singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

there is no variational characterization of the cycles in the method of periodic projections the method of periodic projections consists in iterating projections onto m closed convex subsets of a hilbert space according to a periodic sweeping strategy in the presence of m 3 sets a long standing question going back to the 1960s is whether the limit cycles obtained by such a process can be characterized as the minimizers of a certain functional in this paper we answer this question in the negative projection algorithms for minimizing smooth convex functions over a product of convex sets are also discussed

power spectra and dynamical invariants for delay differential and difference equations abstract the mackey glass and ikeda delay differential equations ddes are models of feedback systems in which attractor dimension is proportional to the delay power spectra from these models are investigated analytically and numerically as the delay increases much beyond the system response time an exponentially decaying spectrum with a near periodic superimposed modulation appears it is shown that each peak in this modulation is associated with a mode of oscillation predicted by linear stability analysis around the fixed point the number of such modes within a characteristic autocorrelation time of the solution agrees with the lyapunov dimension of the attractor the disappearance of this modulation at higher delay values is also explained the decay rate of the spectrum in the mid to high frequency range is found to agree with the sum of the positive lyapunov exponents i e with the metric entropy realistic models with a distribution of delays display similar spectra as their memory kernel narrows into a delta function for very large delay the dde approaches an infinite dimensional continuous time difference equation ctde with each point on a delay interval evolving independently and according to a chaotic discrete time map ctde spectra which have many features of finite delay dde spectra can be computed analytically if the spectrum of this map is known for constant initial functions they have a 1 f 2 form at low frequencies our analysis suggests that the spectrum of the chaotic map is at the origin of the peak shapes in dde spectra at low frequency our findings highlight the relevance of the linear prrperties of ddes to the characterization of their nonlinear properties

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

upwind and midpoint upwind difference methods for time dependent differential difference equations with layer behavior abstract in this paper we have considered an important class of time dependent singularly perturbed convection diffusion problems with retarded terms which often arise in computational neuroscience we used taylors series to approximate the retarded terms and the resulting time dependent singularly perturbed differential equation is approximated using parameters uniform numerical methods based on euler implicit upwind and midpoint upwind finite difference schemes we discretize the continuous problem using implicit euler scheme in the time direction with a constant step size and the resulting system of equations is approximated using upwind and midpoint upwind difference schemes on a piecewise uniform mesh we will prove the uniform convergence of these two schemes numerical experiments support the convergence results

a norm convergence result on random products of relaxed projections in hilbert space suppose x is a hilbert space and c 1 ldots c n are closed convex intersecting subsets with projections p 1 ldots p n suppose further r is a mapping from bbb n onto 1 ldots n that assumes every value infinitely often we prove a more general version of the following result begin quote if the n tuple c 1 ldots c n is innately boundedly regular then the sequence x n defined by x 0 in x mbox arbitrary x n 1 p r n x n mbox for all n geq 0 converges in norm to some point in bigcap i 1 n c i end quote examples without the usual assumptions on compactness are given methods of this type have been used in areas like computerized tomography and signal processing

approximation bfgs methods for nonlinear image restoration we consider the iterative solution of unconstrained minimization problems arising from nonlinear image restoration our approach is based on a novel generalized bfgs method for such large scale image restoration minimization problems the complexity per step of the method is of o nlogn operations and only o n memory allocations are required where n is the number of image pixels based on the results given in carmine di fiore stefano fanelli filomena lepore paolo zellini matrix algebras in quasi newton methods for unconstrained minimization numer math 94 2003 479 500 we show that the method is globally convergent for our nonlinear image restoration problems experimental results are presented to illustrate the effectiveness of the proposed method

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets in this paper we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets we establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semialgebraic convex sets and the dimension of the underlying space we achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets

wavelet synthesis by alternating projections the coefficients of the quadrature mirror filters involved in orthonormal wavelet or wavelet packets signal decompositions are often chosen in an ad hoc manner in order to adapt such a decomposition to the signal being analyzed it may be pertinent to maximize an energy concentration criterion which leads to a constrained optimization problem a simple geometric interpretation of this problem is given and an alternating projection method is proposed to solve it

iterative least squares estimators in nonlinear image restoration the concept of iterative least squares estimation as applied to nonlinear image restoration is considered regarding the convergence analysis of nonlinear iterative algorithms the potential of the global convergence theorem gct is explored the theoretical analysis is performed on a general class of nonlinear algorithms which defines a signal dependent linear mapping of the residual the descent properties of two normed functions are considered furthermore a procedure for the selection of the iteration parameter is introduced the steepest descent sd iterative approach for the solution of the least squares optimization problem is introduced the convergence properties of the particular algorithm are readily derived on the basis of the generalized analysis and the gct the factors that affect the convergence rate of the sd algorithm are thoroughly studied in the case of the sd algorithm structural modifications are proposed and two hybrid sd algorithms attain convergence in a more uniform fashion with respect to their entries in general the algorithms achieve larger convergence rates than the conventional sd technique

particular solutions of forced generalized airy equations differential equations with delayed forcing terms arise naturally in the explicit solution of differential delay equations 11 in our present studies of singular perturbation analyses of boundary value problems for secondorder differential difference equations with a turning point see 2 for earlier results we have had to deal with forcing terms consisting of shifted airy and lommel functions and their derivatives one example of the type of equation which arises is the inhomogeneous airy equation see the appendix for a derivation y xy c ai x a c bi x a c l x a d ai x a d bi x a d3 l x a 1 where a is a constant 0 a co ci di i 1 2 3 are constants ai bi are airy functions and l is a lommel function see 31 satisfying l xl 1 2

