Fast convergence pirkn-type pc methods with adams-type predictors ∗

This paper discusses predictor–corrector iteration schemes (PC iteration schemes) based on direct collocatio–based Runge–Kutt–Nyström corrector methods (RKN corrector methods) for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y′′(t) = f(y(t)) Our approach is to regard the well-known parallel-iterated RKN methods (PIRKN methods) as PC iteration processes in which the simple, low-order last step value predictors are replaced with the high-order Adams-type predictors. Moreover, the param-eters of the new direct collocation-based RKN corrector methods are chosen in such a way that the convergence rate of the considered PC iteration processes is optimized. In this way, we obtain parallel PC methods with fast convergence and high-accurate predictions. Application of the resulting parallel PC methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the parallel and sequential explicit RKN methods from the literature.     

Fast L1–L2 Minimization via a Proximal Operator

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)–\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)–\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)–\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)–\(L_2\) based on a difference-of-convex approach in the numerical experiments.     

Some variants of the Chebyshev–Halley family of methods with fifth order of convergence

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards.

The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation.     

Multi-level Monte Carlo methods with the truncated Euler–Maruyama scheme for stochastic differential equations

The truncated Euler–Maruyama method is employed together with the Multi-level Monte Carlo method to approximate expectations of some functions of solutions to stochastic differential equations (SDEs). The convergence rate and the computational cost of the approximations are proved, when the coefficients of SDEs satisfy the local Lipschitz and Khasminskii-type conditions. Numerical examples are provided to demonstrate the theoretical results.     

Stability and convergence analysis of implicit–explicit one-leg methods for stiff delay differential equations

The purpose of this paper is devoted to studying the implicit–explicit (IMEX) one-leg methods for stiff delay differential equations (DDEs) which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods.     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Reliable H ∞ filter design for a class of mixed-delay Markovian jump systems with stochastic nonlinearities and multiplicative noises via delay-partitioning method

This paper is concerned with the problem of reliable H _?? filter design for a class of mixeddelay Markovian jump systems with stochastic nonlinearities and multiplicative noises. The mixed delays comprise both discrete time-varying delay and distributed delay. The stochastic nonlinearities in the form of statistical means cover several well-studied nonlinear functions. And the multiplicative disturbances are in the form of a scalar Gaussian white noise with unit variance. Furthermore, the failures of sensors are quantified by a variable varying in a given interval. A filter is designed to guarantee that the dynamics of the estimation error is asymptotically mean-square stable. Sufficient conditions for the existence of such a filter are obtained by using a new Lyapunov-Krasovskii functional and delaypartitioning method. Then a linear matrix inequality (LMI) approach for designing such a reliable H _?? filter is presented. Finally, the effectiveness of the proposed approach is demonstrated by a numerical example.     

Solving the multi-compartment capacitated location routing problem with pickup–delivery routes and stochastic demands

This paper considers an advanced capacitated location routing problem in a distribution network with multiple pickup and delivery routes, and each customer placing random multi-item demands on it. The pickup and delivery services need two fleets of vehicles and will form two different sets of routes. However, the unpredictability of variation in the multi-item demands makes the routing of multi-compartment vehicles to accommodate such demands complex. To solve this multifaceted problem, a new process employing the TABU search is proposed in this research study. This proposed approach includes three stages: location selection, customer assignment, and vehicle routing. The innovative concept is to divide all customers into assignment-determined and assignment-undetermined groups in order to narrow down the search area of a solution domain so the TABU search can be more efficient and effective. Two sets of benchmarks are then generated to verify the quality of the proposed method. According to the experiment results, the proposed solution process can both resolve the problems and yield good results in a reasonable amount of computing time. The analysis of the solution process parameters is also provided. In addition, the comparisons between stochastic demand and deterministic demand cases are calculated and discussed as well.     

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from \({\mathcal {O}}(N^2)\) to \({\mathcal {O}}(N)\) and the computational complexity from \({\mathcal {O}}(N^3)\) to \({\mathcal {O}}(N\log N)\) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.     

An adaptive control architecture for leader–follower multiagent systems with stochastic disturbances and sensor and actuator attacks

In this paper, we develop a novel distributed adaptive control architecture for addressing networked multiagent systems subject to stochastic exogenous disturbances with compromised sensor and actuators. Specifically, for a class of linear leader–follower multiagent systems, we develop a new structure of the neighbourhood synchronisation error for the control design protocol of each follower. The proposed control algorithm addresses time-varying multiplicative sensor attacks on the leader state measurements. In addition, the framework addresses time-varying multiplicative actuator attacks on the followers that do not have a communication link with the leader and additive actuator attacks on all follower agents in the network. The proposed adaptive controller guarantees uniform ultimate boundedness of the state tracking error for each agent in a mean-square sense.     

A manufacturer–buyer integrated inventory model with stochastic lead times for delivering equal- and/or unequal-sized batches of a lot

Although the subject of manufacturer–buyer integrated inventory management with deterministic lead times has received a lot of attention from researchers, the corresponding problem with stochastic lead times has been given comparatively little consideration. Recently, it has been treated in the case of an exponential distribution of lead times with the lot transferred in equal-sized batches (sub-lots). In this treatment the buyer orders the next batch when his/her stock level falls to a certain reorder point, allowing for shortages and complete backordering. The total cost benefit of solving the problem using an integrated inventory system instead of independent ones had been demonstrated. However, rather than an exponential distribution, a normal distribution of lead times seems to provide a better fit to the problem. Moreover, synchronization of the integrated production flow by generalizing the method of transferring batches of the lot might lead to a lower total cost. Based on these notions, we develop here a manufacturer–buyer integrated inventory model with a normal distribution of lead times for delivering equal- and/or unequal-sized batches of a lot. Then a solution technique to the model and hence a solution algorithm are presented. The potential benefit of the present method is illustrated with solutions of some numerical problems. The sensitivities of the solutions to variations in the parameter values are also studied.     

