Linking Microscopic and Macroscopic Models for Evolution: Markov Chain Network Training and Conservation Law Approximations

In this paper, a general framework for the analysis of a connection between the training of artificial neural networks via the dynamics of Markov chains and the approximation of conservation law equations is proposed. This framework allows us to demonstrate an intrinsic link between microscopic and macroscopic models for evolution via the concept of perturbed generalized dynamic systems. The main result is exemplified with a number of illustrative examples where efficient numerical approximations follow directly from network-based computational models, viewed here as Markov chain approximations. Finally, stability and consistency conditions of such computational models are discussed.Comment: 21 pages, 5 figure

Coupling Control and Human-Centered Automation in Mathematical Models of Complex Systems

In this paper we analyze mathematically how human factors can be effectively incorporated into the analysis and control of complex systems. As an example, we focus our discussion around one of the key problems in the Intelligent Transportation Systems (ITS) theory and practice, the problem of speed control, considered here as a decision making process with limited information available. The problem is cast mathematically in the general framework of control problems and is treated in the context of dynamically changing environments where control is coupled to human-centered automation. Since in this case control might not be limited to a small number of control settings, as it is often assumed in the control literature, serious difficulties arise in the solution of this problem. We demonstrate that the problem can be reduced to a set of Hamilton-Jacobi-Bellman equations where human factors are incorporated via estimations of the system Hamiltonian. In the ITS context, these estimations can be obtained with the use of on-board equipment like sensors/receivers/actuators, in-vehicle communication devices, etc. The proposed methodology provides a way to integrate human factor into the solving process of the models for other complex dynamic systems.Comment: 19 page

Solving Stochastic Differential Equations with Jump-Diffusion Efficiently: Applications to FPT Problems in Credit Risk

The first passage time (FPT) problem is ubiquitous in many applications. In finance, we often have to deal with stochastic processes with jump-diffusion, so that the FTP problem is reducible to a stochastic differential equation with jump-diffusion. While the application of the conventional Monte-Carlo procedure is possible for the solution of the resulting model, it becomes computationally inefficient which severely restricts its applicability in many practically interesting cases. In this contribution, we focus on the development of efficient Monte-Carlo-based computational procedures for solving the FPT problem under the multivariate (and correlated) jump-diffusion processes. We also discuss the implementation of the developed Monte-Carlo-based technique for multivariate jump-diffusion processes driving by several compound Poisson shocks. Finally, we demonstrate the application of the developed methodologies for analyzing the default rates and default correlations of differently rated firms via historical data.Comment: Keywords: Default Correlation, First Passage Time, Multivariate Jump-Diffusion Processes, Monte-Carlo Simulation, Multivariate Uniform Sampling Metho

Efficient estimation of default correlation for multivariate jump-diffusion processes

Evaluation of default correlation is an important task in credit risk analysis. In many practical situations, it concerns the joint defaults of several correlated firms, the task that is reducible to a first passage time (FPT) problem. This task represents a great challenge for jump-diffusion processes (JDP), where except for very basic cases, there are no analytical solutions for such problems. In this contribution, we generalize our previous fast Monte-Carlo method (non-correlated jump-diffusion cases) for multivariate (and correlated) jump-diffusion processes. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of other applications, including credit risk and option pricing.Comment: Keywords: Default correlation, First passage time problem, Monte Carlo simulatio

Monte-Carlo Simulations of the First Passage Time for Multivariate Jump-Diffusion Processes in Financial Applications

Many problems in finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the first passage time problem in the context of multivariate (and correlated) jump-diffusion processes. The developed technique provide an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of the default rates and default correlations of several different, but correlated firms via a set of empirical data.Comment: Keywords: First passage time; Monte Carlo simulation; Multivariate jump-diffusion processes; Credit ris

Coupled Effects in Quantum Dot Nanostructures with Nonlinear Strain and Bridging Modelling Scales

