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Irreversible multi-scale diffusions: time scales and model reduction
Abstract
In this work, we consider non-reversible multi-scale stochastic processes, described by stochastic differential equations, for which we review theory on the convergence behaviour to equilibrium and mean first exit times. Relations between these time scales for non-reversible processes are established, and, by resorting to a control theoretic formulation of the large deviations action functional, even the consideration of hypo-elliptic processes is permitted. The convergence behaviour of the processes is studied in a lot of detail, in particular with respect to initial conditions and temperature. Moreover, the behaviour of the conditional and marginal distributions during the relaxation phase is monitored and discussed as we encounter unexpected behaviour. In the end, this results in the proposal of a data-based partitioning into slow and fast degrees of freedom. In addition, recently proposed techniques promising accelerated convergence to equilibrium are examined and a connection to appropriate model reduction approaches is made. For specific examples this leads to either an interesting alternative formulation of the acceleration procedure or structural insight into the acceleration mechanism. For the model order reduction technique of effective dynamics, which uses conditional expectations, error bounds for non-reversible slow-fast stochastic processes are obtained. A comparison with the reduction method of averaging is undertaken, which, for non-reversible processes, possibly yields different reduced equations. For Ornstein-Uhlenbeck processes sufficient conditions are derived for the two methods (effective dynamics and averaging) to agree in the infinite time scale separation regime. Additionally, we provide oblique projections which allow for the sampling of conditional distributions of non-reversible Ornstein-Uhlenbeck processes.
... Since B 12 is a constant matrix, we only need to consider V arρε t, 12 , which is well-defined since the time-dependent covariance for ρ ε t is well-defined. The latter follows because of the bounds on the limiting covariance (see [53,Ex. 2.14] for an explicit formula of the time-dependent covariance for ρ ε t ). ...
This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Leli\`evre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples.
In this paper, we revisit energy-based concepts of controllability and reformulate them for control-affine nonlinear systems perturbed by white noise. Specifically, we discuss the relation between controllability of deterministic systems and the corresponding stochastic control systems in the limit of small noise and in the case in which the target state is a measurable subset of the state space. We derive computable expression for hitting probabilities and mean first hitting times in terms of empirical Gramians, when the dynamics is given by a Hamiltonian system perturbed by dissipation and noise, and provide an easily computable expression for the corresponding controllability function as a function of a subset of the state variables.
In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Planck equations in terms of large-deviation rate functionals.
We consider the exit event from a metastable state for the overdamped Langevin dynamics . Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring-Kramers formula, in the small temperature regime . We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring-Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently small in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring-Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process.
We analyse and interpret the effects of breaking detailed balance on the convergence to equilibrium of interacting particle systems and their hydrodynamic scaling limits. For finite systems of interacting particles, we review existing results showing that irreversible processes converge faster to their steady state than reversible ones. We show how this behaviour appears in the hydrodynamic limit of such processes, as described by macroscopic fluctuation theory, and we provide a quantitative expression for the acceleration of convergence in this setting. We give a geometrical interpretation of this acceleration, in terms of currents that are antisymmetric under time-reversal and orthogonal to the free energy gradient, which act to drive the system away from states where (reversible) gradient-descent dynamics result in slow convergence to equilibrium.
Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a N-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.
Importance sampling is a widely used technique to reduce the variance of a
Monte Carlo estimator by an appropriate change of measure. In this work, we
study importance sampling in the framework of diffusion process and consider a
change of measure which is realized by adding a control force to the original
dynamics. For certain exponential type expectation, the corresponding control
force of the optimal change of measure leads to a zero-variance estimator and
is related to the solution of a Hamilton-Jacobi-Bellmann equation. We prove
that for a certain class of multiscale diffusions, the control force obtained
from the limiting dynamics is asymptotically optimal, and we provide an error
bound for the importance sampling estimators under such suboptimal controls. We
also discuss two other situations in which one can approximate the optimal
control force by solving simplified dynamics. We demonstrate our approximation
strategy with several numerical examples and discuss its application to
large-scale systems, e.g. from molecular dynamics or material science.
