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- Robust Optimization (4)
- primal heuristic (4)
- Large Neighborhood Search (3)
- Navier-Stokes equations (3)
- molecular dynamics (3)
- stochastic programming (3)
- ANOVA decomposition (2)
- FPTAS (2)
- Knapsack Problem (2)
- Markov models (2)
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.
In many fields of physics, chemistry and biology the characterization of dynamical processes
between states or species is of fundamental interest. The central mathematical function in such sit-
uations is the committor probability - a generalized reaction coordinate that measures the progress
of the process of interest as the probability of proceeding towards the target state rather than re-
lapsing to the source state. Here, we present methodology for the efficient computation of com-
mittor probabilities for large-scale systems, such as, for example simuations of biomolecular fold-
ing. A method is derived for computing the committor for discrete state spaces using eigenvectors
with expressions for the sensitivity and a Bayesian error model for the committor. The concepts
are illustrated on two examples of diffusive dynamics with a very large number of states: a two-
dimensional model potential with three minima, and a three-dimensional model representing
protein-ligand binding. The method can finally be used to compute committor probabilities in-
cluding error estimations for medium and large system sizes allowing access to the apparatus of
transition path theory and its applications.
The LIBOR market model is very popular for pricing inter-
est rate derivatives, but is known to have several pitfalls. In addition, if
the model is driven by a jump process, then the complexity of the drift
term is growing exponentially fast (as a function of the tenor length). In
this work, we consider a Levy-driven LIBOR model and aim at developing accurate and efficient log-Levy approximations for the dynamics of
the rates. The approximations are based on truncation of the drift term
and Picard approximation of suitable processes. Numerical experiments
for FRAs, caps and swaptions show that the approximations perform
very well. In addition, we also consider the log-Levy approximation of
annuities, which offers good approximations for high volatility regimes.
We derive and study a dynamical model for suspensions of negatively buoyant particles on an incline. Our theoretical model includes the settling/sedimentation due to gravity as well as the resuspension of particles induced by shear-induced migration, leading to disaggregation of the dense sediment layer. Out of the three different regimes observed in the experiments, we focus on the so-called settled case, where the particles settle out of the flow, and two distinct fronts, liquid and particle, form. Using an approach relying on asymptotics, we systematically connect our dynamic model with the previously developed equilibrium theory for particle-laden flows. We show that the resulting transport equations for the liquid and the particles are of hyperbolic type, and study the dilute limit, for which we derive the analytic solution. We also carry out a systematic experimental study of the settled regime, focusing on the motion of the liquid and the particle fronts. Finally, we carry out numerical simulations of our transport equations. We show that the model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental 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.
In this paper, we study the dual representation for generalized multiple stopping problems,
hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by
integer valued adapted processes and refraction periods modeled by stopping times. As
such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b),
Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise
options, which either take into consideration a refraction period or volume constraints, but
not both simultaneously. We also allow more flexible cashflow structures than the additive
structure in the above references. For example some exponential utility problems are covered
by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm
for constructing confidence intervals for the price of multiple exercise options and exemplify
it by a numerical study on the pricing of a swing option in an electricity market.
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.
Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities
(2012)
In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise
control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to
the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained. These rates coincide with
rates obtained by numerical experiments, which are included in the paper.
We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
We discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.
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.
Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models.
We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model.
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.
Modelling, parameter identification, and simulation play an important rôle in Systems Biology. In recent years, various software packages have been established for scientific use in both licencing types, open source as well as commercial. Many of these codes are based on inefficient and mathematically outdated algorithms. By introducing the package BioPARKIN recently developed at ZIB, we want to improve this situation significantly. The development of the software BioPARKIN involves long standing mathematical ideas that, however, have not yet entered the field of Systems Biology, as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks. BioPARKIN originates from the package PARKIN, written by P.Deuflhard and U.Nowak, that has been applied successfully for parameter identification in physical chemistry for many years.
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.
