Probabilistic approximations of ODEs based bio-pathway dynamics

Bio-chemical networks are often modeled as systems of ordinary differential equations (ODEs). Such systems will not admit closed form solutions and hence numerical simulations will have to be used to perform analyses. However, the number of simulations required to carry out tasks such as parameter estimation can become very large. To get around this, we propose a discrete probabilistic approximation of the ODEs dynamics. We do so by discretizing the value and the time domain and assuming a distribution of initial states w.r.t. the discretization. Then we sample a representative set of initial states according to the assumed initial distribution and generate a corresponding set of trajectories through numerical simulations. Finally, using the structure of the signaling pathway we encode these trajectories compactly as a dynamic Bayesian network.This approximation of the signaling pathway dynamics has several advantages. First, the discretized nature of the approximation helps to bridge the gap between the accuracy of the results obtained by ODE simulation and the limited precision of experimental data used for model construction and verification. Second and more importantly, many interesting pathway properties can be analyzed efficiently through standard Bayesian inference techniques instead of resorting to a large number of ODE simulations. We have tested our method on ODE models of the EGF-NGF signaling pathway [1] and the segmentation clock pathway [2]. The results are very promising in terms of accuracy and efficiency.     

A second order ODE is sufficient for modelling of many periodic signals

Which is the minimum order an autonomous non-linear ordinary differential equation (ODE) needs to have to be able to model a periodic signal? This question is motivated by recent research on periodic signal analysis, where non-linear ODEs are used as models. The results presented here show that a second order ODE is sufficient for a large class of periodic signals. More precisely, conditions on a periodic signal are established that imply the existence of an ODE that has the periodic signal as a solution. A criterion that characterizes the above class of periodic signals by means of the overtone contents of the signals is also presented. The reason why higher order ODEs are sometimes needed is illustrated with geometric arguments. Extensions of the theoretical analysis to cases with orders higher than two are developed using this insight.     

节点转发消息能力动态估计的缓存替换策略

机会网络中的节点以“存储-携带-转发”的方式完成消息转发,消息需要在中继节点缓存较长时间以等待通信机会,高效的缓存替换策略能够提高有限缓存空间的利用率。提出一种基于节点转发消息能力的自适应缓存替换策略。通过动态地感知当前的消息传输状态,并根据其在此节点中的停留时间和消息转发状态,估计节点对该消息的转发能力,进而调整消息的转发以及删除优先级。结果表明所提出的缓存替换策略能够有效提高消息的成功投递率,并大幅度地降低网络负载率。     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

Computable Error Bounds for Coefficients Perturbation Methods

The coefficients perturbation method (CPM) is a numerical technique for solving ordinary differential equations (ODE) associated with initial or boundary conditions. The basic principle of CPM is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as the conditions associated to it. In this paper we shall develop formulae for calculating tight error bounds for CPM when this technique is applied to second order linear ODEs. Unlike results reported in the literature, ours do not require any a priori information concerning the exact error function or its derivative. The results of this paper apply in particular to the Tau Method and to any approximation procedure equivalent to it. The convergence of the derived bounds is also discussed, and illustrated numerically. Received April 5, 2002; revised June 11, 2002 Published online: December 12, 2002     

A computationally efficient technique for transient analysis of repairable Markovian systems

A technique to design efficient methods using a combination of explicit (non-stiff) and implicit (stiff) ODE methods for numerical transient analysis of repairable Markovian systems is proposed. Repairable systems give rise to stiff Markov chains due to extreme disparity between failure rates and repair rates. Our approach is based on the observation that stiff Markov chains are non-stiff for an initial phase of the solution interval. A non-stiff ODE method is used to solve the model for this phase and a stiff ODE method is used to solve the model for the rest of the duration until the end of solution interval. A formal criterion to determine the length of the non-stiff phase is described. A significant outcome of this approach is that the accuracy requirement automatically becomes a part of model stiffness. Two specific methods based on this approach have been implemented. Both the methods use the Runge-Kutta-Fehlberg method as the non-stiff method. One uses the TR-BDF2 method as the stiff method while the other uses an implicit Runge-Kutta method as the stiff method. Numerical results obtained from solving dependability models of a multiprocessor system and an interconnection network are presented. These results show that the methods obtained using this approach are much more efficient than the corresponding stiff methods which have been proposed to solve stiff Markov models.     

Parallel solution of modular ODEs with application to rolling bearing dynamics

In this paper, we study a model of rolling that yields a system of differential equations with a low degree of dependence between the different bodies and, hence, a Jacobian with a sparse structure. We utilize this specific topology in the integration of the ODEs with a stiff solver. Due to the weak coupling between the different machine elements in this rolling bearing model, the procedure for obtaining a solution can be performed efficiently on parallel computers. In this paper, we study the integrator CVODE, modified to enable a parallel integration of different subsystems; this entails function evaluations, the Jacobian computations and the solution of the linear systems of equations. The benefit of using parallel computers for this problem is verified by several test cases run on MIMD computers.     

