This article is concerned with the averaging principle and its extensions
for stochastic dynamical systems with fast and slow degrees of
freedom. It is demonstrated how the \conventional" averaging principle
results from asymptotic multiscale analysis, how one can construct
an indicator for its (in-)appropriateness, and how, if inappropriate, it
may be extended into an improved approximation. The conventional
scheme contains averages over the entire accessible state space of the
fast degrees of freedom and may thus fail if these fast degrees of freedom
exhibit long-term (auto-)correlations. In contrast, the improved
scheme combines several conditional averages with a Markov jump process
that is designed to represent the
ipping process between the
conditional averages and thus incorporates the important long-term
correlations. All important steps of the derivation are illustrated by
numerical experiments. Application to problems from molecular dynamics
is discussed.
We consider a particle constrained to a submanifold ? of the
configuration space Rm. Using that the notion of holonomic constraints coincides
with integrability of the corresponding vector field, we show how this property
naturally determines local coordinates on ?. We give a rigorous justification for
the calculation of the mean force along a constrained coordinate, and we provide
a concise geometrical interpretation of the different contributions to the mean
force in terms of the unconstrained vector field and extrinsic curvature properties
of ? in Rm. Our approach gives rise to a Hybrid Monte-Carlo based algorithm
that can be used to compute the mean force acting on selected coordinates in the
context of thermodynamic free energy statistics.
We consider the problem of automatically extracting simplified models out of complex high--dimensional and t
ime--dependent data. The simplified model is given by a linear Langevin equation with time--varying coeffici
ents. The reduced model may still be high--dimensional, but it is physically intuitive and much easier to in
terpret than the original data. In particular we can distinguish whether dynamical effects are influenced b
y friction, noise, or deterministic motion. The parameters for the reduced model are obtained by a robust an
d efficient numerical predictor--corrector scheme which relies on analytical solutions to a maximum-likeliho
od problem provided the time steps between successive observations are not too large. If the data set is ver
y heterogeneous the time series is better described not by a single model, but by a collection of reduced mo
dels. This scenario is accounted for by embedding the parameter estimation procedure into the framework of h
idden Markov models, i.e., we decompose the data into several subsets, each of which gives rise to an approp
riate linear Langevin model. The switching between the local model is done by a Markov jump process. The opt
imal decomposition into submodels can then be regarded as one global Langevin model with piecewise constant
coefficients. We illustrate the performance of the algorithm by means of several examples. Especially we foc
us on the numerical error as a function of the time step of the observation sequence.
We consider the problem of automatically extracting simplified models out of complex high-dimensional and time-dependent data. The simplified model is given by a linear Langevin equation with time-varying coefficients. The reduced model may still be high-dimensional, but it is physically intuitive and much easier to interpret than the original data. In particular we can distinguish whether certain dynamical effects are influenced by friction, noise, or systematic drift.
The parameters for the reduced model are obtained by a robust and efficient numerical predictor-corrector scheme which relies on analytical solutions to a maximum-likelihood problem provided the time steps between successive observations are not too large. Our approach emphasizes the specific hypoelliptic structure of the Langevin equation given high-dimensional observation data, and therefore can be considered as complemetary to the procedure recently proposed in \emph{Horenko et al. (submitted SIAM MMS, 2007)} by one of the authors, or to the problem of incomplete (one-dimensional) observations \emph{Pokern et al. (submitted to JRSSB, 2007)}. If the data set is very heterogeneous the time series is better described not by a single model, but by a collection of reduced models. This scenario is accounted for by embedding the parameter estimation procedure into the framework of hidden Markov models which it is particularly suited to treat high-dimensional data. That is, we decompose the data into several subsets, each of which gives rise to an appropriate linear Langevin model, where the switching between the local model is done by a Markov jump process. The optimal decomposition into submodels can then be regarded as one global Langevin model with piecewise constant coefficients. We illustrate the performance of the algorithm by means of several examples. Especially we focus on the numerical error as a function of the time step of the observation sequence.
We comment on two different notions of the thermodynamical free energy
that are used in Hamiltonian molecular dynamics. Both concepts have
different scopes of applications as was pointed out recently in the
context of high--friction Langevin dynamics. We show that problems
that rely on either definition can be treated in a uniform way using
constrained molecular dynamics. Not only proves this useful in
designing algorithms that sample the free energy landscape, but it
also clarifies the relation between seemingly contradictory results
that are present in the literature.
We present a formal procedure for structure-preserving model reduction of linear second-order control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems.
It turns out that the Hamiltonian structure, stability and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.
We derive a Crooks-Jarzynski-type identity for computing free energy differences between metastable states that is based on nonequilibrium diffusion processes. Furthermore we outline a brief derivation of an infinite-dimensional stochastic partial differential equation that can be used to efficiently generate the ensemble of trajectories connecting the metastable states.
We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics admit a splitting into fast and slow subspaces and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.
We study balanced model reduction of partially-observed linear stochastic differential equa- tions of Langevin type. Balancing the equations of motion gives rise to a singularly perturbed system of equations with slow and fast degrees of freedom, and we prove that in the limit of the fast variables becoming infinitely fast, the solutions converge to the solution of a reduced-order Langevin equation. We illustrate the method with several numerical examples and discuss the relation to model reduction of deterministic control systems that have an underlying Hamiltonian structure.
