In addition, it is worth mentioning that methods based on averaging, such as HMM and the equation-free method, could work for systems with arbitrary stiff potentials too; however, besides the difficulty in identifying the slow variable, as well as the necessity for using a smaller timestep (i.e., mesoscopic, as opposed to the macroscopic ones used by many of the above methods designed for quadratic stiff potentials), there has been no success in making these generic multiscale ap- proaches symplectic for Hamiltonian systems. In their original form, these methods are based on the averaging of the instantaneous drifts of the slow variable, which breaks symplecticity in all variables. On the other hand, variants that preserve structures other than the symplecticity on all variables have been successfully pro- posed. By using Verlet/leap-frog macro-solvers, methods that are symplectic on slow variables (when those variables can be identified) have been proposed in the framework of HMM (Heterogeneous Multiscale Method) in [252, 56]. A ‘reversible averaging’ method has been proposed in  for mechanical systems with sep- arated fast and slow variables. More recently, a reversible multiscaleintegration method for mechanical systems was proposed in  in the context of HMM. After tracking down the slow variables, this method enforces reversibility in all variables as an optimization constraint at each coarse step when minimizing the distance between the effective drift obtained from the micro-solver and the drift of the macro-solver. We also refer to  for a symmetric HMM for mechanical systems with stiff potentials of the form 1 P ν
Bensoussan, A.: Lectures on stochastic control. In: Mitter, S.K., Moro, A. (eds.) Nonlinear filtering and stochastic control, Proceedings of the 3rd 1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Cortona, July 1–10, 1981 pp. 1–62. Lecture Notes in Mathematics 972. Springer-Verlag, Berlin (1982)
On the other hand, the deterministic approximation can lose an important information in many stochasticsystems. While it usually provides a good approximation of the process mean value, it ignores completely other properties, for instance, variance, bimodality, tail behaviour, etc. Kurtz (1976) provided a second approximation which retains the stochastic nature by means of the diffusion process. The same equations had became popular in Chemistry under the name of Langevin equations due to the contribution by Gillespie (2000). These equations have been used in many works as a computational trick to speed up simulations of the original process. While this computational approach to the diffusion approximation proved to be fruit- ful, such interpretation hides in part the richness and the importance of the result by Kurtz (1976). Moreover, stochastic reaction networks attracted a renewed interest recently (Ander- son and Kurtz, 2015; ´ Erdi and Lente, 2014; Santill´ an, 2014; Ullah and Wolkenhauer, 2011). New motivations come both from the application in the system biology, demonstrating the emergence of the stochastic effects at small scales, and from new theoretical investigations that allowed to extend mathematical results, previously known in the deterministic setting only, to the stochastic world (Anderson et al., 2017a, 2010; Cappelletti and Wiuf, 2016).
Continuous time Markov Chain models are widely used to model physical and biological processes (e.g., see [1,3]). These models are typically used when dealing with dy- namic systems involving low species count. However, be- cause simulations are quite expensive for large population stochasticsystems, in such models one often wishes to know whether or not the stochastic system can be approx- imated by a deterministic one when the population size is sufficiently large. Theory established by Kurtz (e.g. [11– 15]) gives a way to construct a deterministic system to approximate density dependent continuous time Markov Chains as the population size grows large (this result is often called Kurtz’s limit theorem). In general, deter- ministic systems are much easier to analyze compared to stochasticsystems. Techniques, such as parameter estima- tion methods, are well developed for deterministicsystems, whereas parameter estimation is much more difficult in a stochastic framework. For example, in , the authors developed a method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz’s limit theory coupled with inverse problem methods developed for deterministic dynamical systems and illustrated these ideas in the context of disease dynamics. The methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach detailed there leads to a deterministic approximation ob- tained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov
Abstract . In the past years it has become evident that stochastic eﬀects in regulatory networks play an important role, leading to an increasing in stochastic modelling attempts. In contrast, metabolic networks involving large numbers of molecules are most often modelled deterministically. Going to- wards the integration of diﬀerent model systems, gen-regulatory networks become part of a larger model system including signalling pathways and metabolic networks. Thus, the question arises of how to eﬃciently and accurately simulation such coupled or hybrid systems. We present an algorithmic approach for the simulation of hybrid stochastic and deterministic reaction models that allows for adaptive step-size integration of the deterministic equations while at the same time accurately tracing the stochastic reaction events. We present a mathematical derivation of the hybrid system on the stochastic process level, and present numerical examples that outline the power of hybrid simulations.
Stability properties of constrained stochastic processes are useful in many appli- cations arising from computing, telecommunications, and manufacturing systems. In this chapter, we study a family of constrained Markov modulated diffusion processes that arise in the heavy traffic analysis of multiclass queueing networks. We estab- lish positive recurrence and geometric ergodicity properties for such processes under suitable stability conditions on a related deterministic dynamical system (see ,  and ). Results of this chapter will be used in Chapter 4 to study the convergence of invariant measures for Markov modulated open queueing networks in heavy traffic. It will be shown there that under suitable conditions, the invariant measure for the queueing length process converges weakly to the invariant measure of a constrained Markov modulated diffusion process of the form studied in the current chapter.
