Recently the data-driven learning of dynamic systems has become a promising approach because no physical knowledge is needed. Pure machine learning approaches such as Gaussian process regression (GPR) learns a dynamic model from data, with all physical knowledge about the system discarded. This goes from one extreme, namely methods based on optimizing parametric physical models derived from physical laws, to the other. GPR has high flexibility and is able to model any dynamics as long as they are locally smooth, but can not generalize well to unexplored areas with little or no training data. The analytic physical model derived under assumptions is an abstract approximation of the true system, but has global generalization ability. Hence the optimal learning strategy is to combine GPR with the analytic physical model. This paper proposes a method to learn dynamic systems using GPR with analytic ordinary differential equations (ODEs) as prior information. The one-time-step integration of analytic ODEs is used as the mean function of the Gaussian process prior. The total parameters to be trained include physical parameters of analytic ODEs and parameters of GPR. A novel method is proposed to simultaneously learn all parameters, which is realized by the fully Bayesian GPR and more promising to learn an optimal model. The standard Gaussian process regression, the ODE method and the existing method in the literature are chosen as baselines to verify the benefit of the proposed method. The predictive performance is evaluated by both one-time-step prediction and long-term prediction. By simulation of the cart-pole system, it is demonstrated that the proposed method has better predictive performances.

This paper investigates open-loop Stackelberg games for a class of stochastic systems with multiple players. First, the necessary conditions for the existence of an open-loop Stackelberg strategy set are established using the stochastic maximum principle. Such conditions can be represented as solvability conditions for cross-coupled forward-backward stochastic differential equations (CFBSDEs). Second, in order to obtain the open-loop strategy set, a computational algorithm based on a four-step scheme is developed. A numerical example is then demonstrated to show the validity of the proposed method.

In this paper, a stochastic asymptotic stabilization method is proposed for deterministic input-affine control systems, which are randomized by including Gaussian white noises in control inputs. The sufficient condition is derived for the diffusion coefficients so that there exist stochastic control Lyapunov functions for the systems. To illustrate the usefulness of the sufficient condition, the authors propose the stochastic continuous feedback law, which makes the origin of the Brockett integrator become globally asymptotically stable in probability.

In forthcoming sensor networks, a multitude of sensor nodes deployed over a large geographical area for monitoring traffic, climate, etc. are expected to become an inevitable infrastructure. Clustering algorithms play an important role in aggregating a large volume of data that are produced continuously by the huge number of sensor nodes. In such networks, equal-sized multi-hop clusters which include an equal number of nodes are useful for efficiency and resiliency. In addition, scalability is important in such large-scale networks. In this paper, we mathematically design a decentralized equal-sized clustering algorithm using a partial differential equation based on the Fourier transform technique, and then design its protocol by discretizing the equation. We evaluated through simulations the equality of cluster sizes and the resiliency against packet loss and node failure in two-dimensional perturbed grid topologies.

In the near future, decentralized network systems consisting of a huge number of sensor nodes are expected to play an important role. In such a network, each node should control itself by means of a local interaction algorithm. Although such local interaction algorithms improve system reliability, how to design a local interaction algorithm has become an issue. In this paper, we describe a local interaction algorithm in a partial differential equation (or PDE) and propose a new design method whereby a PDE is derived from the solution we desire. The solution is considered as a pattern of nodes' control values over the network each of which is used to control the node's behavior. As a result, nodes collectively provide network functions such as clustering, collision and congestion avoidance. In this paper, we focus on a periodic pattern comprising sinusoidal waves and derive the PDE whose solution exhibits such a pattern by exploiting the Fourier method.

In this letter, a novel adaptive total variation (ATV) model is proposed for image inpainting. The classical TV model is a partial differential equation (PDE)-based technique. While the TV model can preserve the image edges well, it has some drawbacks, such as staircase effect in the inpainted image and slow convergence rate. By analyzing the diffusion mechanism of TV model and introducing a new edge detection operator named difference curvature, we propose a novel ATV inpainting model. The proposed ATV model can diffuse the image information smoothly and quickly, namely, this model not only eliminates the staircase effect but also accelerates the convergence rate. Experimental results demonstrate the effectiveness of the proposed scheme.

