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     

Runge-Kutta neural network for identification of dynamical systemsin high accuracy

This paper proposes Runge-Kutta neural networks (RKNNs) for identification of unknown dynamical systems described by ordinary differential equations (i.e., ordinary differential equation or ODE systems) with high accuracy. These networks are constructed according to the Runge-Kutta approximation method. The main attraction of the RKNNs is that they precisely estimate the changing rates of system states (i.e., the right-hand side of the ODE x =f(x)) directly in their subnetworks based on the space-domain interpolation within one sampling interval such that they can do long-term prediction of system state trajectories. We show theoretically the superior generalization and long-term prediction capability of the RKNNs over the normal neural networks. Two types of learning algorithms are investigated for the RKNNs, gradient-and nonlinear recursive least-squares-based algorithms. Convergence analysis of the learning algorithms is done theoretically. Computer simulations demonstrate the proved properties of the RKNNs.     

L1 adaptive control for general partial differential equation (PDE) systems

This paper addresses the L₁ adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.     

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.     

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.     

Deterministic learning of completely resonant nonlinear wave systems with Dirichlet boundary conditions

In this paper, we investigate the approximation of completely resonant nonlinear wave systems via deterministic learning. The plants are distributed parameter systems (DPS) describing homogeneous and isotropic elastic vibrating strings with fixed endpoints. The purpose of the paper is to approximate the infinite-dimensional dynamics, rather than the parameters of the wave systems. To solve the problem, the wave systems are first transformed into finite-dimensional dynamical systems described by ordinary differential equation (ODE). The properties of the finite-dimensional systems, including the convergence of the solution, as well as the dominance of partial system dynamics according to point-wise measurements, are analyzed. Based on the properties, second, by using the deterministic learning algorithm, an approximately accurate neural network (NN) approximation of the the finite-dimensional system dynamics is achieved in a local region along the recurrent trajectories. Simulation studies are included to demonstrate the effectiveness of the proposed approach.     

Control of a 2 × 2 coupled linear hyperbolic system sandwiched between 2 ODEs

Motivated by an engineering application in cable mining elevators, we address a new problem on stabilization of 2×2 coupled linear first‐order hyperbolic PDEs sandwiched between 2 ODEs. A novel methology combining PDE backstepping and ODE backstepping is proposed to derive a state‐feedback controller without high differential terms. The well‐posedness and invertibility properties of the PDE backstepping transformation are proved. All states, including coupled linear hyperbolic PDEs and 2 ODEs, are included in the closed‐loop exponential stability analysis. Moreover, boundedness and exponential convergence of the designed controller are proved. The performance is investigated via numerical simulation.     

A Maple package to find first order differential invariants of 2ODEs via a Darboux approach

Here we present an implementation of a semi-algorithm to find elementary first order differential invariants (elementary first integrals) of a class of rational second order ordinary differential equations (rational 2ODEs). The algorithm was developed in Duarte and da Mota (2009)  [18]; it is based on a Darboux-type procedure, and it is an attempt to construct an analog (generalization) of the method built by Prelle and Singer (1983)  [6] for rational first order ordinary differential equations (rational 1ODEs). to deal, this time, with 2ODEs. The FiOrDi package presents a set of software routines in Maple for dealing with rational 2ODEs. The package presents commands permitting research investigations of some algebraic properties of the ODE that is being studied.     

Performance modeling of epidemic routing

In this paper, we develop a rigorous, unified framework based on ordinary differential equations (ODEs) to study epidemic routing and its variations. These ODEs can be derived as limits of Markovian models under a natural scaling as the number of nodes increases. While an analytical study of Markovian models is quite complex and numerical solution impractical for large networks, the corresponding ODE models yield closed-form expressions for several performance metrics of interest, and a numerical solution complexity that does not increase with the number of nodes. Using this ODE approach, we investigate how resources such as buffer space and the number of copies made for a packet can be traded for faster delivery, illustrating the differences among various forwarding and recovery schemes considered. We perform model validations through simulation studies. Finally we consider the effect of buffer management by complementing the forwarding models with Markovian and fluid buffer models.     

Synchronization of diffusively-coupled limit cycle oscillators

We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction–diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.     

小数据集条件下基于双重约束的BN参数学习

针对小数据集条件下的贝叶斯网络（Bayesian network,BN）参数学习问题,提出了一种基于双重约束的贝叶斯网络参数学习方法. 首先,对网络中的参数进行分析并将网络中的参数划分为: 父节点组合状态相同而子节点状态不同的参数和父节点组合状态不同而子节点状态相同的参数;然后,针对第一类参数提出了一种新的基于Beta分布拟合的贝叶斯估计方法,而针对第二类参数利用已有的保序回归估计方法进行学习,进而实现了对网络中参数的双重约束学习;最后,通过仿真实例说明了基于双重约束的参数学习方法对小数据集条件下贝叶斯网络参数学习精度提高的有效性.     

Dynamic approximation of spatiotemporal receptive fields in nonlinear neural field models

This article presents an approximation method to reduce the spatiotemporal behavior of localized activation peaks (also called "bumps") in non-linear neural field equations to a set of coupled ordinary differential equations (ODEs) for only the amplitudes and tuning widths of these peaks. This enables a simplified analysis of steady-state receptive fields and their stability, as well as spatiotemporal point spread functions and dynamic tuning properties. A lowest-order approximation for peak amplitudes alone shows that much of the well-studied behavior of small neural systems (e.g., the Wilson-Cowan oscillator) should carry over to localized solutions in neural fields. Full spatiotemporal response profiles can further be reconstructed from this low-dimensional approximation. The method is applied to two standard neural field models: a one-layer model with difference-of-gaussians connectivity kernel and a two-layer excitatory-inhibitory network. Similar models have been previously employed in numerical studies addressing orientation tuning of cortical simple cells. Explicit formulas for tuning properties, instabilities, and oscillation frequencies are given, and exemplary spatiotemporal response functions, reconstructed from the low-dimensional approximation, are compared with full network simulations.     

