Showing papers on "Convergence (routing) published in 1991"

Convergence of approximation schemes for fully nonlinear second order equations

Abstract: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied. It is proved that any monotone, stable, and consistent scheme converges (to the correct solution), provided that there exists a comparison principle for the limiting equation. Several examples are given where the result applies. >

Distribution system power flow analysis-a rigid approach

Abstract: This approach is oriented toward applications in three phase distribution system operational analysis rather than planning analysis. The solution method is the optimally ordered triangular factorization Y/sub BUS/ method (implicit Z/sub BUS/ Gauss method) which not only takes advantage of the sparsity of system equations but also has very good convergence characteristics on distribution problems. Detailed component models are needed for all system components in the simulation. Utilizing the phase frame representation for all network elements, a program called Generalized Distribution Analysis Systems, with a number of features and capabilities not found in existing packages, has been developed for large-scale distribution system simulations. The system being analyzed can be balanced or unbalanced and can be a radial, network, or mixed-type distribution system. Furthermore, because the individual phase representation is employed for both system and component models, the system can comprise single, double, and three-phase systems simultaneously. >

Convergence estimates for multigrid algorithms without regularity assumptions

Abstract: A new technique for proving rate of convergence estimates of multi- grid algorithms for symmetric positive definite problems will be given in this paper. The standard multigrid theory requires a "regularity and approxima- tion" assumption. In contrast, the new theory requires only an easily verified approximation assumption. This leads to convergence results for multigrid re- finement applications, problems with irregular coefficients, and problems whose coefficients have large jumps. In addition, the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications.

Convergence of quasi-Newton matrices generated by the symmetric rank one update

Abstract: Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasi-Newton formulae

Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model

Abstract: The aim of the paper is to demonstrate the numerical feasibility of 4-D variational assimilation using a multilevel primitive-equation model. The experiments consist in minimizing the distance between the model solution and the observations. The gradient of the cost function thus defined is computed by integrating the adjoint of the model. Here, assimilations are performed using model-generated observations. In a preliminary set of experiments, assimilations were performed assuming that observations consisting of a full-model-state vector are available only at the end of the assimilation period. The numerical convergence of the method is proved and the results are meteorologically realistic. The use of the Machenhauer nonlinear normal mode initialization scheme and its adjoint turned out to have hastened the convergence and to have controlled to some extent the amount of gravity waves appearing in the solution. We identify a loss of conditioning of the minimization problem with an increase in the length of the assimilation period. The presence of horizontal diffusion in the model has the effect of degrading the convergence. The second set of experiments evaluates the impact of observations distributed over the whole assimilation period. Through different senarios of sets of observations, we demonstrate the efficiency of the 4-D variational approach in extracting the information contained in the dynamics of the model, together with the information contained in the observations. In particular, observing only the small scales of the flow leads to a good reconstruction of both small scales and large scales. Observations of the mass-field evolution lead to a good reconstruction of the vorticity field in mid latitudes but less so in the tropics. Increased resolution in the model in the experiments was found to have a negative impact on the speed of convergence of the minimization algorithm.

Convergence of learning algorithms with constant learning rates

Abstract: The behavior of neural network learning algorithms with a small, constant learning rate, epsilon , in stationary, random input environments is investigated. It is rigorously established that the sequence of weight estimates can be approximated by a certain ordinary differential equation, in the sense of weak convergence of random processes as epsilon tends to zero. As applications, backpropagation in feedforward architectures and some feature extraction algorithms are studied in more detail. >

Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions

Abstract: Linear discrete-time systems affected by both parameter and input uncertainties are considered. The problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned speed of convergence is investigated. It is shown that the problem has a solution if and only if there exists a certain Lyapunov function which does not belong to a pre-assigned class of functions (i.e. the quadratic ones) but it is determined by the target set in which ultimate boundedness is desired. One of the advantages of this approach is that one can handle systems with control constraints. No matching assumptions are made. For systems with linearly constrained uncertainties, it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets. An extension of the technique to continuous-time systems is presented. >

Robust identification and interpolation in H

Abstract: We consider system identification in H∞ in the framework proposed by Helmicki, Jacobson and Nett. An algorithm using the Jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems. It is shown that this, and similar identification algorithms, can be successfully combined with a model reduction procedure to produce low-order models. Connections with the Nevanlinna-Pick interpolation problem are explored, and an algorithm is given in which the identified model interpolates the given noisy data. Some numerical results are provided for illustration. Finally, the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions.