In recent papers it has been demonstrated that sampling a Gibbs distribution
from an appropriate time-irreversible Langevin process is, from several points
of view, advantageous when compared to sampling from a time-reversible one.
Adding an appropriate irreversible drift to the overdamped Langevin equation
results in a larger large deviations rate function for the empirical measure of
the process, a smaller variance for the long time average of observables of the
process, as well as a larger spectral gap. In this work, we concentrate on
irreversible Langevin samplers with a drift of increasing intensity. The
asymptotic variance is monotonically decreasing with respect to the growth of
the drift and we characterize its limiting behavior. For a Gibbs measure whose
potential has one or more critical points, adding a large irreversible drift
results in a decomposition of the process in a slow and fast component with
fast motion along the level sets of the potential and slow motion in the
orthogonal direction. This result helps understanding the variance reduction,
which can be explained at the process level by the induced fast motion of the
process along the level sets of the potential. Correspondingly the limit of the
asymptotic variance is the asymptotic variance of the limiting slow motion
which is a diffusion process on a graph.
The text contains a review and new results on stochastic averaging via mixing bounds.
The singular perturbation approximation technique for model reduction is related to the direct truncation technique if the system model to be reduced is stable, minimal, and internally balanced. It is shown that these two methods constitute two fully compatible model reduction techniques for a continuous-time system, and both methods yield a stable, minimal, and internally balanced reduced-order system with the same L∞-norm error bound on the reduction. Although the upper bound for both reductions is the same, the direct truncation method tends to have smaller errors at high frequencies and larger errors at low frequencies, whereas the singular perturbation approximation method will display the opposite character. It is also shown that a certain bilinear mapping not only preserves the balanced structure between a continuous-time system and an associated discrete-time system, but also preserves the slow singular perturbation approximation structure. Hence, the continuous-time results on the singular perturbation approximation of balanced systems are easily extended to the discrete-time case. Examples are used to show the compatibility of and the differences in the two reduction techniques for a balanced system
We consider non-reversible perturbations of reversible diffusions that do not
alter the invariant distribution and we ask whether there exists an optimal
perturbation such that the rate of convergence to equilibrium is maximized. We
solve this problem for the case of linear drift by proving the existence of
such optimal perturbations and by providing an easily implementable algorithm
for constructing them. We discuss in particular the role of the prefactor in
the exponential convergence estimate. Our rigorous results are illustrated by
numerical experiments.
For a Markov process the detailed balance condition is equivalent to the
time-reversibility of the process. For stochastic differential equations
(SDE's) time discretization numerical schemes usually destroy the property of
time-reversibility. Despite an extensive literature on the numerical analysis
for SDE's, their stability properties, strong and/or weak error estimates,
large deviations and infinite-time estimates, no quantitative results are known
on the lack of reversibility of the discrete-time approximation process. In
this paper we provide such quantitative estimates by using the concept of
entropy production rate, inspired by ideas from non-equilibrium statistical
mechanics. The entropy production rate for a stochastic process is defined as
the relative entropy (per unit time) of the path measure of the process with
respect to the path measure of the time-reversed process. By construction the
entropy production rate is nonnegative and it vanishes if and only if the
process is reversible. Crucially, from a numerical point of view, the entropy
production rate is an {\em a posteriori} quantity, hence it can be computed in
the course of a simulation as the ergodic average of a certain functional of
the process (the so-called Gallavotti-Cohen (GC) action functional). We compute
the entropy production for various numerical schemes such as explicit
Euler-Maruyama and explicit Milstein's for reversible SDEs with additive or
multiplicative noise. Additionally, we analyze the entropy production for the
BBK integrator of the Langevin processes. We show that entropy production is an
observable that distinguishes between different numerical schemes in terms of
their discretization-induced irreversibility. Furthermore, our results show
that the type of the noise critically affects the behavior of the entropy
production rate.