Stability and performance analysis of linear positive systems with delays using input–output methods

It is known that input–output approaches based on scaled small-gain theorems with constant D-scalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input–output approaches provide, in general, sufficient conditions only. Using these results, we provide simple alternative proofs for many of the existing results on the stability of linear positive systems with discrete/distributed/neutral time-invariant/-varying delays and linear difference equations. In particular, we give a simple proof for the characterisation of diagonal Riccati stability for systems with discrete-delays and generalise this equation to other types of delay systems. The fact that all those results can be reproved in a very simple way demonstrates the importance and the efficiency of the input–output framework for the analysis of linear positive systems. The approach is also used to derive performance results evaluated in terms of the L ₁-, L ₂- and L _∞-gains. It is also flexible enough to be used for design purposes.     

Maximum power tracking of a generic photovoltaic system via a fuzzy controller and a two-stage DC–DC converter

This study presents a new two-stage DC–DC converter for maximum power point tracking (MPPT) and a voltage boost of a generic photovoltaic (PV) system. An intelligent MPPT of PV system based on fuzzy logic control (FLC) is presented to adaptively design the proposed fuzzy controlled MPPT controller (FC-MPPTC) while a voltage boost controller (VBC) is used to fix the output voltage to a voltage level that is higher than the required operating voltage to the back-end grid impedance. Modeling and simulation on the PV system and the DC–DC converter circuit are achieved by state-space and the software Powersim. The PV string considered has the rated power around 600?VA under varied partial shadings. The FC-MPPTC and VBC are designed and realized by a DSP module (TMS320F2812) to adjust the duty cycle in the two-stage DC–DC converter. A special FLC algorithm is forged to render an MPPT faster and more accurate than conventional MPPT technique, perturb and observe (P&O). The simulations are intended to validate the performance of the proposed FC-MPPTC. Experiments are conducted and results show that MPPT can be achieved in a fast pace and the efficiency reaches over 90?%, even up to 96?%. It is also found that the optimized tracking speed of the proposed FC-MPPTC is in fact more stable and faster than the general P&O method with the boost voltage capable of offering a stable DC output.     

Exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction–diffusion terms

The problem of mean square exponential stability for a class of impulsive stochastic fuzzy cellular neural networks with distributed delays and reaction–diffusion terms is investigated in this paper. By using the properties of M-cone, eigenspace of the spectral radius of nonnegative matrices, Lyapunov functional, Itô’s formula and inequality techniques, several new sufficient conditions guaranteeing the mean square exponential stability of its equilibrium solution are obtained. The derived results are less conservative than the results recently presented in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008). In fact, the systems discussed in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008) are special cases of ours. Two examples are presented to illustrate the effectiveness and efficiency of the results.     

Investing in lead-time variability reduction in a collaborative vendor–buyer supply chain model with stochastic lead time

This study deals with investing in lead-time variability reduction problems for the integrated vendor–buyer supply chain system with partial backlogging under stochastic lead time. We consider that lead time variability can be reduced through further investment; more specifically, a logarithmic investment function is used that allows investment to be made to reduce lead-time variability. By using the proposed supply chain model, considerable savings can be achieved to increase the competitive edge. The objective is to derive the optimal production/ordering strategy, and the best investment policy to minimize joint total cost. A computer code using the software, Mathematica, is developed to derive the optimal solution. Furthermore, we discuss the sensitivity of the optimal solution together with the changes of the values of the parameters associated with the model for decision-making. Various numerical examples are given to illustrate the results.     

Stability and bifurcation of a reaction–diffusion predator–prey model with non-local delay and Michaelis–Menten-type prey-harvesting

We investigate a reaction–diffusion predator–prey system with homogeneous Neumann boundary conditions and non-local delay due to predator gestation. By analysing the corresponding characteristic equations, we establish the local stability of the positive steady state. We also discuss the existence of Hopf bifurcations at the positive steady state. We derive sufficient conditions for the global stability of the positive steady state of the proposed problem using the Lyapunov functional. Numerical simulations illustrate the results and reveal that as the discrete delay τ increases, the species may tend to extinction. Changes in the harvesting effort E and non-local delay β can transform an unstable system into a stable one.     

Combining simulated annealing with Lagrangian relaxation and weighted Dantzig–Wolfe decomposition for integrated design decisions in wireless sensor networks

A Wireless Sensor Network (WSN) is the outcome of the collaborative effort of multi-functional, low-power, low-cost, tiny electronic devices called sensors. Their ability to work autonomously provides a distributed environment capable to monitor even remote or inaccessible areas, which explains the wide application range of WSNs. There are four main issues in the design of a WSN: determining sensor locations (deployment), scheduling sensors, finding sink locations, and obtaining sensor-to-sink data routes. Sensors have very limited energy resources and their efficient management becomes critical for elongating network lifetime. As a result, most of the works on optimal WSN design are concerned with efficient energy usage. Unfortunately, only a few of them use an integrated approach and try to address these four issues simultaneously. In this work we also follow this line of research and develop first a monolithic mixed-integer linear programming model that maximizes network lifetime by optimally determining sensor and sink locations, sensor-to-sink data flows, active and stand-by periods of the sensors subject to data flow conservation, energy consumption and budget usage constraints. Then we propose a nested solution method consisting of two procedures: simulated annealing that performs search for the best sink locations in the outer level and Lagrangian relaxation based heuristic employed with weighted Dantzig–Wolfe decomposition for the multiplier update in the inner level, which determines sensor locations, activity schedules of the sensors and data flows routes. We demonstrate the efficiency and accuracy of the new approach on randomly generated instances by extensive numerical experiments.