Stable adaptive control and recursive identification using radial Gaussian networks

Abstract: Previous work has provided the theoretical foundations of a constructive design procedure for uniform approximation of smooth functions to a chosen degree of accuracy using networks of Gaussian radial basis functions. This construction and the guaranteed uniform bounds were shown to provide the basis for stable adaptive neurocontrol algorithms for a class of nonlinear plants. The authors detail and extend these ideas in three directions. First, some practical details of the construction are provided, explicitly illustrating the relation between the free parameters in the network design and the degree of approximation error on a particular set. Next, the original adaptive control algorithm is modified to permit incorporation of additional prior knowledge of the system dynamics, allowing the neurocontroller to operate in parallel with conventional fixed or adaptive controllers. Finally, it is shown how the Gaussian network construction may also be utilized in recursive identification algorithms with similar guarantees of stability and convergence. >

On the global convergence of trust region algorithms using inexact gradient information

Abstract: Trust region algorithms are an important class of methods that can be used to solve unconstrained optimization problems. Strong global convergence results are demonstrated for a class of methods where the gradient values are approximated rather than computed exactly, provided they obey a simple relative error condition. No requirement is made that gradients be recomputed to successively greater accuracy after unsuccessful iterations.

Neural controller based on back-propagation algorithm

Abstract: The paper investigates the possibility of using a simple approximation for evaluating the error which must be back-propagated to allow a neural net to learn to control a plant in an adaptive way. The algorithm is based on an approximation of the Jacobian of the plant. The method is applied to five simulations. The first two simulations allow a comparison between the proposed algorithm and the standard back-propagation, for which the error to be back-propagated is precisely known. The results for the two methods show equivalent performances, and equivalent convergence time, for the test problems. This shows that the rate of convergence of the neural net does not seem to depend crucially on the values of the Jacobian. The last three simulations investigate the possibility of online adaptive learning. The results show that control based on some approximation of theJacobian is possible for a neural network.

Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions

Abstract: Unspecified Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: http://doi.org/10.5167/uzh-22758 Originally published at: Chipot, M; Fila, M; Quittner, P (1991). Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Mathematica Universitatis Comenianae. New Series, 60(1):35-103. Acta Math. Univ. Comenianae Vol. LX, 1(1991), pp. 35–103 35 STATIONARY SOLUTIONS, BLOW UP AND CONVERGENCE TO STATIONARY SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS M. CHIPOT, M. FILA AND P. QUITTNER

Data reconciliation in generalized linear dynamic systems

Abstract: A generalized linear dynamic model or singular model, for which the standard state space representation and the Kalman filtering cannot be applied, is used to develop a new algorithm to solve the linear dynamic material balance problem. This algorithm is based on the method developed in the steady-state case and leads to a recursive scheme, which is very useful in real-time processing. It reduces the computational problem such as singularities and round-off errors that may occur in complex systems. Convergence conditions are given and verified for the dynamic material balance case.

An aggregate function method for nonlinear programming

Abstract: This paper presents a new method, called the "aggregate function" method, for solvingnonlinear programming problems. At first, we use the "maximum" constraint in place of theoriginal constraint set to convert a multi-constrained optimization problem to a non-smoothbut singly constrained problem; we then employ the surrogate constraint concept and themaximum entropy principle to derive a smooth function, by which the non-smooth maximumconstraint is approximated and the original problem is converted to a smooth and singly con-strained problem; furthermore, we develop a multiplier penalty algorithm. The presentalgorithm has merits of stable and fast convergence and ease of computer implementation,and is particularly suitable to solving a nonlinear programming problem with a large num-ber of constraints.

On the rate of convergence of a partially asynchronous gradient projection algorithm

Abstract: Recently, Bertsekas and Tsitsiklis proposed a partially asynchronous implementation of the gradient projection algorithm of Goldstein and Levitin and Polyak for the problem of minimizing a differentiable function over a closed convex set. In this paper, the rate of convergence of this algorithm is analyzed. It is shown that if the standard assumptions hold (that is, the solution set is nonempty and the gradient of the function is Lipschitz continuous) and (i) the isocost surfaces of the objective function, restricted to the solution set, are properly separated and (ii) a certain multifunction associated with the problem is locally upper Lipschitzian, then this algorithm attains a linear rate of convergence.

On extensions of Polyak's averaging approach to stochastic approximation

Abstract: Stochastic approximation algorithms are considered and Polyak's averaging approach (cf [1]) is revisited Under much weaker conditions, convergence and rate of convergence results are developed In lieu of uncorrelated noise, φ-mixing type of random disturbances are treated By means of weak convergence methods, it is shown that the multistage algorithms via averaging have asymptotically optimal convergence speed and are efficient procedures

Cell centered and cell vertex multigrid schemes for the Navier-Stokes equations

Abstract: Two efficient and robust finite-volume multigrid schemes for solving the Navier-Stokes equations are investigated These schemes employ either a cell centred or a cell vertex discretisation technique An explicit Runge-Kutta algorithm is used to advance the solution in time Acceleration techniques are applied to obtain faster steady-state convergence Accuracy and convergence of the schemes are examined Computational results for transonic airfoil flows are essentially the same, even for a coarse mesh Both schemes exhibit very good convergence rates for a broad range of artificial dissipation coefficients

Solving multistage stochastic networks: An application of scenario aggregation

Abstract: The scenario aggregation algorithm is specialized for stochastic networks. The algorithm determines a solution that does not depend on hindsight and accounts for the uncertain environment depicted by a number of appropriately weighted scenarios. The solution procedure decomposes the stochastic program to its constituent scenario subproblems, thus preserving the network structure. Computational results are reported demonstrating the algorithm's convergence behavior. Acceleration schemes are discussed along with termination criteria. The algorithm's potential for execution on parallel multiprocessors is discussed.