Large time asymptotics of the solutions to non-symmetric Fokker- Planck type equations are studied by extending the entropy method to this case. We present a modified Bakry-Emery criterion that yields covergence of the solution to the steady state in relative entropy with an explicit exponen- tial rate. In parallel it also implies a logarithmic Sobolev inequality w.r.t. the steady state measure. Explicit examples illustrate that skew-symmetric per- turbations in the Fokker Planck operator can "help" to improve the constant in such a logarithmic Sobolev inequality.
We give a new formula expressing the components of the mean force in terms of a conditional expectation which can be computed by Blue Moon sampling. This generalizes to the vectorial case a formula first derived by Ruiz-Montero et al. [Mol. Phys. 1997, 90, 925] for a scalar reaction coordinate. We also discuss how to compute this conditional average by means of constrained stochastic dynamics which, unlike the usual constrained molecular dynamics, introduces no bias. Finally, we give a new perspective on bias removal by using constrained molecular dynamics.
We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.
We consider operators of Kramers–Fokker–Planck type in the semi-classical limit such that the exponent of the associated Maxwellian
is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete
asymptotics of the exponentially small splitting between the first two eigenvalues.
On considère des opérateurs du type de Kramers–Fokker–Planck dans la limite semi-classique tels que l’exposant du maxwellien
associé soit une fonction de Morse avec deux minima et un point selle. Sous des hypothèses supplémentaires convenables on
établit un développement asymptotique complet de l’écart exponentiellement petit entre les deux premières valeurs propres.
We consider a diffusion on a potential landscape which is given by a smooth
Hamiltonian in the regime of low temperature
. We proof the Eyring-Kramers formula for the optimal constant in
the Poincar\'e (PI) and logarithmic Sobolev inequality (LSI) for the associated
generator of the diffusion.
The proof is based on a refinement of the two-scale approach introduced by
Grunewald, Otto, Westdickenberg and Villani; and of the mean-difference
estimate introduced by Chafa\"i and Malrieu. The Eyring-Kramers formula follows
as a simple corollary from two main ingredients: The first one shows that the
PI and LSI constant of the diffusion restricted to metastable regions
corresponding to the local minima scales well in . This mimics the
fast convergence of the diffusion to metastable states. The second ingredient
is the estimation of a mean-difference by a weighted transport distance. It
contains the main contribution to the PI and LSI constant, resulting from
exponentially long waiting times of jumps between metastable states of the
diffusion.
We present how entropy estimates and logarithmic Sobolev inequalities on the
one hand, and the notion of quasi-stationary distribution on the other hand,
are useful tools to analyze metastable overdamped Langevin dynamics, in
particular to quantify the degree of metastability. We discuss the interest of
these approaches to estimate the efficiency of some classical algorithms used
to speed up the sampling, and to evaluate the error introduced by some
coarse-graining procedures. This paper is a summary of a plenary talk given by
the author at the ENUMATH 2011 conference.
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. Comment: 21 pages
The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable , where x describes the configuration of the system in a high-dimensional space , and is a smooth function with value in (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on , the equilibrium properties on are completely determined by the free energy. On the other hand, the question of the effective dynamics on is much more difficult to address. Starting from an overdamped Langevin equation on , we propose an effective dynamics for using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.
This introduction to multiscale methods gives readers a broad overview of the many uses and applications of the methods. The book begins by setting the theoretical foundations of the subject area, and moves on to develop a unified approach to the simplification of a wide range of problems which possess multiple scales, via perturbation expansions; differential equations and stochastic processes are studied in one unified framework. The book concludes with an overview of a range of theoretical tools used to justify the simplified models derived via the perturbation expansions. The presentation of the material is particularly suited to the range of mathematicians, scientists and engineers who want to exploit multiscale methods in applications. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable readers to build their own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter. All of the twenty-one chapters are supplemented with exercises.
This volume is a research expository article on the applied mathematics of turbulent dynamical systems through the paradigm of modern applied mathematics. It involves the blending of rigorous mathematical theory, qualitative and quantitative modeling, and novel numerical procedures driven by the goal of understanding physical phenomena which are of central importance to the field. The contents cover general framework, concrete examples, and instructive qualitative models. Accessible open problems are mentioned throughout.
