Differential variational inequalities

International audienceThis paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems , and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensiona

Evolutionary-based sparse regression for the experimental identification of duffing oscillator

In this paper, an evolutionary-based sparse regression algorithm is proposed and applied onto experimental data collected from a Duffing oscillator setup and numerical simulation data. Our purpose is to identify the Coulomb friction terms as part of the ordinary differential equation of the system. Correct identification of this nonlinear system using sparse identification is hugely dependent on selecting the correct form of nonlinearity included in the function library. Consequently, in this work, the evolutionary-based sparse identification is replacing the need for user knowledge when constructing the library in sparse identification. Constructing the library based on the data-driven evolutionary approach is an effective way to extend the space of nonlinear functions, allowing for the sparse regression to be applied on an extensive space of functions. The results show that the method provides an effective algorithm for the purpose of unveiling the physical nature of the Duffing oscillator. In addition, the robustness of the identification algorithm is investigated for various levels of noise in simulation. The proposed method has possible applications to other nonlinear dynamic systems in mechatronics, robotics, and electronics

Prediction of Daily PM2.5 Concentration in China Using Data-Driven Ordinary Differential Equations

Accurate reporting and forecasting of PM2.5 concentration are important for improving public health. In this paper, we propose a daily prediction method of PM2.5 concentration by using data-driven ordinary differential equation (ODE) models. Specifically, based on the historical PM2.5 concentration, this method combines genetic programming and orthogonal least square method to evolve the ODE models, which describe the transport of PM2.5 and then uses the data-driven ODEs to predict the air quality in the future. Experiment results show that the ODE models obtain similar prediction results as the typical statistical model, and the prediction results from this method are relatively good. To our knowledge, this is the first attempt to evolve data-driven ODE models to study PM2.5 prediction

Two Species Evolutionary Game Model of User and Moderator Dynamics

We construct a two species evolutionary game model of an online society consisting of ordinary users and behavior enforcers (moderators). Among themselves, moderators play a coordination game choosing between being "positive" or "negative" (or harsh) while ordinary users play prisoner's dilemma. When interacting, moderators motivate good behavior (cooperation) among the users through punitive actions while the moderators themselves are encouraged or discouraged in their strategic choice by these interactions. We show the following results: (i) We show that the $\omega$-limit set of the proposed system is sensitive both to the degree of punishment and the proportion of moderators in closed form. (ii) We demonstrate that the basin of attraction for the Pareto optimal strategy $(\text{Cooperate},\text{Positive})$ can be computed exactly. (iii) We demonstrate that for certain initial conditions the system is self-regulating. These results partially explain the stability of many online users communities such as Reddit. We illustrate our results with examples from this online system.Comment: 8 pages, 4 figures, submitted to 2012 ASE Conference on Social Informatic

Fully Automated Myocardial Infarction Classification using Ordinary Differential Equations

Portable, Wearable and Wireless electrocardiogram (ECG) Systems have the potential to be used as point-of-care for cardiovascular disease diagnostic systems. Such wearable and wireless ECG systems require automatic detection of cardiovascular disease. Even in the primary care, automation of ECG diagnostic systems will improve efficiency of ECG diagnosis and reduce the minimal training requirement of local healthcare workers. However, few fully automatic myocardial infarction (MI) disease detection algorithms have well been developed. This paper presents a novel automatic MI classification algorithm using second order ordinary differential equation (ODE) with time varying coefficients, which simultaneously captures morphological and dynamic feature of highly correlated ECG signals. By effectively estimating the unobserved state variables and the parameters of the second order ODE, the accuracy of the classification was significantly improved. The estimated time varying coefficients of the second order ODE were used as an input to the support vector machine (SVM) for the MI classification. The proposed method was applied to the PTB diagnostic ECG database within Physionet. The overall sensitivity, specificity, and classification accuracy of 12 lead ECGs for MI binary classifications were 98.7%, 96.4% and 98.3%, respectively. We also found that even using one lead ECG signals, we can reach accuracy as high as 97%. Multiclass MI classification is a challenging task but the developed ODE approach for 12 lead ECGs coupled with multiclass SVM reached 96.4% accuracy for classifying 5 subgroups of MI and healthy controls

Survey on Modelling Methods Applicable to Gene Regulatory Network

Gene Regulatory Network (GRN) plays an important role in knowing insight of cellular life cycle. It gives information about at which different environmental conditions genes of particular interest get over expressed or under expressed. Modelling of GRN is nothing but finding interactive relationships between genes. Interaction can be positive or negative. For inference of GRN, time series data provided by Microarray technology is used. Key factors to be considered while constructing GRN are scalability, robustness, reliability and maximum detection of true positive interactions between genes. This paper gives detailed technical review of existing methods applied for building of GRN along with scope for future work

Learning Equilibria in Games by Stochastic Distributed Algorithms

We consider a class of fully stochastic and fully distributed algorithms, that we prove to learn equilibria in games. Indeed, we consider a family of stochastic distributed dynamics that we prove to converge weakly (in the sense of weak convergence for probabilistic processes) towards their mean-field limit, i.e an ordinary differential equation (ODE) in the general case. We focus then on a class of stochastic dynamics where this ODE turns out to be related to multipopulation replicator dynamics. Using facts known about convergence of this ODE, we discuss the convergence of the initial stochastic dynamics: For general games, there might be non-convergence, but when convergence of the ODE holds, considered stochastic algorithms converge towards Nash equilibria. For games admitting Lyapunov functions, that we call Lyapunov games, the stochastic dynamics converge. We prove that any ordinal potential game, and hence any potential game is a Lyapunov game, with a multiaffine Lyapunov function. For Lyapunov games with a multiaffine Lyapunov function, we prove that this Lyapunov function is a super-martingale over the stochastic dynamics. This leads a way to provide bounds on their time of convergence by martingale arguments. This applies in particular for many classes of games that have been considered in literature, including several load balancing game scenarios and congestion games

Revealing hidden dynamics from time-series data by ODENet

To understand the hidden physical concepts from observed data is the most basic but challenging problem in many fields. In this study, we propose a new type of interpretable neural network called the ordinary differential equation network (ODENet) to reveal the hidden dynamics buried in the massive time-series data. Specifically, we construct explicit models presented by ordinary differential equations (ODEs) to describe the observed data without any prior knowledge. In contrast to other previous neural networks which are black boxes for users, the ODENet in this work is an imitation of the difference scheme for ODEs, with each step computed by an ODE solver, and thus is completely understandable. Backpropagation algorithms are used to update the coefficients of a group of orthogonal basis functions, which specify the concrete form of ODEs, under the guidance of loss function with sparsity requirement. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet demonstrates its remarkable capability to deal with time-series data. In the end, we apply the ODENet to real actin aggregation data observed by experimentalists, and it shows an impressive performance as well

Cycles in Nonlinear Macroeconomics

The monograph is concerned with some key problems of the theory of nonlinear economic dynamics. The authors' concept consists in analyzing the problem of structural instability of economic systems within the framework of the synergetic paradigm. As examples, the classical models of macroeconomics are considered. The authors present the results of the study of the phenomenon of self-organization in open and nonequilibrium economic systems. The generation of limit cycles, as well as of more complex periodic structures, is discussed; the character of their stability is examined.Comment: 100 pages, 16 figures; the figures correcte

Splitting schemes for poroelasticity and thermoelasticity problems

In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations are presented. Approximation in space is based on the finite element method. We construct splitting schemes and give some numerical comparisons for typical poroelasticity problems. The results of numerical simulation of a 3D problem are presented. Special attention is given to using hight performance computing systems.Comment: 19 pages, 8 figure