Materials by design is a core driver in enhancing sustainability and improving effi- ciency in a broad spectrum of industries. To this end, thermo-mechanical processes and many of the underlying phenomena were studied extensively in the context of specific cases. The goal of this thesis is threefold: First, we aim to establish a novel numerical model on the micro- and mesoscale that captures dynamic recrys- tallization in a generalized framework. Based on the inheritance of the idea of state switches, we term this scheme Field-Monte-Carlo Potts method. We employ a fi- nite deformation framework in conjunction with a continuum-scale crystal plasticity formulation and extend the idea of state switches to cover both grain migration and nucleation. We introduce physically-motivated state-switch rules, based on which we achieve a natural marriage between the deterministic nature of crystal plastic- ity and the stochastic nature of dynamic recrystallization. Using a novel approach to undertake the states-switches in a transient manner, the new scheme benefits from enhanced stability and can, therefore, handle arbitrary levels of anisotropy. We demonstrate this functionality at the example of pure Mg at room temperature, which experiences strong anisotropy through the different hardening behavior on the hc + ai-pyramidal and prismatic slip systems as opposed to the basal slip systems as well as through the presence of twinning as an alternative strain accommodating mechanisms. Building on this generalized approach, we demonstrate spatial con- vergence of the scheme along with the ability to capture the transformation from single- to multi-peak stress-strain behavior.
Several numerical approaches are implemented to find the solutions of Hamiltonian systems, but some of them are inappropriate for long-time simulations. In recent years geometric numerical integration methods are extensively used and gaining popularity for the solution of Hamiltonian problems. A geometrical integrator preserves one or more geometric (physical) property(ies) exactly (i.e. within the roundoff errors). In physical systems energy preservation, symmetries, time-reversal invariance, symplectic structure (for stochastic symplectic perspective we refer to ), an- gular momentum and phase-space volume are some very important and crucial geometric properties.
Abstract. We discuss applications of a recently developed method for model reduction based on linear response the- ory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equa- tions with slow and fast degrees of freedom. The weak cou- pling model reduction method results in general in a non- Markovian system; we therefore discuss the Markovianiza- tion of the system to allow for straightforward numerical integration. We compare the applied method to the equa- tions obtained through homogenization in the limit of large timescale separation between slow and fast degrees of free- dom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more ac- curate results in all test cases, albeit with a higher numerical cost.
Due to the wide range of scales in these solutions, it is extremely challenging to resolve the small scales of the solutions by direct numerical simulation. Tremendous computational resources are required to solve for the small scales of the solution, which makes it prohibitively expensive to solve such problems. Even for today’s computing resources, it is easy to exceed the limit of computer memory or CPU time. Sometimes, from an application perspective, it is often sufficient to predict the macroscopic properties of the multiscalesystems, and therefore, we are interested in the large scale solutions. Furthermore, if we want to find out the information at all scales, we can construct the small scale solutions from the large scale solutions by exploring the coupling between them. Thus, finding an effective equation that governs the large scale solution is very important. It is very difficult to derive an effective equation since the coupling between the small scale solution and the large scale solution is in general nonlinear and nonlocal. SPDEs involving multiple scales become more complicated. We not only need to use a very fine mesh to resolve the small scales of the solution in the physical space, but also need to approximate the solution in the stochastic space of which the dimension could be high. Thus, we need to seek accurate numerical methods for PDEs and SPDEs with multiple scales, and reduce the computational cost.
Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with time-dependent structural coefficients are developed to examine ki- nematic and dynamic problems for a deterministic chaos of N stochastic waves in the two-dimen- sional theory of the Newtonian flows with harmonic velocity. The Dirichlet problems are formu- lated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains for stochastic waves vanishing at infinity. Development of the novel method of solving partial differential equations through decomposition in invariant structures is resumed by using experimental and theoretical computation in Maple™. This computational method generalizes the analytical methods of separation of variables and un- determined coefficients. Exact solutions for the deterministic chaos of upper and lower cumula- tive flows are revealed by experimental computing, proved by theoretical computing, and justified by the system of Navier-Stokes PDEs. Various scenarios of a developed wave chaos are modeled by 3N parameters and 2N boundary functions, which exhibit stochastic behavior.
individual dynamics of a certain parameter set. However, it is not the degree of complexity (e.g., complex periodic versus simple periodic behavior) that determines the transition range. Our results show that it is rather the attractive property of the respective phase space that plays a more important role than the complexity of Ca 21 oscillations. The attractive prop- erties of the phase space have been quantiﬁed by the sum of Lyapunov exponents (the divergence). Our results indicate that at lower divergence the transition from stochastic to deterministic behavior occurs at lower particle numbers, which means that the system is well characterized by ODEs at realistic particle numbers. At higher divergence values the transition occurs at signiﬁcantly higher particle numbers, which indicates the need to employ stochastic modeling. These ﬁndings are in accordance with the experimental ob- servation that apparently stochastic behavior is more com- mon in bursting calcium oscillations during high agonist doses, which corresponds to high divergence value in the corresponding model. The results were also veriﬁed with other models and should apply for many types of bio- chemical models.
