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Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case
(2012)
We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some
problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are
sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schroedinger and the drift-diffusion equation
but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.
The hypergraph assignment problem (HAP) is the generalization of assignments
from directed graphs to directed hypergraphs. It serves, in particular,
as a universal tool to model several train composition rules in vehicle rotation
planning for long distance passenger railways. We prove that even for problems
with a small hyperarc size and hypergraphs with a special partitioned structure
the HAP is NP-hard and APX-hard. Further, we present an extended integer
linear programming formulation which implies, e. g., all clique inequalities.
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-calles Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Lojasiewicz-Simon-type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms $f$, $g$ are real analytic. Moreover, we provide an estimate for the convergence rate.
We investigate a distributed optimal control problem for a phase field
model of Cahn-Hilliard type. The model describes two-species phase segregation
on an atomic lattice under the presence of diffusion; it has been introduced recently in
[4], on the basis of the theory developed in [15], and consists of a system of two
highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply
directly, although the control constraints and the cost functional are of standard type.
We show that the problem admits a solution, and we derive the first-order
necessary conditions of optimality.
A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity
(2012)
In this paper, we prove the existence
and global boundedness from above for a solution to an
integrodifferential model for nonisothermal multi-phase
transitions under nonhomogeneous third type boundary conditions.
The system couples a quasilinear internal energy balance
ruling the evolution of the absolute temperature with a vectorial
integro-differential inclusion governing the vectorial
phase-parameter dynamics. The specific heat and the heat
conductivity k are allowed to depend both on the order parameter
$\chi$ and on the absolute temperature $\teta$ of the system, and
the convex component of the free energy may or may not be
singular. Uniqueness and continuous data dependence are
also proved under additional assumptions.
This paper is devoted to an optimal control problem of Maxwell's equations in the presence of pointwise state constraints. The control is given by a divergence--free three--dimensional vector function representing an applied current density. To cope with the divergence--free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control--to--state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by N\'ed\'elec's curl--conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$--norm. The paper ends by numerical results including a numerical verification of our theoretical results.
An optimal control problem arising in the context of 3D electromagnetic induction heating is investigated. The state equation is given by a quasilinear stationary heat equation coupled with a semilinear time-harmonic eddy current equation. The temperature-dependent electrical conductivity and the presence of pointwise inequality state-constraints represent the main challenge of the paper. In the first part of the paper, the existence and regularity of the state are addressed. The second part of the paper deals with the analysis of the corresponding linearized equation. Some sufficient conditions are presented which guarantee the solvability of the linearized system. The final part of the paper is concerned with the optimal control. The aim of the optimization is to find the optimal voltage such that a desired temperature can be achieved optimally. The corresponding first-order necessary optimality condition is presented.
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time--dependent heat equations coupled with 3D time--harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi--infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.
We consider a shape implant design problem that arises in the context of facial surgery. We introduce a reformulation as an optimal control problem, where the control acts as a boundary force. The state is modelled as a minimizer of a polyconvex hyperelastic energy functional. We show existence of optimal solutions and derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results are presented.
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.
Dynamical averages based on functionals of dynamical trajectories, such as time-correlation func-
tions, play an important role in determining kinetic or transport properties of matter. At temperatures
of interest, the expectations of these quantities are often dominated by contributions from rare events,
making the precise calculation of these quantities by molecular dynamics simulation difficult. Here,
we present a reweighting method for combining simulations from multiple temperatures (or from
simulated or parallel tempering simulations) to compute an optimal estimate of the dynamical prop-
erties at the temperature of interest without the need to invoke an approximate kinetic model (such as
the Arrhenius law). Continuous and differentiable estimates of these expectations at any temperature
in the sampled range can also be computed, along with an assessment of the associated statistical
uncertainty. For rare events, aggregating data from multiple temperatures can produce an estimate
of the desired precision at greatly reduced computational cost compared with simulations conducted
at a single temperature. Here, we describe use of the method for the canonical (NVT) ensemble us-
ing four common models of dynamics (canonical distribution of Hamiltonian trajectories, Andersen
thermostatting, Langevin, and overdamped Langevin or Brownian dynamics), but it can be applied to
any thermodynamic ensemble provided the ratio of path probabilities at different temperatures can be
computed. To illustrate the method, we compute a time-correlation function for solvated terminally-
blocked alanine peptide across a range of temperatures using trajectories harvested using a modified
parallel tempering protocol.