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs. Their integrands are piecewise linear, but neither smooth nor of bounded variation in the sense of Hardy and Krause. We show that under some weak geometric condition on the two-stage model all terms of their
ANOVA decomposition, except the one of highest order, are smooth and, hence, certain Quasi-Monte Carlo algorithms may achieve the optimal rate of convergence $O(n^{-1+\delta})$ with $\delta\in(0,\frac{1}{2})$ and a constant not depending on the dimension if the integrands belong to weighted tensor product Sobolev spaces with properly selected weights. The geometric condition is generically (i.e., almost everywhere) satisfied if the underlying distribution is normal. We also discuss sensitivity
indices and efficient dimensions of two-stage integrands, and suggest a dimension reduction heuristic for such integrands.
Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain, but do not belong to tensor product Sobolev spaces. Motivated by applying Quasi-Monte Carlo methods we show that all terms of their ANOVA decomposition, except the one of highest order, are smooth if the underlying densities are smooth and certain geometric condition is satisfied. The latter condition is generically satisfied in the normal case.
In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.
We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature.
Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov
growth that include surface energies also with large anisotropy as well as the effect of wetting energy,
elastic energy and a randomly perturbed atomic deposition flux.
A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows
for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions.
These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.
Analysis of the second phase of the GMRES convergence for a convection-diffusion model problem
(2012)
It is well konwn that GMRES applied to linear algebraic systems arising from a convection-diffusion model problem that has been discretized by the streamline upwind Petrov-Galerkin (SUPG) method, typically displays two distinct phases of convergence: a slow initial phase followed by a convergence acceleration in the second phase. This paper complements the known results on the length of the initial phase by analyzing how the acceleration in the second phase of convergence is related to the mesh Peclet number and the choice of the stabilization parameter in the SUPG discretization. The analysis is based on some new expressions and bounds for the GMRES residuals, which can be of general interest.
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.
Logical modeling of biological regulatory networks gives rise to a representation of the system's dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multi-valued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step towards meaningful network reduction as well as model refinement methods.
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.
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.
Energetic reasoning is one of the most powerful propagation algorithms in cumulative scheduling. In practice, however, it is not commonly used because it has a high running time and its success highly depends on the tightness of the variable bounds. In order to speed up energetic reasoning, we provide an easy-to-check necessary condition for energetic reasoning to detect infeasibilities. We present an implementation of energetic reasoning that employs this condition and that can be parametrically adjusted to handle the trade-off between solving time and propagation overhead. Computational results on instances from the PSPLIB are provided. These results show that using this condition decreases the running time by more than a half, although more search nodes need to be explored.
This paper discusses adaptive finite element methods (AFEMs) for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.
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.
Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key substances over time. In this work we describe recent improvements of this model, including the introduction of new components, and elimination of time delays. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after a single dose administration of synthetic prostaglandin F2alpha.
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.
Hybrid systems are often used to describe many complex dynamic phenomena by combining multiple modes of
dynamics into whole systems. In this paper, we present a flat Dirichlet process switching (FDPS) model that defines
a prior on mode switching dynamics of hybrid systems. Compared with the classical Markovian jump system (MJS)
models, the FDPS model is nonparametric and can be applied to the hybrid systems with an unbounded number of
potential modes. On the other hand, the probability structure of the new model is simpler and more flexible than the
recently proposed hierarchical Dirichlet process (HDP) based MJS. Furthermore, we develop a Markov chain Monte
Carlo (MCMC) method for estimating the states of hybrid systems with FDPS prior. And the numerical simulations
of a hybrid system in different conditions are employed to show the effectiveness of the proposed approach.
In the paradigm of VON N EUMANN AND M ORGENSTERN, a representation of affine pref-
erences in terms of an expected utility can be obtained under the assumption of weak continu-
ity. Since the weak topology is coarse, this requirement is a priori far from being negligible.