We consider a single-link multi-source network with FAST TCP sources. We adopt a continuous-time dynamic model for FAST TCP sources, and propose a static model to adequately describe the queuing delay dynamics at the link. The proposed model turns out to have a structure that reveals the time-varying network feedback delay, which allows us to analyze FAST TCP with due consideration of the time-varying network feedback delay. Based on the proposed model, we establish sufficient conditions for the boundedness of congestion window of each source and for the global asymptotic stability. The asymptotic stability condition shows that the stability property of each source is affected by all other sources sharing the link. Simulation results illustrate the validity of the sufficient condition for the global asymptotic stability.

This paper proposes a new method to numerically obtain Floquet multipliers which characterize stability of periodic orbits of ordinary differential equations. For sufficiently smooth periodic orbits, we can compute Floquet multipliers using some standard numerical methods with enough accuracy. However, it has been reported that these methods may produce incorrect results under some conditions. In this work, we propose a new iterative method to compute Floquet multipliers using eigenvectors of matrix solutions of the variational equations. Numerical examples show effectiveness of the proposed method.

Given a sparse linear system, A x = b, we can solve the equivalent system P A PT y = P b, x = PT y, where P is a permutation matrix. It has been known that, for example, when P is the RCMK (Reverse Cuthill-Mckee) ordering permutation, the convergence rate of the Krylov subspace method combined with the ILU-type preconditioner is often enhanced, especially if the matrix A is highly nonsymmetric. In this paper we offer a reordering heuristic for accelerating the solution of large sparse linear systems by the Krylov subspace methods with the ILUT preconditioner. It is the LRB (Line Red/Black) ordering based on the well-known 2-point Red-Black ordering. We show that for some model-like PDE (partial differential equation)s the LRB ordered FDM (Finite Difference Method)/FEM (Finite Element Method) discretization matrices require much less fill-ins in the ILUT factorizations than those of the Natural ordering and the RCMK ordering and hence, produces a more accurate preconditioner, if a high level of fill-in is used. It implies that the LRB ordering could outperform the other two orderings combined with the ILUT preconditioned Krylov subspace method if the level of fill-in is high enough. We compare the performance of our heuristic with that of the RCMK (Reverse Cuthill-McKee) ordering. Our test matrices are obtained from various standard discretizations of two-dimensional and three-dimensional model-like PDEs on structured grids by the FDM or the FEM. We claim that for the resulting matrices the performance of our reordering strategy for the Krylov subspace method combined with the ILUT preconditioner is superior to that of RCMK ordering, when the proper number of fill-in was used for the ILUT. Also, while the RCMK ordering is known to have little advantage over the Natural ordering in the case of symmetric matrices, the LRB ordering still can improve the convergence rate, even if the matrices are symmetric.

In this paper we compare various parallel preconditioners such as Point-SSOR (Symmetric Successive OverRelaxation), ILU(0) (Incomplete LU) in the Wavefront ordering, ILU(0) in the Multi-color ordering, Multi-Color Block SOR (Successive OverRelaxation), SPAI (SParse Approximate Inverse) and pARMS (Parallel Algebraic Recursive Multilevel Solver) for solving large sparse linear systems arising from two-dimensional PDE (Partial Differential Equation)s on structured grids. Point-SSOR is well-known, and ILU(0) is one of the most popular preconditioner, but it is inherently serial. ILU(0) in the Wavefront ordering maximizes the parallelism in the natural order, but the lengths of the wavefronts are often nonuniform. ILU(0) in the Multi-color ordering is a simple way of achieving a parallelism of the order N, where N is the order of the matrix, but its convergence rate often deteriorates as compared to that of natural ordering. We have chosen the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver, since for the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with the Multi-Color ordering. By using block version we expect to minimize the interprocessor communications. SPAI computes the sparse approximate inverse directly by least squares method. Finally, ARMS is a preconditioner recursively exploiting the concept of independent sets and pARMS is the parallel version of ARMS. Experiments were conducted for the Finite Difference and Finite Element discretizations of five two-dimensional PDEs with large meshsizes up to a million on an IBM p595 machine with distributed memory. Our matrices are real positive, i.e., their real parts of the eigenvalues are positive. We have used GMRES(m) as our outer iterative method, so that the convergence of GMRES(m) for our test matrices are mathematically guaranteed. Interprocessor communications were done using MPI (Message Passing Interface) primitives. The results show that in general ILU(0) in the Multi-Color ordering and ILU(0) in the Wavefront ordering outperform the other methods but for symmetric and nearly symmetric 5-point matrices Multi-Color Block SOR gives the best performance, except for a few cases with a small number of processors.

