Accelerating Krawczyk-like interval algorithms for the solution of nonlinear systems of equations by using second derivatives

Recently the properties of Krawczyk-like iterative interval methods for the solution of systems of nonlinear equations have been discussed in several papers (e.g. [2], [3], [5], [6]). These methods converge to a solution under relatively weak conditions provided an initial inclusion vector is known. In the present paper we describe a method that improves the convergence speed for an important class of problems by using second partial derivatives. This method is particularly interesting for large systems with a Jacobi matrix whose off-diagonal coefficients are all constant.     

Decay to spatially homogeneous states for the numerical solution of reaction-diffusion systems

The aim of this paper is to discuss the behaviour of the numerical solution of systems of nonlinear reaction-diffusion equations with homogeneous Neumann boundary conditions. We show that, under certain conditions, the solution obtained using known finite difference methods reproduces the behaviour of the exact solution. In particular we prove that the numerical solution decays as time increases to a spatially homogeneous vector, which is a suitably «weighted» mean value of the numerical solution itself.     

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

A new approach to solving systems of linear interval equations based on the generalized procedure of interval extension is proposed. This procedure is based on the treatment of interval zero as an interval centered around zero, and for this reason it is called the “interval extended zero” method. Since the “interval extended zero” method provides a fuzzy solution to interval equations, its interval representations are proposed. It is shown that they may be naturally treated as modified operations of interval division. These operations are used for the modified interval extensions of known numerical methods for solving systems of linear equations and finally for solving systems of linear interval equations. Using a well known example, it is shown that the solution obtained by the proposed method can be treated as an inner interval approximation of the united solution and an outer interval approximation of the tolerable solution, and lies within the range of possible AE-solutions between the extreme tolerable and united solutions. Generally, we can say that the proposed method provides the results which can be treated as approximate formal solutions sometimes referred to as algebraic solutions. Seven known examples are used to illustrate the method’s efficacy and advantages in comparison with known methods providing formal (algebraic) solutions to systems of linear interval equations. It is shown that a new method provides results which are close to the so-called maximal inner solutions (the corresponding method was developed by Kupriyanova, Zyuzin and Markov) and the algebraic solutions obtained by the subdifferential Newton method proposed by Shary. It is important that the proposed method makes it possible to avoid inverted interval solutions. The influence of the system’s size and number of zero entries on the results is analyzed by applying the proposed method to the Leontief input–output model of economics.     

Continuation methods for solving modified discrete-time algebraicRiccati equations

It is well known that the continuation methods have been successfully applied to solve polynomial systems and fixed point problems, etc. In this paper, we consider a discrete-time algebraic Riccati equation with an admissible, low rank, and symmetric perturbation. Our attention is directed primarily to this modified discrete-time algebraic Riccati equation and the numerical method for its solution based on proceeding along the continuation path     

Optimal control of Formula One car energy recovery systems

The utility of orthogonal collocation methods in the solution of optimal control problems relating to Formula One racing is demonstrated. These methods can be used to optimise driver controls such as the steering, braking and throttle usage, and to optimise vehicle parameters such as the aerodynamic down force and mass distributions. Of particular interest is the optimal usage of energy recovery systems (ERSs). Contemporary kinetic energy recovery systems are studied and compared with future hybrid kinetic and thermal/heat ERSs known as ERS-K and ERS-H, respectively. It is demonstrated that these systems, when properly controlled, can produce contemporary lap time using approximately two-thirds of the fuel required by earlier generation (2013 and prior) vehicles.     

A comparative study of awareness methods for peer‐to‐peer distributed virtual environments

The increasing popularity of multi‐player online games is leading to the widespread use of large‐scale Distributed Virtual Environments (DVEs) nowadays. In these systems, peer‐to‐peer (P2P) architectures have been proposed as an efficient and scalable solution for supporting massively multi‐player applications. However, the main challenge for P2P architectures consists of providing each avatar with updated information about which other avatars are its neighbors. This problem is known as the awareness problem. In this paper, we propose a comparative study of the performance provided by those awareness methods that are supposed to fully solve the awareness problem. This study is performed using well‐known performance metrics in distributed systems. Moreover, while the evaluations shown in the literature are performed by executing P2P simulations on a single (sequential) computer, this paper evaluates the performance of the considered methods on actually distributed systems. The evaluation results show that only a single method actually provides full awareness to avatars. This method also provides the best performance results. Copyright © 2008 John Wiley & Sons, Ltd.     

