dc free error control block codes dc free codes and error control ec codes are widely used in digital transmission and storage systems to improve system performance in terms of code rate bit error rate ber and low frequency suppression and to provide a flexible tradeoff between these parameters this paper introduces a new class of codes with both dc control and ec capability the new codes integrate dc free encoding and ec encoding and are decoded by first applying standard ec decoding techniques prior to dc free decoding thereby avoiding the drawbacks that arise when dc free decoding precedes ec decoding the dc free code property is introduced into standard ec codes through multimode coding techniques at the cost of minor loss in ber performance on the additive white gaussian noise channel and some increase in implementation complexity particularly at the encoder this paper demonstrates that a wide variety of ec block codes can be integrated into this dc free coding structure including binary cyclic codes binary primitive bch codes reed solomon codes reed muller codes and some capacity approaching ec block codes such as low density parity check codes and product codes with iterative decoding performance of the new dc free ec block codes is presented

using sampling and simplex derivatives in pattern search methods in this paper we introduce ways of making a pattern search more efficient by reusing previous evaluations of the objective function based on the computation of simplex derivatives e g simplex gradients at each iteration one can attempt to compute an accurate simplex gradient by identifying a sampling set of previously evaluated points with good geometrical properties this can be done using only past successful iterates or by considering all past function evaluations the simplex gradient can then be used to reorder the evaluations of the objective function associated with the directions used in the poll step or to update the mesh size parameter according to a sufficient decrease criterion neither of which requires new function evaluations we present these procedures in detail and apply them to a set of problems from the cuter collection numerical results show that these procedures can enhance significantly the practical performance of pattern search methods

numerical analysis of mathcal v mathcal u decomposition mathcal u gradient and mathcal u hessian approximations advances in bundle methods for nonsmooth optimization have lead to the development of mathcal v mathcal u decompositions the mathcal u gradient and the mathcal u hessian these variational analysis constructs have proven extremely valuable and lead to the development of the superlinearly convergent mathcal v mathcal u algorithm for nonsmooth optimization in this paper we examine these constructs from the viewpoint of derivative free optimization we show that given a finite max function f x max i 0 1 ldots m f i x and a black box which returns function values for each f i it is possible to construct approximations of the mathcal v mathcal u decompositions mathcal u gradient and mathcal u hessian the approximations do not require excessive black box calls and the accuracy of the approximations is directly related to the accuracy of the approximate gradient and hessians for each f i

quantum locally compact metric spaces abstract we introduce the notion of a quantum locally compact metric space which is the noncommutative analogue of a locally compact metric space and generalize to the non unital setting the notion of quantum metric spaces introduced by rieffel we then provide several examples of such structures including the moyal plane compact quantum metric spaces and locally compact metric spaces this paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the non unital setup and offers important insights into noncommutative geometry for non compact quantum spaces

the power spectral density of maximum entropy charge constrained sequences a limit on the absolute value of the running digital sum of a sequence is known as the charge constraint such a limit imposes a spectral null at dc the maximum entropy distribution of a charge constrained sequence is presented a closed form expression for the power spectral density of maximum entropy charge constrained sequences is given and plotted

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

derivative free optimization of expensive functions with computational error using weighted regression we propose a derivative free algorithm for optimizing computationally expensive functions with computational error the algorithm is based on the trust region regression method by conn scheinberg and vicente a r conn k scheinberg and l n vicente ima j numer anal 28 2008 pp 721 748 but uses weighted regression to obtain more accurate model functions at each trust region iteration a heuristic weighting scheme is proposed that simultaneously handles i differing levels of uncertainty in function evaluations and ii errors induced by poor model fidelity we also extend the theory of lambda poisedness and strong lambda poisedness to weighted regression we report computational results comparing interpolation regression and weighted regression methods on a collection of benchmark problems weighted regression appears to outperform interpolation and regression models on nondifferentiable functions and functions with deterministic noise

global convergence of trust region algorithms for convex constrained minimization without derivatives in this work we propose a trust region algorithm for the problem of minimizing a function within a convex closed domain we assume that the objective function is differentiable but no derivatives are available the algorithm has a very simple structure and allows a great deal of freedom in the choice of the models under reasonable assumptions for derivative free schemes we prove global convergence for the algorithm that is to say that all accumulation points of the sequence generated by the algorithm are stationary

on the convergence of the multidirectional search algorithm this paper presents the convergence analysis for the multidirectional search algorithm a direct search method for unconstrained minimization the analysis follows the classic lines of proofs of convergence for gradient related methods the novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary although it is assumed that the function is continuously differentiable over some subset of the domain the proof can be extended to treat most nonsmooth cases of interest the argument breaks down only at points where the derivative exists but is not continuous finally it is shown how a general convergence theory can be developed for an entire class of direct search methods which includes such methods as the factorial design algorithm and the pattern search algorithm that share a key feature of the multidirectional search algorithm