This paper considers numerical methods for stability analyses of periodic solutions of ordinary differential equations. Stability of a periodic solution can be determined by the corresponding monodromy matrix and its eigenvalues. Some commonly used numerical methods can produce inaccurate results of them in some cases, for example, near bifurcation points or when one of the eigenvalues is very large or very small. This work proposes a numerical method using a periodic boundary condition for vector fields, which preserves a critical property of the monodromy matrix. Numerical examples demonstrate effectiveness and a drawback of this method.

There is increasing evidence from in vivo recordings in monkeys trained to respond to stimuli by making left- or rightward eye movements, that firing rates in certain groups of neurons in oculo-motor areas mimic drift-diffusion processes, rising to a (fixed) threshold prior to movement initiation. This supplements earlier observations of psychologists, that human reaction-time and error-rate data can be fitted by random walk and diffusion models, and has renewed interest in optimal decision-making ideas from information theory and statistical decision theory as a clue to neural mechanisms. We review results from decision theory and stochastic ordinary differential equations, and show how they may be extended and applied to derive explicit parameter dependencies in optimal performance that may be tested on human and animal subjects. We then briefly describe a biophysically-based model of a pool of neurons in locus coeruleus, a brainstem nucleus implicated in widespread norepinephrine release. This neurotransmitter can effect transient gain changes in cortical circuits of the type that the abstract drift-diffusion analysis requires. We also describe how optimal gain schedules can be computed in the presence of time-varying noisy signals. We argue that a rational account of how neural spikes give rise to simple behaviors is beginning to emerge.

Computer simulation of cellular process is one of the most important applications in bioinformatics. Since such simulators need huge computational resources, many biologists must use expensive PC/WS clusters. ReCSiP is an FPGA-based, reconfigurable accelerator which aims to realize economical high-performance simulation environment on desktop computers. It can exploit fine-grain parallelism in the target applications by small hardware modules in the FPGA which work in parallel manner. As the first step to implement a simulator of cellular process on ReCSiP, a solver to perform a basic simulation of metabolism was implemented. The throughput of the solver was about 29 times faster than the software on Intel's PentiumIII operating at 1.13 GHz.

Fractional calculus is the generalization of the operators of differential and integration to non-integer order, and a differential equation involving the fractional calculus operators such as d1/2/dt1/2 and d-1/2/dt-1/2 is called the fractional differential equation. They have many applications in science and engineering. But not only its analytical solutions exist only for a limited number of cases, but also, the numerical methods are difficult to solve. In this paper we propose a new numerical method based on the operational matrices of the orthogonal functions for solving the fractional calculus and fractional differential equations. Two classical fractional differential equation examples are included for demonstration. They show that the new approach is simper and more feasible than conventional methods. Advantages of the proposed method include (1) the computation is simple and computer oriented; (2) the scope of application is wide; and (3) the numerically unstable problem never occurs in our method.

With the emerging technology of photonic networks, careful design becomes necessary to make most of the already installed fibre capacity. Appropriate numerical tools are readily available. Usually, these are based on the split-step Fourier method (SSFM), employing the fast Fourier transform (FFT). With N discretization points, the complexity of the SSFM is O(N log2N). For real-world wavelength division multiplexing (WDM) systems, the simulation time can be of the order of days, so any speed improvement would be most welcome. We show that the SSFM is a special case of the so-called collocation method with harmonic basis functions. However, for modelling nonlinear optical waveguides, various other basis function systems offer significant advantages. For calculating the propagation of single soliton-like impulses, a problem-adapted Gauss-Hermite basis leads to a strongly reduced computation time compared to the SSFM . Further, using a basis function system constructed from a scaling function, which generates a compactly supported wavelet, we developed a new and flexible split-step wavelet collocation method (SSWCM). This technique is independent of the propagating impulse shapes, and provides a complexity of the order O(N) for a fixed accuracy. For a typical modelling situation with up to 64 WDM channels, the SSWCM leads to significantly shorter computation times than the standard SSFM.