The similarities between drinking behavior and contagion are used to explore — under two types of social structure in the population: homogenous and heterogeneous mixing— the role of nonlinear social interactions on the dynamics of alcohol con- sumption; quantifying how some model parameters influence the latter dynamics. A deterministic model, assuming homogeneous mixing, is used to derive relative sensitiv- ity functions which explain how the recovery and relapse rates affect the establishment of problem drinkers. A continuos-time Markov chain model is derived from the de- terministic model; stochastic simulations are used to quantify how histograms of the number of problem drinkers depend on the recovery and relapse rates, as these rates are gradually incremented. The impact of lowering the relapse rate, as a result of suc- cessful treatment, at various intervention times, is assessed by stochastic simulations of drinking dynamics among communities with small-world structure; reductions in the average number of problem drinkers are obtained —with some community structures showing more vulnerability to higher levels of prevalence than others. We conclude from sensitivity analyses (of deterministic and stochastic models) that, either: increas- ing the recovery rate; or lowering the relapse rate, are measures with positive effects —they tend to reduce the number of problem drinkers.
Table 1 presents the percentage of sucesses obtained in the comparison between the two processes, where n is the sample size, L is the autocorre- lation lenght, the sample autocorrelation and sample partial autocorrelation metrics (ACFG and PACFG metrics) uses a geometric decay of p = 0.05, in the LNPER metric F for low frequencies corresponds to periodogram ordi- nates from 1 to √ n and F for high frequencies corresponds to periodogram ordinates from √ n + 1 to n/2.
Typically, traffic impact assessment is conducted to evaluate the impacts by proposed developments. In these assessments, trip rates are mostly adapted from local trip generation manual. This study is about evaluation on the current trip rates of commercial land use for three hypermarkets in Malaysia, by comparing deterministic and stochastic approaches. In this study, three deterministic approaches, Trip rate Analysis, Cross-Classification Analysis and Regression Analysis are applied to generate trip rates respectively. Similarly, stochastic approaches using ‘Weibull distribution’ are found appropriate for the collected trip generation data. The obtained mean trip rates have been critically examined for their suitability of use. The results indicate that the four approaches considered in this research produced significant variances from the estimated mean entry trip rates as mentioned in the Trip Generation Manual of Highway Planning Unit (HPU) of Malaysia.Therefore, this study debates the adoption of currently available trip rates for nationwide application and presents suggestions about the direction of future trip generation studies. In this regard, this study presents a framework for the estimation of trip rates for Malaysian conditions, based on the aforementioned four approaches along with some guidelines for their adoption.
Abstract. We present a deterministic selection-mutation model with a discrete trait vari- able. We show that for a selection-mutation birth term the deterministic model has a unique interior equilibrium and we prove that this equilibrium is globally stable. In the pure selec- tion case the outcome is known to be that of competitive exclusion where the subpopulation with the largest growth to mortality ratio will survive and the remaining subpopulations will go to extinction. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth to mortality ratios are small in the pure selection case, it cannot be determined apriori which subpopulation will have the highest probability to survive and win the competition.
With the advent of eye tracking technology, gaze control is perceived to be an alternative way of human-machine interaction. Owing to the nature of eye movement, visually localizing a target is faster than by hand control and less likely to cause fatigue. Because of the connection between eye movement and attention, it is plausible to infer what targets may be of interest to people based on their gaze pattern. However, current design paradigms focus on fitting eye movements to user interfaces initially designed for hand movement. For example, many studies attempted to utilize gaze point as a mouse cursor, which resulted in limited success. To date, gaze-based interaction has been prone to mistakes and it is time consuming compared to other conventional non-keyboard input methods. Therefore, commercial eye tracking systems are almost exclusively used by people with severe disabilities or for military purposes.
To our knowledge, such results for FRPD classes have not been previously discussed in the literature. One reason for this is that much work on natural language processing uses probabilis- tic non-deterministic automata. These describe the same class of stochastic languages as Hidden Markov Models (HMMs) (Vidal et al., 2005a,b). Non-determinism can make a big difference when it comes to parsing and learning. In a determin- istic model M, each string w can be associated with at most one path through M , whereas in non-deterministic M, there can be infinitely many paths for w. This is one reason why methods used for learning HMM are not guaranteed to return a MLE. Since the states are ‘hidden’ one uses meth- ods like Expectation Maximization, which may converge to a local optimum that is not a global optimum (Jurafsky and Martin, 2008; Heinz et al., 2015).