Dynamical fingerprints of macromolecules obtained from experiments often seem to indicate two- or
three state kinetics while simulations typically reveal a more complex picture. Markov state models of
molecular conformational dynamics can be used to predict these dynamical fingerprints and to reconcile
experiment with simulation. This is illustrated on two model systems: a one-dimensional energy surface
and a four-state model of a protein folding equilibrium. We show that (i) there might be no process
which corresponds to our notion of folding, (ii) often the experiment will be insensitive to some of the
processes present in the system, (iii) with a suitable combination the observable and initial conditions in
a relaxation experiment one can selectively measure specific processes. Furthermore, our method can be
used to design experiments such that specific processes appear with large amplitudes. We demonstrate
that for a fluorescence quenching experiment of the MR121-G9-W peptide.
Protein-ligand interactions are essential for nearly all biological processes, and yet the bio-
physical mechanism that enables potential binding partners to associate before specific binding
occurs remains poorly understood. Fundamental questions include which factors influence the
formation of protein-ligand encounter complexes, and whether designated association path-
ways exist. In this article we introduce a computational approach to systematically analyze
the complete ensemble of association pathways and to thus investigate these questions. This
approach is employed here to study the binding of a phosphate ion to the Escherichia coli
Phosphate Binding Protein. Various mutants of the protein are considered and their effects
on binding free energy profiles, association rates and association pathway distributions are
quantified. The results reveal the existence of two anion attractors, i.e. regions that initially
attract negatively charged particles and allow them to be efficiently screened for phosphate
which is specifically bound subsequently. Point mutations that affect the charge on these
attractors modulate their attraction strength and speed up association to a factor of 10 of
the diffusion limit and thus change the association pathways of the phosphate ligand. It is
demonstrated that a phosphate that pre-binds to such an attractor neutralizes its attraction
effect to the environment, making the simultaneous association of a second phosphate ion
unlikely. Our study suggests ways how structural properties can be used to tune molecular
association kinetics so as to optimize the efficiency of binding, and highlights the importance
of kinetic properties.
While studies of protein-ligand association have mostly focused on the native complex and its
stability (binding affinity), relatively little attention has been paid on the association process
that precedes the formation of the complex. Here we review approaches to study the kinet-
ics of association and association mechanisms, i.e. the probability distribution of association
pathways. Selected methods are described that allow these properties to be calculated quan-
titatively from simulation models. We summarize some applications of these methods and
finally propose a model mechanism by which proteins may efficiently screen potential ligands
for those that can be natively bound.
In this paper, we present a Gaussian Markov random field (GMRF) model for the transition
matrices (TMs) of Markov chains (MCs) by assuming the existence of a neighborhood relationship
between states, and develop the maximum a posteriori (MAP) estimators under different obser-
vation conditions. Unlike earlier work on TM estimation, our method can make full use of the
similarity between different states to improve the estimated accuracy, and the estimator can be
performed very efficiently by solving a convex programming problem. In addition, we discuss the
parameter choice of the proposed model, and introduce a Monte Carlo cross validation (MCCV)
method. The numerical simulations of a diffusion process are employed to show the effectiveness
of the proposed models and algorithms.
Optimal Identification of Semi-Rigid Domains in Macromolecules from Molecular Dynamics Simulation
(2012)
Biological function relies on the fact that biomolecules can switch between different conformations and aggregation states.
Such transitions involve a rearrangement of parts of the biomolecules involved that act as dynamic domains. The reliable
identification of such domains is thus a key problem in biophysics. In this work we present a method to identify semi-rigid
domains based on dynamical data that can be obtained from molecular dynamics simulations or experiments. To this end
the average inter-atomic distance-deviations are computed. The resulting matrix is then clustered by a constrained
quadratic optimization problem. The reliability and performance of the method are demonstrated for two artificial peptides.
Furthermore we correlate the mechanical properties with biological malfunction in three variants of amyloidogenic
transthyretin protein, where the method reveals that a pathological mutation destabilizes the natural dimer structure of the
protein. Finally the method is used to identify functional domains of the GroEL-GroES chaperone, thus illustrating the
efficiency of the method for large biomolecular machines.