Superlinearly convergent algorithm for min-max problems

Abstract: Algorithms for solving the problem of minimizing the maximum of a finite number of functions are proposed and analyzed. Quadratic approximations to the functions are employed in the determination of a search direction. Global convergence is proven and it is shown that a quadratic rate of convergence is obtained.

A global convergence theory for the Celis-Dennis-Tapia trust-region algorithm for constrained optimization

Abstract: A global convergence theory for a class of trust-region algorithms for solving the equality constrained optimization problem is presented. This theory is sufficiently general that it holds for any algorithm that generates steps giving at least a fraction of Cauchy decrease in the quadratic model of the constraints, and that uses the augmented Lagrangian as a merit function. This theory is used to establish global convergence of the 1984 Celis–Dennis–Tapia algorithm with a different scheme for updating the penalty parameter. The behavior of the penalty parameter is also discussed.

Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application

Abstract: Multidimensional reflected Brownian motions, also called regulated Brownian motions or simply RBM's, arise as approximate models of queueing networks Thus the stationary distributions of these diffusion processes are of interest for steady-state analysis of the corresponding queueing systems This paper considers two-dimensional semimartingale RBM's with rectangular state space, which include the RBM's that serve as approximate models of finite queues in tandem The stationary distribution of such an RBM is uniquely characterized by a certain basic adjoint relationship, and an algorithm is proposed for numerical solution of that relationship We cannot offer a general proof of convergence, but the algorithm has been coded and applied to special cases where the stationary distribution can be determined by other means; the computed solutions agree closely with previously known results and convergence is reasonably fast Our current computer code is specific to two-dimensional rectangles, but the basic logic of the algorithm applies equally well to any semimartingale RBM with bounded polyhedral state space, regardless of dimension To demonstrate the role of the algorithm in practical performance analysis, we use it to derive numerical performance estimates for a particular example of finite queues in tandem; our numerical estimates of both the throughput loss rate and the average queue lengths are found to agree with simulated values to within about five percent

Computational aspects of sensitivity calculations in linear transient structural analysis

Abstract: A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. Several existing sensitivity calculation methods and two new methods are compared for three example problems. All of the methods considered are computationally efficient enough to be suitable for largeorder finite element models. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the convergence of both response quantities and sensitivities as a function of the number of vectors used. Two types of sensitivity calculation techniques were considered. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. This was found to result in poor convergence of stress sensitivities in several cases. To overcome this poor convergence, two semianalytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in very good convergence of the stress sensitivities. In both techniques the computational cost is much less than would result if the vibration modes were recalculated and then used in an overall finite difference method. A dot over a symbol indicates derivative with respect to time. A superscriptT indicates a transposed matrix.

Finite convergence of algorithms for nonlinear programs and variational inequalities

Abstract: Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.

Kinematic routing using finite elements on a triangular irregular network

Abstract: Automated extraction of geometry for hydraulic routing from digital elevation models (DEM) is a procedure that must be easily accomplished for widespread application of distributed hydraulically based rainfall excess-runoff models. One-dimensional kinematic routing on a regular grid DEM is difficult due to flow division and convergence. Two-dimensional kinematic routing on a triangular irregular network (TIN) surmounts many of these difficulties. Because TIN DEMs typically require far fewer points to represent topography than regular grid DEMs, substantial computational economy is also realized. One-dimensional routing using vector contour data overcomes the grid-based routing disadvantages but often requires several orders of magnitude more storage points than a TIN. The methodology presented in this paper represents a compromise between slightly increased computational complexity and the economy of TIN topographic representation. We take the unique approach of subdividing each topographic triangle (TIN facet) into a set of coplanar triangular finite elements, performing routing on a single facet and then routing the resulting excess hydrograph to downstream facets and channels via upstream boundary conditions. Results indicate that shock conditions are readily handled, computed depths match analytic results to within ±3% and volume balances are typically within 1%. This modeling system illustrates the viability of kinematic routing over a TIN DEM derived directly from digital mapping data.

Convergence analysis of an algorithm for blind equalization

Abstract: The authors consider a key algorithm for blind equalization and derive expressions for the evolution of the equalizer output error trajectory. The authors develop a model to examine the convergence behavior of this algorithm. Suitable approximations are incorporated into the model to facilitate analysis. The validity of these approximations is demonstrated for a typical communication channel. It is shown that even for a channel for which the assumption of Gaussianity of the equalizer input data may not be very good, the analysis presented predicts the convergence behavior reasonably well. >

Three-dimensional structured networks for matrix equation solving

Abstract: Two three-dimensional structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is to first represent a given equation-solving problem by a 3-D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem: then, train the 3-D structured network to match the desired pattern array using some training algorithms; and finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3-D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3-D structured networks. >