We demonstrate that the conventional application of linear models to the analysis of optoelectromechanical properties of nanostructures in bandstructure engineering could be inadequate. The focus of the present paper is on a model based on the coupled Schrodinger-Poisson system where we account consistently for the piezoelectric effect and analyze the influence of different nonlinear terms in strain components. The examples given in this paper show that the piezoelectric effect contributions are essential and have to be accounted for with fully coupled models. While in structural applications of piezoelectric materials at larger scales, the minimization of the full electromechanical energy is now a routine in many engineering applications, in bandstructure engineering conventional approaches are still based on linear models with minimization of uncoupled, purely elastic energy functionals with respect to displacements. Generalizations of the existing models for bandstructure calculations are presented in this paper in the context of coupled effects.Comment: 24 pages, 1 table, 15 figure

First Passage Time for Multivariate Jump-diffusion Stochastic Models With Applications in Finance

The ``first passage-time'' (FPT) problem is an important problem with a wide range of applications in mathematics, physics, biology and finance. Mathematically, such a problem can be reduced to estimating the probability of a (stochastic) process first to reach a critical level or threshold. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we develop a Monte-Carlo-based methodology for the solution of the FPT problem in the context of a multivariate jump-diffusion stochastic process. The developed methodology is tested by using different parameters, the simulation results indicate that the developed methodology is much more efficient than the conventional Monte Carlo method. It is an efficient tool for further practical applications, such as the analysis of default correlation and predicting barrier options in finance.Comment: Keywords: Monte-Carlo simulations, first passage time, multivariate jump-diffusion process; 10 pages, 3 figure

Effect of Internal Viscosity on Brownian Dynamics of DNA Molecules in Shear Flow

The results of Brownian dynamics simulations of a single DNA molecule in shear flow are presented taking into account the effect of internal viscosity. The dissipative mechanism of internal viscosity is proved necessary in the research of DNA dynamics. A stochastic model is derived on the basis of the balance equation for forces acting on the chain. The Euler method is applied to the solution of the model. The extensions of DNA molecules for different Weissenberg numbers are analyzed. Comparison with the experimental results available in the literature is carried out to estimate the contribution of the effect of internal viscosity.Comment: Keywords: effect of internal viscosity, dumbbell model, Brownian dynamics, DNA molecules in shear flo

Thermo-Mechanical Wave Propagation In Shape Memory Alloy Rod With Phase Transformations

Many new applications of ferroelastic materials require a better understanding of their dynamics that often involve phase transformations. In such cases, an important prerequisite is the understanding of wave propagation caused by pulse-like loadings. In the present study, a mathematical model is developed to analyze the wave propagation process in shape memory alloy rods. The first order martensite transformations and associated thermo-mechanical coupling effects are accounted for by employing the modified Ginzburg-Landau-Devonshire theory. The Landau-type free energy function is employed to characterize different phases, while a Ginzburg term is introduced to account for energy contributions from phase boundaries. The effect of internal friction is represented by a Rayleigh dissipation term. The resulted nonlinear system of PDEs is reduced to a differential-algebraic system, and Chebyshev's collocation method is employed together with the backward differentiation method. A series of numerical experiments are performed. Wave propagations caused by impact loadings are analyzed for different initial temperatures. It is demonstrated that coupled waves will be induced in the material. Such waves will be dissipated and dispersed during the propagation process, and phase transformations in the material will complicate their propagation patterns. Finally, the influence of internal friction and capillary effects on the process of wave propagation is analyzed numerically.Comment: Keywords: nonlinear waves, thermo-mechanical coupling, martensite transformations, Ginzburg-Landau theory, Chebyshev collocation metho

Model Reduction Applied to Square to Rectangular Martensitic Transformations Using Proper Orthogonal Decomposition

Model reduction using the proper orthogonal decomposition (POD) method is applied to the dynamics of ferroelastic patches to study the first order square to rectangular phase transformations. Governing equations for the system dynamics are constructed by using the Landau-Ginzburg theory and are solved numerically. By using the POD method, a set of empirical orthogonal basis functions is first constructed, then the system is projected onto the subspace spanned by a small set of basis functions determined by the associated singular values. The performance of the low dimensional model is verified by simulating nonlinear thermo-mechanical waves and square to rectangular transformations in a ferroelastic patch. Comparison between numerical results obtained from the original PDE model and the low dimensional one is carried out.Comment: Keywords: Phase transformation, ferroelastic patch, model reduction, proper orthogonal decomposition, Galerkin projectio