Topics covered include:
· Geophysical flows with rotation, topography, deterministic and random forcing
· New statistical energy principles for general turbulent dynamical systems, with applications
· Linear statistical response theory combined with information theory to cope with model errors
· Reduced low order models
· Recent mathematical strategies for online data assimilation of turbulent dynamical systems as well as rigorous results for finite ensemble Kalman filters
The volume will appeal to graduate students and researchers working mathematics, physics and engineering and particularly those in the climate, atmospheric and ocean sciences interested in turbulent dynamical as well as other complex systems.
We estimate the total variation and Kantorovich distances between transition probabilities of two diffusions with different diffusion matrices and drifts via a natural quadratic distance between the drifts and diffusion matrices. Applications to nonlinear Fokker–Planck–Kolmogorov equations, optimal control and mean field games are given.
Starting from the overdamped Langevin dynamics in , we consider a scalar Markov process which approximates the dynamics of the first component . In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that is a good approximation of is proven in terms of time marginals, under assumptions quantifying the timescale separation between the first component and the other components of . Here, we prove an upper bound on the trajectorial error , for any , under a similar set of assumptions. We also show that the technique of proof can be used to obtain quantitative averaging results.
In the small noise regime, the average transition time between metastable
states of a reversible diffusion process is described at the logarithmic scale
by Arrhenius' law. The Eyring-Kramers formula classically provides a
subexponential prefactor to this large deviation estimate. For irreversible
diffusion processes, the equivalent of Arrhenius' law is given by the
Freidlin-Wentzell theory. In this paper, we compute the associated prefactor
and thereby generalise the Eyring-Kramers formula to irreversible diffusion
processes. In our formula, the role of the potential is played by
Freidlin-Wentzell's quasipotential, and a correction depending on the
non-Gibbsianness of the system along the instanton is highlighted. Our analysis
relies on a WKB analysis of the quasistationary distribution of the process in
metastable regions, and on a probabilistic study of the process in the
neighbourhood of saddle-points of the quasipotential.
Consider the dynamical system in R
r
defined by a smooth vector field b(x): (1.1)
We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on ℝ n . The proof is based on arguments introduced by D. Bakry et al. [J. Funct. Anal. 254, No. 3, 727–759 (2008; Zbl 1146.60058)], but for the sake of completeness, all details are provided.
We investigate the existence of steady states and exponential decay for
hypocoercive Fokker--Planck equations on the whole space with drift terms that
are linear in the position variable. For this class of equations, we first
establish that hypoellipticity of its generator and confinement of the system
is equivalent to the existence of a unique normalised steady state. These two
conditions also imply hypocoercivity, i.e. exponential convergence of the
solution to equilibrium.
Since the standard entropy method does not apply to degenerate parabolic
equations, we develop a new modified entropy method (based on a modified,
non-degenerate entropy dissipation-like functional) to prove this exponential
decay in relative entropy (logarithmic till quadratic) - with a sharp rate.
Furthermore, we compute the spectrum and eigenspaces of the generator as well
as flow-invariant manifolds of Gaussian functions.
Next, we extend our method to kinetic Fokker-Planck equations with a class of
non-quadratic potentials. And, finally, we apply this new method to
non-symmetric, uniformly parabolic Fokker-Planck equations with linear drift.
At least in 2D this always yields the sharp exponential envelopes for the
entropy function. In this case, we obtain even a sharp multiplicative constant
in the decay estimate for the non-symmetric semigroup.