Discrete-time decentralized linear quadratic control for linear time-varying systems

This article addresses the problem of designing a decentralized control solution for a network of agents modeled by linear time-varying (LTV) dynamics, in a discrete-time framework. A general scheme is proposed, in which the problem is formulated as a classical linear quadratic regulator problem, for the global system, subject to a given sparsity constraint on the gain, which reflects the decentralized nature of the network. A method able to compute a sequence of well-performing stabilizing regulator gains is presented and validated resorting to simulations of two randomly generated LTV systems, one stable and the other unstable. Moreover, a tracking solution is developed, building on the solution to the regulator problem. Both methods rely on a closed-form solution, thus they can be computed very rapidly. Similarly to the centralized solution, both the presented methods require that a window of the future system dynamics is known. Both methods are validated resorting to simulations of: (i) a nonlinear network of four interconnected tanks; and (ii) a large-scale nonlinear network of interconnected tanks. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that track the desired output. Finally, both algorithms are scalable, being adequate for implementation in large-scale networks.     

Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems

In this paper, the problem of delay dependent error estimates for waveform relaxation methods applied to Volterra type systems of functional-differential equations of neutral type including systems of differential-algebraic equations is discussed. Under a Lipschitz condition (with delay dependent right-hand side) imposed on the so-called splitting function it is shown how the error estimates depend on the character of delays and that for this reason they are better than the known error estimates for relaxation methods. It is proved that under some assumptions the exact solution can be obtained after a finite number of steps of the iterative process, i.e., we prove that the waveform relaxation methods have the same property as the classical method of steps for solving delay-differential equations with nonvanishing delays. We also show the convergence of the waveform relaxation method without assuming that the spectral radius of the corresponding matrix related to the Lipschitz coefficients for the neutral argument is less than one.     

Spatial and temporal constraints in variational correspondence methods

This paper proposes a new method for optical-flow and stereo estimation based on the inclusion of both spatial and temporal constraints in a variational framework. These constraints bound the solution based on a priori information, or in other words, based on what is known of a possible solution or how it is expected to change temporally. This knowledge can be something that (a) is known since the geometrical properties of the scene are known or (b) is deduced by a higher-level algorithm capable of inferring this information. In the latter case, the constraint terms enable the exchange of information between high- and low-level vision systems: the high-level system makes a hypothesis of a possible scene setup which is then tested by the low-level one, recurrently. Since high-level vision systems incorporate knowledge of the world that surrounds us, this kind of hypothesis testing loop between the high- and low-level vision systems should converge to a more coherent solution.     

Identification of structural systems using naturally induced vibration data in the presence of measurement noise

The identification of structural systems using naturally induced vibration data in the presence of measurement noise by the method of instrumental variables is studied. It is well known that when measurement noise is present least squares methods will yield an inconsistent estimator. This leads us to consider methods which will yield a consistent estimator in the presence of noise. The maximum likelihood method provides a solution to the method, but is difficult to implement in the case of large systems because of the amount of computation required. In this paper we present the application of the method of instrument variables for the identification of the parameters of structural systems excited by white noise in the presence of white measurement noise. The equations required for the application of the method to a structural system and the resulting consistent estimator are derived. Although the concept of instrumental variables is not new, the application of this method to problems of structural systems is sufficiently attractive to justify its presentation. The results of simulation experiments which verify the theoretical development are presented.     

SCPIP – an efficient software tool for the solution of structural optimization problems

This paper describes SCPIP, a FORTRAN77 subroutine that has been proven to be a reliable implementation of convex programming methods in an industrial environment. Convex approximation methods like the method of moving asymptotes are used nowadays in many software packages for structural optimization. They are known to be efficient tools for the solution of design problems, in particular if displacement dependent constraints like stresses occur. A major advantage over many but not all classical approaches of mathematical programming is that at an iteration point a local model is formulated. For the solution of such a model no further function and gradient evaluations are necessary besides those at the current iteration point. The first versions of convex approximation methods used all a dual approach to solve the subproblems which is still a very efficient algorithm to solve problems with at most a medium number of constraints. But it is not efficient for problems with many constraints. An alternative is the use of an interior point method for the subproblem solution. This leads to more freedom in the definition of the linear systems where most of the computing time to solve the subproblems is spent. In consequence, large-scale problems can be handled more efficiently.     