In this paper, we present a numerical method with guaranteed accuracy to solve initial value problems (IVPs) of normal form simultaneous first order ordinary differential equations (ODEs) which have wide domain. Our method is based on the algorithm proposed by Kashiwagi, by which we can obtain inclusions of exact values at several discrete points of the solution curve of ODEs. The method can be regarded as an extension of the Lohner's method. But the algorithm is not efficient for equations which have wide domain, because the error bounds become too wide from a practical point of view. Our purpose is to produce tight bounds even for such equations. We realize it by combining Kashiwagi's algorithm with the mean value form. We also consider the wrapping effects to obtain tighter bounds.

We consider a dynamically reconfigureable network where dynamically changing traffic is offered. Rearrangement and adjustment of network capacity can be performed to maintain Quality of Service (QoS) requirements for different traffic classes in the dynamic traffic environment. In this work, we consider the case of a single, dynamic traffic class scenario in a loss mode environment. We have developed a numerical, analytical tool which models the dynamically changing network traffic environment using a time-varying, fluid-flow, differential equation; of which we can use to study the impact of adaptive capacity adjustment control schemes. We present several capacity adjustment control schemes including schemes which use blocking and system utilization as means to calculate when and how much adjustment should be made. Through numerical studies, we show that a purely blocking-based capacity adjustment control scheme with a preset adjustment value can be very sensitive to capacity changes and can lead to network instability. We also show that schemes, that uses system utilization as a means to calculate the amount of capacity adjustment needed, is consistently stable for the load scenarios considered. Finally, we introduce a minimum time interval threshold between adjustments, which can avoid network instability, in the cases where the results showed that capacity adjustment had been performed too often.

We analyze the dynamics of self-organizing cortical maps under the influence of external stimuli. We show that if the map is a contraction, then the system has a unique equilibrium which is globally asymptotically stable; consequently the system acts as a stable encoder of external input stimuli. The system converges to a fixed point representing the steady-state of the neural activity which has as an upper bound the superposition of the spatial integrals of the weight function between neighboring neurons and the stimulus autocorrelation function. The proposed theory also includes nontrivial interesting solutions.

This work presents a further development of the approach to modelling thermal (i.e. carrier-velocity-fluctuation) noise in semiconductor devices proposed in papers by the present authors. The basic idea of the approach is to apply classical theory of Ito's stochastic differential equations (SDEs) and stochastic diffusion processes to describe noise in devices and circuits. This innovative combination enables to form consistent mathematical basis of the noise research and involve a great variety of results and methods of the well-known mathematical theory in device/circuit design. The above combination also makes our approach completely different, on the one hand, from standard engineering formulae which are not associated with any consistent mathematical modelling and, on the other hand, from the treatments in theoretical physics which are not aimed at device/circuit models and design. (Both these directions are discussed in more detail in Sect. 1). The present work considers the bipolar transistor compact model derived in Ref. [2] according to theory of Ito's SDEs and stochastic diffusion processes (including celebrated Kolmogorov's equations). It is shown that the compact model is transformed into the Ito SDE system. An iterative method to determine noisy currents as entries of the stationary stochastic process corresponding to the above Ito system is proposed.

In this paper, we propose a plausible software reliability growth model by applying a mathematical technique of stochastic differential equations. First, we extend a basic differential equation describing the average behavior of software fault-detection processes during the testing phase to a stochastic differential equation of ItÔ type, and derive a probability distribution of its solution processes. Second, we obtain several software reliability measures from the probability distribution. Finally, applying a method of maximum-likelihood we estimate unknown parameters in our model by using available data in the actual software testing procedures, and numerically show the stochastic behavior of the number of faults remaining in the software system. Further, the model is compared among the existing software reliability growth models in terms of goodness-of-fit.