Discrete-state Markov (or master equation) models provide a useful simplified representation for
characterizing the long-time statistical evolution of biomolecules in a manner that allows direct
comparison with experiments as well as the elucidation of mechanistic pathways for an inherently
stochastic process. A vital part of meaningful comparison with experiment is the characterization of
the statistical uncertainty in the predicted experimental measurement, which may take the form of
an equilibrium measurement of some spectroscopic signal, the time-evolution of this signal following
a perturbation, or the observation of some statistic (such as the correlation function) of the equilib-
rium dynamics of a single molecule. Without meaningful error bars (which arise due to the finite
quantity of data used to construct the model), there is no way to determine whether the deviations
between model and experiment are statistically meaningful. Previous work has demonstrated that
a Bayesian method that enforces microscopic reversibility can be used to characterize the correlated
uncertainties in state-to-state transition probabilities (and functions thereof) for a model inferred from
molecular simulation data. Here, we extend this approach to include the uncertainty in observables
that are functions of molecular conformation (such as surrogate spectroscopic signals) characteriz-
ing each state, permitting the full statistical uncertainty in computed spectroscopic experiments to be
assessed. We test the approach in a simple model system to demonstrate that the computed uncer-
tainties provide a useful indictor of statistical variation, and then apply it to the computation of the
fluorescence autocorrelation function measured for a dye-labeled peptide previously studied by both
experiment and simulation.
Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe
long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great
amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into
microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data
to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current
perspective of Markov models and explain in short their theoretical basis and assumptions. We
describe transition path theory which allows the entire ensemble of protein folding pathways to be
investigated and that combines naturally with Markov models. Experimental observations can be
naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally
observable timescales can be equipped with an understanding of the structural rearrangement
processes that take place at these timescales. The concepts of this paper are illustrated by a
simple kinetic model of protein folding.
Large-scale stochastic models are relevant in many different fields such as com- putational biology, finance, social sciences, communication and traffic networks. In order to both efficiently simulate and analyze such models and to understand the essential properties of the sys- tem, it is desirable to have model reduction techniques that much reduce the dimensionality of the model while at the same time preserving the system’s essential dynamical properties. In this paper, a general model reduction technique for the class of discrete space and time Hidden Markov Models is presented, thereby also including the more special class discrete Markov Chains. The method is illustrated on some model applications.
Actin is a major structural protein of the eukaryotic cytoskeleton and enables cell motility.
Here, we present a model of the actin filament (F-actin) that incorporates the global structure
of the recently published model by Oda et al. but also conserves internal stereochemistry. A
comparison is made using molecular dynamics simulation of the model with other recent F-
actin models. A number of structural determents such as the protomer propeller angle, the
number of hydrogen bonds and the structural variation among the protomers are analyzed.
The MD comparison is found to reflect the evolution in quality of actin models over the last
six years. In addition, simulations of the model are carried out in states with both ADP or
ATP bound and local hydrogen-bonding differences characterized. The results point to the
significance of a direct interaction of Gln137 with ATP for activation of ATPase activity after
the G-to-F-actin transition.
This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.
Rapid Branching
(2012)
We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.
Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
(2012)
This paper deals with the mathematical modeling and simulation of
crystal growth processes by the so-called Czochralski method and related methods,
which are important industrial processes
to grow large
bulk single crystals of semiconductor materials such as, e.g., gallium arsenide
(GaAs) or silicon (Si) from the melt.
In particular, we investigate a recently developed
technology in which traveling magnetic fields are applied in order to
control
the behavior of the turbulent melt flow. Since numerous different physical effects
like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via
radiation, etc., play an important role in the process, the corresponding mathematical
model leads to an extremely difficult system of initial-boundary value problems for
nonlinearly coupled partial differential equations. In this paper, we describe a mathematical
model that is under use for the simulation of real-life growth scenarios, and we give an overview
of mathematical results and numerical simulations that have been obtained for it in recent years.