Foreword Stephen P. Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng Introduction to MCMC, Charles J. Geyer A short history of Markov chain Monte Carlo: Subjective recollections from in-complete data, Christian Robert and George Casella Reversible jump Markov chain Monte Carlo, Yanan Fan and Scott A. Sisson Optimal proposal distributions and adaptive MCMC, Jeffrey S. Rosenthal MCMC using Hamiltonian dynamics, Radford M. Neal Inference and Monitoring Convergence, Andrew Gelman and Kenneth Shirley Implementing MCMC: Estimating with confidence, James M. Flegal and Galin L. Jones Perfection within reach: Exact MCMC sampling, Radu V. Craiu and Xiao-Li Meng Spatial point processes, Mark Huber The data augmentation algorithm: Theory and methodology, James P. Hobert Importance sampling, simulated tempering and umbrella sampling, Charles J.Geyer Likelihood-free Markov chain Monte Carlo, Scott A. Sisson and Yanan Fan MCMC in the analysis of genetic data on related individuals, Elizabeth Thompson A Markov chain Monte Carlo based analysis of a multilevel model for functional MRI data, Brian Caffo, DuBois Bowman, Lynn Eberly, and Susan Spear Bassett Partially collapsed Gibbs sampling & path-adaptive Metropolis-Hastings in high-energy astrophysics, David van Dyk and Taeyoung Park Posterior exploration for computationally intensive forward models, Dave Higdon, C. Shane Reese, J. David Moulton, Jasper A. Vrugt and Colin Fox Statistical ecology, Ruth King Gaussian random field models for spatial data, Murali Haran Modeling preference changes via a hidden Markov item response theory model, Jong Hee Park Parallel Bayesian MCMC imputation for multiple distributed lag models: A case study in environmental epidemiology, Brian Caffo, Roger Peng, Francesca Dominici, Thomas A. Louis, and Scott Zeger MCMC for state space models, Paul Fearnhead MCMC in educational research, Roy Levy, Robert J. Mislevy, and John T. Behrens Applications of MCMC in fisheries science, Russell B. Millar Model comparison and simulation for hierarchical models: analyzing rural-urban migration in Thailand, Filiz Garip and Bruce Western
Introduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincare inequalities.- 5.Logarithmic Sobolev inequalities.- 6.Sobolev inequalities.- Part III Related functional, isoperimetric and transportation inequalities: 7.Generalized functional inequalities.- 8.Capacity and isoperimetry-type inequalities.- 9.Optimal transportation and functional inequalities.- Part IV Appendices: A.Semigroups of bounded operators on a Banach space.- B.Elements of stochastic calculus.- C.Some basic notions in differential and Riemannian geometry.- Notations and list of symbols.- Bibliography.- Index.
The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced.
The old sections devoted to the smile issue in the LIBOR market model have been enlarged into several new chapters. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered.
The fast-growing interest for hybrid products has led to new chapters. A special focus here is devoted to the pricing of inflation-linked derivatives.
The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.
The first part of the paper is concerned with a version of Freidlin-Wentzell exit theorems suitable for control theoretic applications. In the second part the general results are specialized to the case of linear systems and two types of stabilizability questions are considered.
The calculation of free‐energy differences is one of the main challenges in computational biology and biochemistry. Umbrella sampling, biased molecular dynamics (MD), is one of the methods that provide free energy along a reaction coordinate. Here, the method is derived in a historic overview and is compared with related methods like thermodynamic integration, slow growth, steered MD, or the Jarzynski‐based fast‐growth technique. In umbrella sampling, bias potentials along a (one‐ or more‐dimensional) reaction coordinate drive a system from one thermodynamic state to another (e.g., reactant and product). The intermediate steps are covered by a series of windows, at each of which an MD simulation is performed. The bias potentials can have any functional form. Often, harmonic potentials are used for their simplicity. From the sampled distribution of the system along the reaction coordinate, the change in free energy in each window can be calculated. The windows are then combined by methods like the weighted histogram analysis method or umbrella integration. If the bias potential is adapted to result in an even distribution between the end states, then this whole range can be spanned by one window (adaptive‐bias umbrella sampling). In this case, the free‐energy change is directly obtained from the bias. The sampling in each window can be improved by replica exchange methods; either by exchange between successive windows or by running additional simulations at higher temperatures. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 932–942 DOI: 10.1002/wcms.66
This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods
Let dx( t ) = d( t,w )dW( t ) + b( t,w )dt,d\xi \left( t \right) = \delta \left( {t,\omega } \right)dW\left( t \right) + \beta \left( {t,\omega } \right)dt,
where
( W( t ),\mathfrakFt )\left( {W\left( t \right),\mathfrak{F}_t } \right)
is a Wiener process,
\mathfrakF\mathfrak{F}
processes such that dx( t ) = s( t,x( t ) )dW( t ) + b( t,x( t ) )dtdx\left( t \right) = \sigma \left( {t,x\left( t \right)} \right)dW\left( t \right) + b\left( {t,x\left( t \right)} \right)dt
with non-random coefficients which admits a weak solutionx(t) having the same one-dimensional probability distribution as
s( t,x ) = ( E( ddT ( t )|x( t ) = x ) )\tfrac12 ,b( t,x ) = E( b( t )|x( t ) = x ).\sigma \left( {t,x} \right) = \left( {E\left( {\delta \delta ^T \left( t \right)|\xi \left( t \right) = x} \right)} \right)^{\tfrac{1}{2}} ,b\left( {t,x} \right) = E\left( {\beta \left( t \right)|\xi \left( t \right) = x} \right).