Unsupervised kernel least mean square algorithm for solving ordinary differential equations

In this paper a novel method is introduced based on the use of an unsupervised version of kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The algorithm is unsupervised because here no desired signal needs to be determined by user and the output of the model is generated by iterating the algorithm progressively. However, there are several new approaches in literature to solve ODEs but the new approach has more advantages such as simple implementation, fast convergence and also little error. Furthermore, it is also a KLMS with obvious characteristics. In this paper the ability of KLMS is used to estimate the answer of ODE. First a trial solution of ODE is written as a sum of two parts, the first part satisfies the initial condition and the second part is trained using the KLMS algorithm so as the trial solution solves the ODE. The accuracy of the method is illustrated by solving several problems. Also the sensitivity of the convergence is analyzed by changing the step size parameters and kernel functions. Finally, the proposed method is compared with neuro-fuzzy [21] approach.     

New solutions for ordinary differential equations

This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the related higher-order ODE is integrated. Fixing the constant of integration, one then uses existing methods to solve the integrated ODE. By construction, each solution of the integrated ODE yields a solution of the given ODE. Moreover, it is shown when the general solution of an integrated ODE yields either the general solution or a family of particular solutions of the given ODE. As an example, new solutions are obtained for an important class of nonlinear oscillator equations. All solutions presented in this paper cannot be obtained using the current Maple ODE solver.     

Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models

A method is developed that allows the verified integration of ODEs based on local modeling with high-order Taylor polynomials with remainder bound. The use of such Taylor models of order n allows convenient automated verified inclusion of functional dependencies with an accuracy that scales with the (n + 1)-st order of the domain and substantially reduces blow-up.Utilizing Schauder's fixed point theorem on certain suitable compact and convex sets of functions, we show how explicit nth order integrators can be developed that provide verified nth order inclusions of a solution of the ODE. The method can be used not only for the computation of solutions through a single initial condition, but also to establish the functional dependency between initial and final conditions, the so-called flow of the ODE. The latter can be used efficiently for a substantial reduction of the wrapping effect.Examples of the application of the method to conventional initial value problems as well as flows are given. The orders of the integration range up to twelve, and the verified inclusions of up to thirteen digits of accuracy have been demanded and obtained.     

Periodic signal analysis by maximum likelihood modeling of orbits of nonlinear ODEs

This paper treats a new approach to the problem of periodic signal estimation. The idea is to model the periodic signal as a function of the state of a second-order nonlinear ordinary differential equation (ODE). This is motivated by Poincare theory, which is useful for proving the existence of periodic orbits for second-order ODEs. The functions of the right-hand side of the nonlinear ODE are then parameterized by a multivariate polynomial in the states, where each term is multiplied by an unknown parameter. A maximum likelihood algorithm is developed for estimation of the unknown parameters, from the measured periodic signal. The approach is analyzed by derivation and solution of a system of ODEs that describes the evolution of the Cramer-Rao bound over time. This allows the theoretically achievable accuracy of the proposed method to be assessed in the ideal case where the signals can be exactly described by the imposed model. The proposed methodology reduces the number of estimated unknowns, at least in cases where the actual signal generation resembles that of the imposed model. This in turn is expected to result in an improved accuracy of the estimated parameters.     

Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.     

CHEMSODE: a stiff ODE solver for the equations of chemical kinetics

The ODEs describing a chemical kinetics system can be very stiff and are the most computationally costly part of most reactive flow simulations. Research areas ranging from combustion to climate modeling are often limited by their ability to solve these chemical ODE systems both accurately and efficiently. These problems are commonly treated with an implicit numerical method due to the stiffness that is usually present. The implicit solution technique introduces a large amount of computational overhead necessary to solve the nonlinear algebraic system derived from the implicit time-stepping method. In this paper, a code is presented that avoids much of the usual overhead by preconditioning the implicit method with an iterative technique. This results in a class of time-stepping method that is explicit and very stable for chemical kinetics problems.     

Dependability modeling of Software Defined Networking

Software Defined Networking (SDN) is a new network design paradigm that aims at simplifying the implementation of complex networking infrastructures by separating the forwarding functionalities (data plane) from the network logical control (control plane). Network devices are used only for forwarding, while decisions about where data is sent are taken by a logically centralized yet physically distributed component, i.e., the SDN controller. From a quality of service (QoS) point of view, an SDN controller is a complex system whose operation can be highly dependent on a variety of parameters, e.g., its degree of distribution, the corresponding topology, the number of network devices to control, and so on. Dependability aspects are particularly critical in this context. In this work, we present a new analytical modeling technique that allows us to represent an SDN controller whose components are organized in a hierarchical topology, focusing on reliability and availability aspects and overcoming issues and limitations of Markovian models. In particular, our approach allows to capture changes in the operating conditions (e.g., in the number of managed devices) still allowing to represent the underlying phenomena through generally distributed events. The dependability of a use case on a two-layer hierarchical SDN control plane is investigated through the proposed technique providing numerical results to demonstrate the feasibility of the approach.     