In this paper a bottom-up approach of automatic simplification of a railway network is presented. Starting from a very detailed, microscopic level, as it is used in railway simulation, the network is transformed by an algorithm to a less detailed level (macroscopic network), that is sufficient for long-term planning and optimization. In addition running and headway times are rounded to a pre-chosen time discretization by a special cumulative method, which we will present and analyse in this paper. After the transformation we fill the network with given train requests to compute an optimal slot allocation. Then the optimized schedule is re-transformed into the microscopic level and can be simulated without any conflicts occuring between the slots. The algorithm is used to transform the network of the very dense Simplon corridor between Swiss and Italy. With our aggregation it is possible for the first time to generate a profit maximal and conflict free timetable for the corridor across a day by a simultaneously optimization run.
The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.
Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.
We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the "regularity" of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.
Vehicle rotation planning is a fundamental problem in rail transport. It decides how the railcars, locomotives, and carriages are operated in order to implement the trips of the timetable. One important planning requirement is operational regularity, i.e., using the rolling stock in the same way on every day of operation. We propose to take regularity into account by modeling the vehicle rotation planning problem as a minimum cost hyperassignment problem (HAP). Hyperassignments are generalizations of assignments from directed graphs to directed hypergraphs. Finding a minimum cost hyperassignment is NP-hard. Most instances arising from regular vehicle rotation planning, however, can be solved well in practice. We show that, in particular, clique inequalities strengthen the canonical LP relaxation substantially.
We investigate a nonstandard phase field
model of Cahn-Hilliard type. The model, which was introduced in
[16], describes two-species phase segregation and consists of a
system of two highly nonlinearly coupled PDEs. It has been studied
recently in
[5], [6] for the case of homogeneous Neumann
boundary conditions. In this paper, we investigate the case that the
boundary condition for one of the unknowns of the system is of third
kind and nonhomogeneous. For the resulting system, we show
well-posedness, and we study optimal boundary control
problems. Existence of optimal controls is shown, and the first-order
necessary optimality conditions are derived. Owing to the strong
nonlinear couplings in the PDE system, standard arguments of optimal
control theory do not apply directly, although the control constraints
and the cost functional will be of standard type.
This paper is concerned with a diffusion model of phase-field type, consisting
of a {parabolic} system of two partial differential equations{,} interpreted as balances
of microforces and microenergy{, for two unknowns: the problem's order parameter $\rho$}
and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper \cite{CGPS3}, we proved that this problem is \wepo\ and investigated the \loti\ \bhv\ of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\eps$
tends to $0$. We prove convergence of
$(\varepsilon,\delta)-$solutions to the corresponding solutions for
the case $\eps =0$, whose long-time behavior we characterize; in the
proofs, we employ compactness and monotonicity arguments.
We study the expansion of the eigenfunctions of Schrödinger operators with smooth confinement potentials in Hermite functions; confinement potentials are potentials that become unbounded at infinity. The key result is that such eigenfunctions and all their derivatives decay more rapidly than any exponential function under some mild growth
conditions to the potential and its derivatives. Their expansion in Hermite functions converges therefore very fast, super-algebraically.
The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces
(2012)
The paper continues the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr\"odinger operators and their
exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order $\vartheta$ for $\vartheta<3/4$. The bound $3/4$ is best possible and can neither be reached nor surpassed. Such results are important for the study
of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.
We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem.
We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.
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 Raleigh coefficient. It is shown that this Raleigh 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.
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.
While seemingly straightforward in principle, the reliable estimation of rate constants is seldom easy in practice. Numerous issues, such as the complication of poor reaction coordinates, cause obvious approaches to yield unreliable estimates. When a reliable order parameter is available, the reactive flux theory of Chandler allows the rate constant to be extracted from the plateau region of an appropriate reactive flux correlation function. However, when applied to real data from single- molecule experiments or molecular dynamics simulations, the reactive flux correlation function requires the numerical differentiation of a noisy empirical correlation function, which can result in an unacceptably poor estimate of the rate and pathological dependence on the sampling interval. We present a modified version of this theory which does not require numerical derivatives, allowing rate constants to be robustly estimated from the time-correlation function directly. We illustrate the approach using single-molecule passive force spectroscopy measurements of an RNA hairpin.