optimal pulse shape design for digital communication systems by projections onto convex sets recent developments in dsp hardware has made it possible to use almost any pulse shape for digital data trans mission this resulted in a search for algorithms capable of constructing pulse shapes matching to the properties of a given channel the projection onto convex sets technique is suitable for the solution of this problem particularly because of its ffexibility in modeling a variety of constraints the use of the technique for the optimal pulse shape design problem is demonstrated through a detailed example from power line communications i introduction ince nyquist s invention of the intersymbol interference isi the optimal pulse shaping problem for digital data transmission over channels with various properties has been investigated by a number of researchers 1 7 the main differences between various digital nyquist pulse design al gorithms published in the literature are the type of the filter fir or iir they choose to generate the pulse shape and the formulation and solution of the optimization problem some of these approaches use general optimization techniques and they can handle additional constraints if needed however addition of each new constraint requires reformulation of the problem and the solution algorithm the projections onto convex sets pocs technique is an optimization method introduced in the 1960 s 8 a number of successful applications of the pocs technique in the signal processing area have been reported in the recent years 9 ll the pocs technique is attractive for optimum pulse shape design particularly because of its modular way of formulating problems with many constraints in this technique the addition of new constraints or removal of some of the existing ones does not require reformulation it is also possible to handle certain inconsistent requirements in the pocs formulation 12 if there are nonintersecting convex sets it is possible to find the vector which is as close as possible to one set of constraints while satisfying the rest

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

alternating direction method of multipliers for nonlinear image restoration problems in this paper we address the total variation tv based nonlinear image restoration problems in nonlinear image restoration problems an original image is corrupted by a spatiallyinvariant blur the build in nonlinearity in imaging system and the additive gaussian white noise we study the objective function consisting of the nonlinear least squares data fitting term and the tv regularization term of the restored image by making use of the structure of the objective function an efficient alternating direction method of multipliers can be developed for solving the proposed model the convergence of the numerical scheme is also studied numerical examples including nonlinear image restoration and high dynamic range imaging are reported to demonstrate the effectiveness of the proposed model and the efficiency of the proposed numerical scheme

e uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with small delay abstract non standard finite difference methods nsfdms now a days are playing an important role in solving the real life problems governed by odes and or by pdes many differential models of sciences and engineerings for which the existing methodologies do not give reliable results these nsfdms are solving them competitively to this end in this paper we consider second order linear singularly perturbed differential difference equations using the second of the five non standard modeling rules of mickens r e mickens nonstandard finite difference models of differential equations world scientific singapore 1994 the new finite difference methods are obtained for the particular cases of these problems this rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation s both theoretically and numerically we show that these nsfdms are e uniformly convergent

an alternating projection that does not converge in norm this paper proves that the method of alternating projections between two closed convex intersecting sets does not always converge in norm weak convergence was established by bregman soviet math dokl 6 1965 688 but the status of norm convergence was undetermined an explicit counterexample is provided

singular perturbation analysis of boundary value problems for differential difference equations a study is made of a class of boundary value problems for linear second order differential difference equations in which the highest order derivative is multiplied by a small parameter consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems the resulting analysis leads to several novel features which are not present in problems without differences

a new class of quasi newtonian methods for optimal learning in mlp networks in this paper we present a new class of quasi newton methods for an effective learning in large multilayer perceptron mlp networks the algorithms introduced in this work named lqn utilize an iterative scheme of a generalized bfgs type method involving a suitable family of matrix algebras l the main advantages of these innovative methods are based upon the fact that they have an o nlogn complexity per step and that they require o n memory allocations numerical experiences performed on a set of standard benchmarks of mlp networks show the competitivity of the lqn methods especially for large values of n

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

matrix algebras in quasi newton methods for unconstrained minimization in this paper a new class of quasi newton methods named qn is introduced in order to solve unconstrained minimization problems the novel approach which generalizes classical bfgs methods is based on a hessian updating formula involving an algebra of matrices simultaneously diagonalized by a fast unitary transform the complexity per step of qn methods is o n log n thereby improving considerably bfgs computational efficiency moreover since qn s iterative scheme utilizes single indexed arrays only o n memory allocations are required global convergence properties are investigated in particular a global convergence result is obtained under suitable assumptions on f numerical experiences 7 confirm that qn methods are particularly recommended for large scale problems

monotone operator theory in convex optimization several aspects of the interplay between monotone operator theory and convex optimization are presented the crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized we review the properties of subdifferentials as maximally monotone operators and in tandem investigate those of proximity operators as resolvents in particular we study new transformations which map proximity operators to proximity operators and establish connections with self dual classes of firmly nonexpansive operators in addition new insights and developments are proposed on the algorithmic front

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

singular perturbation analysis of boundary value problems for differential difference equations iii turning point problems this paper continues a study of a class of boundary value problems for linear second order differential difference equations in which the second order derivative is multiplied by a small parameter siam j appl math 42 1982 pp 502 531 45 1985 pp 687 707 the previous papers focused on problems involving boundary and interior layer phenomena rapid oscillations and resonance behavior the problems studied here have solutions which exhibit turning point behavior i e transition regions between rapid oscillations and exponential behavior the presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval a combination of exact solutions singular perturbation methods and numerical computations are used in these studies