In this work, we replace the assumption of weak continuity by monotonicity. More precisely,
on the space of lotteries on an interval of the real line, it is shown that any affine preference
order which is monotone with respect to the first stochastic order admits a representation in
terms of an expected utility for some nondecreasing utility function. As a consequence, any
affine preference order on the subset of lotteries with compact support, which is monotone
with respect to the second stochastic order, can be represented in terms of an expected util-
ity for some nondecreasing concave utility function. We also provide such representations
for affine preference orders on the subset of those lotteries which fulfill some integrability
conditions. The subtleties of the weak topology are illustrated by some examples.
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.
We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.
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 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.
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.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
Recently, Khuller, Moss and Naor presented a greedy algorithm for the budgeted maximum coverage problem. In this note, we observe that this algorithm also approximates a special case of set-union knapsack problem within a constant factor. In the special case, an element is a member of less than a constant number of subsets. This guarantee naturally extends to densest k-subgraph problem on graphs of bounded degree.
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.
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.
We propose a new approach to competitive analysis by introducing the novel concept of online approximation schemes. Such scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our method can also be applied to many other problems and yields a completely new and interesting view on online algorithms.
A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on an adapted mesh which consists of triangles and parametric quadrilaterals in 2D. The number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the distance of the neighboring inclusions. While being related to network methods, the approach can tackle more general settings. We present an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. Moreover, it is demonstrated that the assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.
This study presents a differential equation model for the feedback mechanisms between
Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing
Hormone (LH), development of follicles and corpus luteum, and the production of estradiol
(E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual
cycle. In contrast to other models, this model does not involve delay differential equations
and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster
simulation times and efficient parameter identification. These steps were essential to tackle
the task of developing a mathematical model for the administration of GnRH analogues. The
focus of this paper is on model development for GnRH receptor binding and the integration of
a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH
antagonist Cetrorelix into the menstrual cycle model. The final mathematical model describes
the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women.
Moreover, it correctly predicts the changes in the cycle following single and multiple dose
administration of Nafarelin or Cetrorelix at different stages in the cycle.
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.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
In incompressible flows with vanishing
normal velocities at the boundary, irrotational forces in the momentum
equations should be balanced
completely by the pressure gradient.
Unfortunately, nearly all available discretization methods for incompressible flows violate this property.
The origin of the problem is that discrete velocity approximations
of incompressible flows are usually not
divergence-free. Hence, the use of divergence-free velocity reconstructions is
proposed wherever an $L^2$ scalar product appears in the discrete
variational formulation.
The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids.
It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element
increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
A mathematical model for instationary magnetization
processes is considered, where the underlying spatial domain
includes electrically conducting and nonconducting regions. The
model accounts for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil. By a theorem of Showalter on degenerate parabolic
equations, theorems on existence, uniqueness, and regularity of the
solution to the associated Maxwell integrodifferential system are
proved.
We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLIB we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.
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.
Diffusion processes are relevant for a variety of phenomena in the natural sciences, including
diffusion of cells or biomolecules within cells, diffusion of molecules on a membrane or surface,
diffusion of a molecular conformation within a complex energy landscape. Many experimental
tools exist now to track such diffusive motions in single cells or molecules, including high-resolution
light microscopy, optical tweezers, fluorescence quenching, and Förster resonance energy transfer
(FRET). Experimental observations are most often indirect and incomplete: (1) They do not
directly reveal the potential or diffusion constants that govern the diffusion process, (2) they have
limited time and space resolution, and (3) the highest-resolution experiments do not track the
motion directly but rather probe it stochastically by recording single events, such as photons,
whose properties depend on the state of the system under investigation.
Here, we propose a general Bayesian framework to model diffusion processes with nonlinear
drift based on incomplete observations as generated by various types of experiments. A maximum
penalized likelihood estimator is given as well as a Gibbs sampling method that allows to estimate
the trajectories that have caused the measurement, the nonlinear drift or potential function and
the noise or diffusion matrices, as well as uncertainty estimates of these properties. The approach
is illustrated on numerical simulations of FRET experiments where it is shown that trajectories,
potentials and diffusion constants can be efficiently and reliably estimated even in cases with little
statistics or non-equilibrium measurement conditions.