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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.
In this paper we analyze decompositions of reversible nearly uncoupled
Markov chains into rapidly mixing subchains. We state upper
bounds on the 2nd eigenvalue for restriction and stochastic complementation
chains of reversible Markov chains, as well as a relation between
them. We illustrate the obtained bounds analytically for bunkbed
graphs, and furthermore apply them to restricted Markov chains that
arise when analyzing conformation dynamics of a small biomolecule.
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 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.
Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics
of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have
seen widespread use in recent years. This approach has many appealing characteristics compared
to straightforward molecular dynamics simulation and analysis, including the potential to mitigate
the sampling problem by extracting long-time kinetic information from short trajectories and the
ability to straightforwardly calculate expectation values and statistical uncertainties of various
stationary and dynamical molecular observables. In this article, we summarize the current state of
the art in generation and validation of MSMs and give some important new results. We describe
an upper bound for the approximation error made by modeling molecular dynamics with an MSM
and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast
to previous practice, it becomes clear that the best MSM is not obtained by the most metastable
discretization, but the MSM can be much improved if non-metastable states are introduced near
the transition states. Moreover, we show that it is not necessary to resolve all slow processes
by the state space partitioning, but individual dynamical processes of interest can be resolved
separately. We also present an efficient estimator for reversible transition matrices and a robust
test to validate that an MSM reproduces the kinetics of the molecular dynamics data.
Markov State Models (MSMs) have become the tool of choice to analyze large amounts of molec-
ular dynamics data by approximating them as a Markov jump process between suitably predefined
states. Here we investigate ”Core Set MSMs”, a new type of MSMs that builds on metastable core
sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core
Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transi-
tion Path Theory (TPT). As a result, Core Set MSMs can now be used to extract phenomenological
rate constants between the metastable sets of the system and to approximate the evolution of certain
key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed
and illustrated on a model potential and the torsion angle dynamics of Alanine dipeptide.
We shortly review the uncoupling-coupling method, a Markov chain
Monte Carlo based approach to compute statistical properties of systems like
medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the
decomposition of reversible nearly uncoupled Markov chains into rapidly mixing
subchains. We show how the underlying scheme of uncoupling-coupling can also be
applied to stochastic differential equations where it can be translated into a domain
decomposition technique for partial differential equations.
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 report on a novel approach to the automatic identification
of metastable states from long term simulation of complex
molecular systems. The new approach is based on a hierarchical concept
of metastability: metastable states are understood as subsets of
state or configuration space from which the dynamics exits only very rarely;
subsets with the smallest exit probabilities are of most interest, their
further decomposition then may reveal subsets from which exiting
is less but comparably difficult for the system under investigation.
The article gives a survey of the theoretical foundation of
the approach and its algorithmic realization that generalizes
the well-known concept of Hidden Markov Models.
The performance of the resulting algorithm are illustrated by
application to a 100 ns simulation of penta-alanine with explicit water.
We demonstrate the resulting metastable states allow to
reveal the conformation dynamics of the moelcule.
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.
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.
Folding and conformational changes of macromolecules often require the generation of large amounts of simulation data that are difficult to ana- lyze. Markov state models (MSMs) address this challenge by providing a systematic way to decompose the state space of the molecular system into substates and to estimate a transition matrix containing the transi- tion probabilities between these substates. This transition matrix can be analyzed to reveal the metastable, i.e. long-living, states of the system, its slowest relaxation timescales and transition pathways and rates e.g. from unfolded to folded, or from dissociated to bound states. To reduce the technical burden of constructing such MSMs we provide the software framework EMMA (available at https://simtk.org/home/emma) to con- struct, validate and analyse such Markov State Models.