Tracking in a class of nonminimum-phase systems with nonlinear internal dynamics via sliding mode control using method of system center

A method of asymptotic output tracking in a class of causal nonminimum-phase uncertain nonlinear systems is considered. Local asymptotic stability of output tracking-error dynamics are provided for the specified class of systems with the nonlinear hyperbolic at the origin internal dynamics forced by a reference output profile and external disturbances defined by a known linear exosystem. The nonlinear vector field of the internal dynamics is expanded in a power series, obtained at a selected operational point in the internal dynamics state space. The presented technique employs some linear algebraic methods and sliding mode control approach. The solution is a complete constructive algorithm.     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

Refinement for user interface designs

Formal approaches to software development require that we correctly describe (or specify) systems in order to prove properties about our proposed solution prior to building it. We must then follow a rigorous process to transform our specification into an implementation to ensure that the properties we have proved are retained. Different transformation, or refinement, methods exist for different formal methods, but they all seek to ensure that we can guide the transformation in a way which preserves the desired properties of the system. Refinement methods also allow us to subsequently compare two systems to see if a refinement relation exists between the two. When we design and build the user interfaces of our systems we are similarly keen to ensure that they have certain properties before we build them. For example, do they satisfy the requirements of the user? Are they designed with known good design principles and usability considerations in mind? Are they correct in terms of the overall system specification? However, when we come to implement our interface designs we do not have a defined process to follow which ensures that we maintain these properties as we transform the design into code. Instead, we rely on our judgement and belief that we are doing the right thing and subsequent user testing to ensure that our final solution remains useable and satisfactory. We suggest an alternative approach, which is to define a refinement process for user interfaces which will allow us to maintain the same rigorous standards we apply to the rest of the system when we implement our user interface designs.     

Nonlinearity problem in the numerical solution of superstiff Cauchy problems

For the numerical solution of Cauchy stiff initial problems, many schemes have been proposed for ordinary differential equation systems. They work well on linear and weakly nonlinear problems. The article presents a study of a number of well-known schemes on nonlinear problems (which include, for example, the problem of chemical kinetics). It is shown that on these problems, the known numerical methods are unreliable. They require a sufficient step reducing at some critical moments, and to determine these moments, sufficiently reliable algorithms have not been developed. It is shown that in the choice of time as an argument, the difficulty is associated with the boundary layer. If the length of the integral curve arc is taken as an argument, difficulties are caused by the transition zone between the boundary layer and regular solution.     

Design of retarded fractional delay differential systems using the method of inequalities

Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems （RFDDSs） by the method of inequalities, in which the design problem is formulated so that it is suitable for solution by numerical methods. Zakian＇s original formulation, which was first proposed in connection with rational systems, is extended to the case of RFDDSs. In making the use of this formulation possible for RFDDSs, the associated stability problems are resolved by using the stability test and the numerical algorithm for computing the abscissa of stability recently developed by the authors. During the design process, the time responses are obtained by a known method for the numerical inversion of Laplace transforms. Two numerical examples are given, where fractional controllers are designed for a time-delay and a heat-conduction plants.     

Reliability modelling and evaluation of phased mission systems

A reliability model for phased mission systems is developed. First, the mission phase change times are assumed to be known in advance. Next, these times are assumed as random variables and a method of solution is presented. The solution procedure is illustrated by an illustrative example. It is seen that for large systems, an analytical expression for mission system reliability becomes quite involved and requires a computation procedure.     

A homotopy method for exact output tracking of some non‐minimum phase nonlinear control systems

This paper presents a method for non‐causal exact dynamic inversion for a class of non‐minimum phase nonlinear systems, which seems to be an alternative to those existing in the literature. This method is based on a homotopy procedure that allows to find a ‘small’ periodic solution of a desired equation by a continuous deformation of a known periodic solution of a simpler auxiliary system. This method allows to face the exact output tracking problem for some non‐minimum phase systems that are well known in the literature, such as the inverted pendulum, the motorcycle and the CTOL aircraft. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient parallelization of the iterative solution of a coupled fluid–structure interaction problem

In this paper we consider the parallelization of the generation and iterative solution of coupled linear systems modelling the interaction of an acoustic field in a fluid medium with an elastic structure immersed in the fluid. The particular case studied is that of a hollow steel sphere in water. The aim of the work is to speed up the generation and solution of the systems. We describe the methods used, which involve special sparse storage arrangements and a novel application of a sparse approximate inverse preconditioning technique, and present results showing that the methods are very effective in terms of speeding up the generation and iterative solution of the systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Solutions of minimum time problem and minimum fuel problem for discrete linear admissible control systems

In this paper the minimum time problem and the minimum fuel problem for discrete linear admissible control systems arc formulated (is one linear programming problem with two different objective functions. It is found that the obtained problem is very similar to the dual form of the linear programming problem for the discrete linear L₁ approximation problem. An efficient dual simplex algorithm for the latter problem is used with the necessary modification* to obtain the solution of either of the former problems. Numerical results show that the present, method compares favourably with other known methods.