A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

A Waveform Relaxation method as applied to a linear system of ODEs is the Picard iteration for a linear Volterra integral equation of the second kind ({\cal I} - {\cal K})y = b \eqno (1) called Waveform Relaxation second kind equation. A corresponding Waveform Relaxation Runge-Kutta method is the Picard iteration for a discretized version ({\cal I} - {\cal K}_l )y_l = b_l \eqno (2) of the integral equation (1), where y l is the continuous solution of the original linear system of ODE provided by the so called limit method. We consider a W-cycle multigrid method, with Picard iteration as smoothing step, for iteratively computing y l . This multigrid method belongs to the class of multigrid methods of the second kind as described in Hackbusch [3, chapter 16]. In the paper we prove that the truncation error after one iteration is of the same order of the discretization error y l @ y of the limit method and the truncation error after two iterations has order larger than the discretization error. Thus we can see the multigrid method as a new numerical method for solving the original linear system of ODE which provides, after one iteration, a continuous solution of the same order of the solution of the limit method, and after two iterations, a solution with asymptotically the same error of the solution of the limit method. On the other hand the computational cost of the multigrid method is considerably smaller than the limit method.     

一种基于节点位置余弦相似度的机会网络转发算法

针对机会网络中数据送达率较低的问题,文中根据节点历史接触信息即节点相遇次数、相遇时间长度、节点关系稳定性来计算节点转发效用值。首先选择通信范围内效用值最大的邻居节点作为初始转发节点,再根据余弦相似度选择其他转发节点,使得转发节点能够尽可能均匀地分布。在此基础上,提出一种基于节点位置余弦相似度的机会网络转发算法(Opportunistic Network forwarding algorithm based on Node Cosine Similarity,ONNCS)。该算法使得转发节点能够均匀地分布,因此数据报文能够尽快地被转发到目的节点。实验结果表明,ONNCS具有较高的转发成功率和较低的转发能耗,转发成功率高出其他算法5%～8%。     

Efficient Broadcasting Using Network Coding and Directional Antennas in MANETs

In this paper, we consider the issue of efficient broadcasting in mobile ad hoc networks (MANETs) using network coding and directional antennas. Network coding-based broadcasting focuses on reducing the number of transmissions each forwarding node performs in the multiple source/multiple message broadcast application, where each forwarding node combines some of the received messages for transmission. With the help of network coding, the total number of transmissions can be reduced compared to broadcasting using the same forwarding nodes without coding. We exploit the usage of directional antennas to network coding-based broadcasting to further reduce energy consumption. A node equipped with directional antennas can divide the omnidirectional transmission range into several sectors and turn some of them on for transmission. In the proposed scheme using a directional antenna, forwarding nodes selected locally only need to transmit broadcast messages, original or coded, to restricted sectors. We also study two extensions. The first extension applies network coding to both dynamic and static forwarding node selection approaches. In the second extension, we design two approaches for the single source/single message issue in the network coding-based broadcast application. Performance analysis via simulations on the proposed algorithms using a custom simulator and ns2 is presented.     

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.     

Effect of node churn on frame interval for peer-to-peer video streaming with data-block synchronization mechanism

Coolstreaming is a mesh based peer-to-peer (P2P) video streaming system in which single video stream is decomposed into multiple sub-streams. A client-peer node retrieves the sub-streams from multiple parent-peer nodes, combining them into the original video stream. Each client-peer node has two buffers, a synchronization buffer and a cache buffer, and arriving data blocks are synchronized at the synchronization buffer and then forwarded to the cache buffer. In this buffering system, data-block synchronization is important to guarantee high video quality. In this paper, we consider the effect of churn on the performance of data-block synchronization scheme with which data blocks are simultaneously forwarded just after all the data blocks composing a macro data block arrive at the synchronization buffer. It is assumed that data blocks belonging to a sub-stream arrive at the client-peer node according to an interrupted Poisson process. The synchronization buffer is modeled as a multiple-buffer queueing system with homogeneous interrupted Poisson processes, and the mean forwarding interval is derived. Numerical examples show that the average forwarding interval increases as parent-peer nodes leave more frequently.     

Loss rates for stochastic fluid models

We introduce loss rates, a novel class of performance measures for Markovian stochastic fluid models and discuss their applications potential. We derive analytical expressions for loss rates and describe efficient methods for their evaluation. Further, we study interesting asymptotic properties of loss rates for large size of the buffer, which are crucial for identifying the Quality of Service requirements guaranteed for each user. We illustrate the theory with a numerical example.