Computational techniques for hybrid system verification

This paper concerns computational methods for verifying properties of polyhedral invariant hybrid automata (PIHA), which are hybrid automata with discrete transitions governed by polyhedral guards. To verify properties of the state trajectories for PIHA, the planar switching surfaces are partitioned to define a finite set of discrete states in an approximate quotient transition system (AQTS). State transitions in the AQTS are determined by the reachable states, or flow pipes, emitting from the switching surfaces according to the continuous dynamics. This paper presents a method for computing polyhedral approximations to flow pipes. It is shown that the flow-pipe approximation error can be made arbitrarily small for general nonlinear dynamics and that the computations can be made more efficient for affine systems. The paper also describes CheckMate, a MATLAB-based tool for modeling, simulating and verifying properties of hybrid systems based on the computational methods previously described.     

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.     

Set-Theoretic Estimation of Hybrid System Configurations

Hybrid systems serve as a powerful modeling paradigm for representing complex continuous controlled systems that exhibit discrete switches in their dynamics. The system and the models of the system are nondeterministic due to operation in uncertain environment. Bayesian belief update approaches to stochastic hybrid system state estimation face a blow up in the number of state estimates. Therefore, most popular techniques try to maintain an approximation of the true belief state by either sampling or maintaining a limited number of trajectories. These limitations can be avoided by using bounded intervals to represent the state uncertainty. This alternative leads to splitting the continuous state space into a finite set of possibly overlapping geometrical regions that together with the system modes form configurations of the hybrid system. As a consequence, the true system state can be captured by a finite number of hybrid configurations. A set of dedicated algorithms that can efficiently compute these configurations is detailed. Results are presented on two systems of the hybrid system literature.      

Reduced Modeling of Unknown Trajectories

This paper deals with model order reduction of parametrical dynamical systems. We consider the specific setup where the distribution of the system’s trajectories is unknown but the following two sources of information are available: (i) some “rough” prior knowledge on the system’s realisations; (ii) a set of “incomplete” observations of the system’s trajectories. We propose a Bayesian methodological framework to build reduced-order models (ROMs) by exploiting these two sources of information. We emphasise that complementing the prior knowledge with the collected data provably enhances the knowledge of the distribution of the system’s trajectories. We then propose an implementation of the proposed methodology based on Monte-Carlo methods. In this context, we show that standard ROM learning techniques, such e.g., proper orthogonal decomposition or dynamic mode decomposition, can be revisited and recast within the probabilistic framework considered in this paper. We illustrate the performance of the proposed approach by numerical results obtained for a standard geophysical model.     

Performance optimisation strategies for automatically generated FPGA accelerators for biomedical models

Biomedical modelling that is mathematically described by ordinary differential equations (ODEs) is often one of the most computationally intensive parts of simulations. With high inherent parallelism, hardware acceleration based on field programmable gate array has great potential to increase the computational performance of the ODE model integration while being very power efficient. ODE‐based Domain‐specific Synthesis Tool is a tool we proposed previously to automatically generate the complete hardware/software co‐design framework for computing biomedical models based on CellML. Although it provides remarkable performance improvement and high energy efficiency compared with CPUs and GPUs, there is still a great potential for optimisation. In this paper, we investigate a set of optimisation strategies including compiler optimisation, resource fitting and balancing, and multiple pipelines. They all have in common that they can be performed automatically and hence can be integrated in our domain‐specific high level synthesis tool. We evaluate the optimised hardware accelerator modules generated by ODE‐based Domain‐specific Synthesis Tool on real hardware based on their resource usage, processing speed and power consumption. The results are compared with single threaded and multi‐core CPUs with/without Streaming SIMD Extension (SSE) optimisation and a graphics card. The results show that the proposed optimisation strategies provide significant performance improvement and result in even more energy‐efficient hardware accelerator modules. Furthermore, the resources of the target field programmable gate array device can be more efficiently utilised in order to fit larger biomedical models than before. Copyright © 2015 John Wiley & Sons, Ltd.     

A preconditioning strategy for microwave susceptibility in ferromagnets

3D numerical simulations of ferromagnetic materials can be compared with experimental results via microwave susceptibility. In this paper, an optimised computation of this microwave susceptibility for large meshes is proposed. The microwave susceptibility is obtained by linearisation of the Landau and Lifchitz equations near equilibrium states and the linear systems to be solved are very ill-conditioned. Solutions are computed using the conjugate gradient method for the normal equation (CGN Method). An efficient preconditioner is developed consisting of a projection and an approximation of an “exact” preconditioner in the set of circulant matrices. Control of the condition number due to the preconditioning and evolution of the singular value decomposition are shown in the results.     

A reduced-order framework applied to linear systems withconstrained controls

A major issue in the control of dynamical systems is the integration of both technological constraints and some dynamic performance requirements in the design of the control system. The authors show in this work that it is possible to solve a class of constrained control problems of linear systems by using a reduced-order system obtained by the projection of the trajectories of the original system onto a subspace associated with the undesirable open-loop eigenvalues. The class of regulation schemes considered uses full state feedback to guarantee that any trajectory emanating from a given polyhedral set of admissible initial states remains in that set. This set of admissible states is said to be positively invariant with respect to the closed-loop system. The authors also address the important issues of numerical stability and complexity of the computations     

Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.