Single-molecule force spectroscopy has proven to be a powerful tool for studying the kinetic be- havior of biomolecules. Through application of an external force, conformational states with small or transient populations can be stabilized, allowing them to be characterized and the statistics of in- dividual trajectories studied to provide insight into biomolecular folding and function. Because the observed quantity (force or extension) is not necessarily an ideal reaction coordinate, individual ob- servations cannot be uniquely associated with kinetically distinct conformations. While maximum- likelihood schemes such as hidden Markov models have solved this problem for other classes of single-molecule experiments by using temporal information to aid in the inference of a sequence of distinct conformational states, these methods do not give a clear picture of how precisely the model parameters are determined by the data due to instrument noise and finite-sample statistics, both sig- nificant problems in force spectroscopy. We solve this problem through a Bayesian extension that allows the experimental uncertainties to be directly quantified, and build in detailed balance to fur- ther reduce uncertainty through physical constraints. We illustrate the utility of this approach in characterizing the three-state kinetic behavior of an RNA hairpin in a stationary optical trap.
In multicriteria optimization, a compromise solution is a feasible solution whose
cost vector minimizes the distance to the ideal point w.r.t. a given norm. The coor-
dinates of the ideal point are given by the optimal values for the single optimization
problem for each criterion.
We show that the concept of compromise solutions ts nicely into the existing
notion of Pareto optimality: For a huge class of norms, every compromise solution
is Pareto optimal, and under certain conditions on the norm all Pareto optimal so-
lution are also a compromise solution, for an appropriate weighting of the criteria.
Furthermore, under similar conditions on the norm, the existence of an FPTAS for
compromise solutions guarantees the approximability of the Pareto set.
These general results are completed by applications to classical combinatorial
optimization problems. In particular, we study approximation algorithms for the
multicriteria shortest path problem and the multicriteria minimum spanning tree
problem. On the one hand, we derive approximation schemes for both problems, on
the other hand we show that for the latter problem simple approaches like local search
and greedy techniques do not guarantee good approximation factors.
We solve Skorokhod's embedding problem for Brownian motion with linear drift $(W_t+ \kappa t)_{t\geq 0}$ by means of techniques of stochastic control theory. The search for a stopping time
$T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ possessing the first
moment is based on solutions of backward stochastic differential equations of quadratic
type. This new approach generalizes an approach by Bass [Bas] of the classical version of
Skorokhod's embedding problem using martingale representation techniques.
We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$
with a unique stable point at the origin. We perturb the system by L\'evy noise of
intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}.
The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially
light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. We study the first exit problem for
the trajectories of the solutions of the stochastic differential equation from the interval
$[-1, 1]$. In the small noise limit $\varepsilon\to 0$ we determine the law and the mean value of the
first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.
In a strong constant electric field, a dielectric particle immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude. We derive the governing equations for this experiment, and ascertain
that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations:
explicit estimates are derived for their period and amplitude, effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
Markov state models have become very popular for the description of conformation dynamics of molecules over
long timescales. The construction of such models requires a partitioning of the configuration space such that the discretization
can serve as an approximation of metastable conformations. Since the computational complexity for the construction of a
Markov state model increases quadratically with the number of sets, it is desirable to obtain as few sets as necessary. In this
paper we propose an algorithm for the adaptive refinement of an initial coarse partitioning. A spectral clustering method is
applied to the final partitioning to detect the metastable conformations. We apply this method to the conformation analysis of
a model tri-peptide molecule, where metastable beta- and gamma-turn conformations can be identified.
Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.
We introduce nonsmooth Schur-Newton methods for the solution of the nonlinear discrete saddle-point problems arising from discretized vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The discrete problems are obtained by semi-implicit discretization in time and a first order finite element discretization in space. We incorporate the linear constraints that enforce solutions to stay on the Gibbs simplex using Lagrangian multipliers and prove existence of these multipliers under the assumption of a non-trivial initial condition for the order parameters.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod to an elastic continuum. The coupling conditions accommodate for the difference in dimension between the two models. Also, the Cosserat rod model incorporates director variables, which are not present in the elastic continuum model. Two alternative coupling conditions are proposed, which correspond to two different configuration trace spaces. For both we show existence of solutions of the coupled problems. We also derive the corresponding conditions for the dual variables and interpret them in mechanical terms.