence of a driving field and linear response to such a field, with a well-defined low-field mobility satisfying the Einstein relation.