We propose a nonequilibrium sampling method for computing free energy profiles along a given reaction coordinate. The method consists of two parts: a controlled Langevin sampler that generates nonequilibrium bridge paths conditioned by the reaction coordinate, and Jarzynski’s formula for reweighting the paths. Our derivation of the equa- tions of motion of the sampler is based on stochastic perturbation of a controlled dissipative Hamiltonian system, for which we prove Jarzynski’s identity as a special case of the Feynman-Kac formula. We illustrate our method by means of a suitable numerical example and briefly discuss issues of optimally choosing the control protocol for the reaction coordinate.
We revisit the problem of the linear response of a constrained mechanical system. In doing so we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville’s theorem and the Green-Kubo relations naturally follow.
We propose a robust and efficient numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant measure is precisely the cell approximation of the Boltzmann distribution of the original process. Moreover the resulting jump process preserves the detailed balance property of the original stochastic process.
he generalized Langevin equation is useful for modeling a wide
range of physical processes. Unfortunately its parameters,
especially the memory function, are difficult to determine for
nontrivial processes. In this paper, relations between a
time-discrete generalized Langevin model and discrete multivariate
autoregressive (AR) or autoregressive moving average models (ARMA)
are established. This allows a wide range of discrete linear
methods known from time series analysis to be applied. In
particular, the determination of the memory function {\it via} the
order of the respective AR or ARMA model is addressed. The method
is illustrated on a one-dimensional test system and subsequently
applied to the molecular dynamics of a biomolecule which exhibits
an interesting relationship between the solvent method used, the
molecular conformation and the depth of the memory.
We consider a particle constrained to a submanifold ? of the configuration space
Rm. Using that the notion of holonomic constraints coincides with integrability of the corresponding
vector field, we show how this property naturally determines local coordinates on ? . We give a
rigorous justification for the calculation of the mean force along the constrained coordinates, and
we provide a concise geometrical interpretation of the different contributions to the mean force.
Our approach gives rise to a generalisation of the Fixman Theorem which is well known and widely
used in molecular dynamics applications. It further allows for working out a Hybrid Monte-Carlo
based algorithm that can be used to compute arbitrary statistical quantities from constrained
simulations such as the mean force in the context of thermodynamic free energy statistics.
We address the problem of computing canonical ensembles of Hamiltonian systems that are subject to holonomic constraints. For this purpose we introduce a hybrid Monte-Carlo scheme that allows for sampling configurational expectation values in the canonical ensemble without bias, proving ergodicity for the proposed scheme. In doing so, we extend recent ideas of Canc\`es \emph{et al.} (\emph{M2AN, Vol. 41, No. 2, pp. 351-389, 2007}) who could prove a law of large numbers for unconstrained molecular systems with a separable Hamiltonian employing a least-action principle. The sampling scheme is illustrated by means of calculating the free energy profile of a small peptide.
We present a formal procedure for structure-preserving model reduction of linear second-order control problems. Second-order equations appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design to mention just a few. However typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper we adopt the framework of port-Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. We explore two possible routes to truncation of a balanced port-Hamiltonian system: one is by imposing holonomic constraints, the other one proceeds by a singular perturbation argument using an explicit scaling of the small Hankel singular values. In both cases the reduced system turns out to be port-Hamiltonian. Moreover the procedure preserves stability and passivity. For the singularly perturbed system we prove convergence of the corresponding transfer functions.
or stable linear input-output systems, the method of balanced truncation (B.C.~Moore, {\em IEEE Trans. Auto. Contr.} {\bf AC-26}, 17--32, 1981) consists in finding a coordinate transformation such that modes which are least sensitive to the external input (controllability) also give the least output (observability) and therefore can be neglected. A drawback is that projecting the original equations of motion onto the subspace of interest typically fails to preserve the problem's physical structure, e.g., if the original equations are of second-order form or Hamiltonian. For Hamiltonian systems, a natural way of restricting a system to a subspace is by means of constraints, and we show, employing singular perturbation arguments, that balanced truncation can be done in a structure-preserving fashion. The thus obtained reduced Hamiltonian system preserves stability and passivity and satisfies the usual balanced truncation error bound.
We study Balanced Truncation for stochastic differential equations. In doing so, we adopt ideas from large deviations theory and discuss notions of controllability and obervability for dissipative Hamiltonian systems with degenerate noise term, also known as Langevin equations. For partially-observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.
e propose an algorithm for the fast and efficient simulation of polymers represented by
chains of hard spheres. The particles are linked by holonomic bond constraints.
While the motion of the polymers is free (i.e., no collisions occur) the equations
of motion can be easily integrated using a collocation-based partitioned Gauss-Runge-Kutta method.
The method is reversible, symplectic and preserves energy. Moreover the numerical scheme allows the integration using much longer time steps than any explicit integrator such as the popular Verlet method. If polymers collide the point of impact can be determined to arbitrary precision by simple nested intervals. Once the collision point is known the impulsive contribution can be computed analytically. We illustrate our approach by means of a suitable numerical example.