We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal.6, 587–600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases. The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information.
Let A=∑i,j=1NqijDij+∑i,j=1NbijxjDi be a possibly degenerate Ornstein–Uhlenbeck operator in RN and assume that the associated Markov semigroup has an invariant measure μ. We compute the spectrum of A in Lμp for 1⩽p<∞.
We present a method of balancing for nonlinear systems which is an extension of balancing for linear systems in the sense that it is basd on the input and output energy of a system. It is a local result, but gives ‘broader’ results than we obtain by just linearizing the system. Furthermore, the relation with balancing of the linearization is dealt with. We propose to use the method as a tool for nonlinear model reduction and investigate some of the properties of the reduced system.
Large deviations theory is concerned with asymptotic estimates of probabilities of rare events associated with stochastic
processes. A stochastic control approach to large deviations is outlined. Both problems of small random perturbations and
large deviations from ergodicity are considered. For large deviations of Markov diffusion processes, PDE — viscosity solution
methods are mentioned. Another stochastic control formulation, applicable to a broad range of large deviations problems is
due to Dupuis and Ellis. This approach reduces many aspects of large deviations to the theory of weak convergence of probability
measures.
A method is presented for computing the average solution of problems which are too complicated for adequate resolution, but where information about the statistics of the solution is available. The method involves computing average derivatives by interpolation based on linear regression, and an updating of a measure constrained by the available crude information. Examples are given. 1 Outline of goal and method There are many problems in science whose solution is described by a set of differential equations, but where the solution of these equations is so complicated that it cannot be found in practice, even numerically, because it cannot be properly resolved. An accurate numerical solution requires that the problem be well-resolved, i.e, that enough variables ("degrees of freedom") be retained in the calculation to represent all relevant features of the solution. Well-known examples where good resolution cannot be achieved in practice include turbulence and various problems in st...
Let be a given probability density proportional to in a high-dimensional Euclidean space . The diffusion is often used to sample from . Instead of , we consider diffusions with smooth drift b(x) and having equilibrium . First we study some general properties and then concentrate on the Gaussian case, namely, with a strictly negative-definite real matrix D and with a stability matrix B; that is, the real parts of the eigenvalues of B are strictly negative. Using the rate of convergence of the covariance of X(t) [or together with EX(t)] as the criterion, we prove that, among all such b(x), the drift Dx is the worst choice and that improvement can be made if and only if the eigenvalues of D are not identical. In fact, the convergence rate of the covariance is , where is the maximum of the real parts of the eigenvalues of B and the infimum of over all such B is . If, for example, a "circulant" drift is added to Dx, then for essentially all D, the diffusion with this modified drift has a better convergence rate.
Three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one. The first one states the smoothness of an invariant diffusion density with respect to a parameter. The second establishes a similar smoothness of the solution of the Poisson equation in R-d. The third one states a diffusion approximation result, or in other words an averaging of singularly perturbed diffusion for "fully coupled SDE systems" or "SDE systems with complete dependence.".