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

constrained optimization involving expensive function evaluations a sequential approach abstract this paper presents a new sequential method for constrained nonlinear optimization problems the principal characteristics of these problems are very time consuming function evaluations and the absence of derivative information such problems are common in design optimization where time consuming function evaluations are carried out by simulation tools e g fem cfd classical optimization methods based on derivatives are not applicable because often derivative information is not available and is too expensive to approximate through finite differencing the algorithm first creates an experimental design in the design points the underlying functions are evaluated local linear approximations of the real model are obtained with help of weighted regression techniques the approximating model is then optimized within a trust region to find the best feasible objective improving point this trust region moves along the most promising direction which is determined on the basis of the evaluated objective values and constraint violations combined in a filter criterion if the geometry of the points that determine the local approximations becomes bad i e the points are located in such a way that they result in a bad approximation of the actual model then we evaluate a geometry improving instead of an objective improving point in each iteration a new local linear approximation is built and either a new point is evaluated objective or geometry improving or the trust region is decreased convergence of the algorithm is guided by the size of this trust region the focus of the approach is on getting good solutions with a limited number of function evaluations

on trust region methods for unconstrained minimization without derivatives we consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function then a new vector of variables is calculated by minimizing the current model within a trust region techniques are described for adjusting the trust region radius and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model particular attention is given to quadratic models with diagonal second derivative matrices because numerical experiments show that they are often more efficient than full quadratic models for general objective functions finally some recent research on the updating of full quadratic models is described briefly using fewer interpolation equations than before the resultant freedom is taken up by minimizing the frobenius norm of the change to the second derivative matrix of the model a preliminary version of this method provides some very promising numerical results

frames and grids in unconstrained and linearly constrained optimization a nonsmooth approach this paper describes a class of frame based direct search methods for unconstrained and linearly constrained optimization a template is described and analyzed using clarke s break nonsmooth calculus this provides a unified and simple approach to earlier results for grid and frame based methods and also provides partial convergence results when the objective function is not smooth undefined in some places or both the template also covers many new methods which combine elements of previous ideas using frames and grids these new methods include grid based simple descent algorithms which allow moving to points off the grid at every iteration and can automatically control the grid size provided function values are available the concept of a grid is also generalized to that of an admissible set which allows sets for example with circular symmetries the method is applied to linearly constrained problems using a simple barrier approach

bayesian regularization neural networks for optimizing fluid flow processes this paper details the application of neural networks and evolutionary strategies for shape optimization problems these commonly grouped under the name soft computing methods are quite recent methods and in demand for application to engineering optimization tasks for complex problems like in non convex optimization such heuristic techniques are able to outperform conventional optimization methods we show that the use of progressive network models can yield satisfactory results when optimizing engineering relevant applications numerical examples are given

provably convergent multifidelity optimization algorithm not requiring high fidelity derivatives functions does not alter the con vergence proof of the optimization algorithm and is shown to be robust to poor low delity information the algorithm is compared with an unconstrained single delity quasi newton algorithm and two

on the convergence of grid based methods for unconstrained optimization the convergence of direct search methods for unconstrained minimization is examined in the case where the underlying method can be interpreted as a grid or pattern search over successively refined meshes an important aspect of the main convergence result is that translation rotation scaling and shearing of the successive grids are allowed

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

codes for zero spectral density at zero frequency corresp in pulse amplitude modulation pam digital transmission systems line encoding is used for shaping the spectrum of the encoded symbol sequence to suit the frequency characteristics of the transmission channel in particular it is often required that the encoded symbol sequence have a zero mean and spectral density vanishing at zero frequency we show that the finite running digital sum condition is a necessary and sufficient condition for this to occur the result holds in particular for alphabetic codes which are the most widely used line codes

spectral lines of codes given as functions of finite markov chains spectral lines of signals that are given by functions of finite markov chains are investigated a problem of characterizing encoders in such a manner that the messages emitted from these encoders have some amount of information about clock independent of the source statistics is considered necessary and sufficient conditions are established for the encoded message to have a spectral line of a given amplitude for every transition probability matrix of the underlying markov chain several other related results about spectral lines are given a biased coboundary condition is defined at a frequency f and it is shown that this condition is necessary and sufficient for the encoded message to have a spectral density null at f

performance of efficient balanced codes the problem of appraising the spectral performance of codes based on a new algorithm for generating zero disparity codewords presented by d e knuth 1986 is addressed in order to get some insight into the efficiency of knuth s construction technique the authors evaluate the spectral properties of its code streams the structure of knuth codes allows the derivation a simple expression for an approximation to the sum of variance of these codes this quantity plays a key role in the spectral performance characterization of dc balanced codes the authors evaluate this expression and compare the sum variance of knuth codes with the sum variance of the polarity bit codes for fixed redundancy under the premise that the sum variance can serve as a quantity to judge the width of the spectral notch the authors conclude that codes based on knuth s algorithm offer less spectral suppression than polarity bit codes with the same redundancy

thb splines multi patch parameterization for multiply connected planar domains via template segmentation abstract given a planar multiply connected domain we provide a multi patch parameterization exploiting a segmentation algorithm and the construction of templates the segmentation is a fundamental step which allows to split the input domain into patches topologically equivalent to quadrilaterals templates are prototypes of segmentation that will be chosen according to the shape of the given domain we will compute a bijective harmonic mapping h defined between the input and the chosen template the segmentation will be transferred from the prototype template to the input by approximating the inverse mapping templates are equipped with a multi patch structure and each patch i is parametrized individually by a geometry mapping g i the final multi patch parameterization of is achieved exploiting the combination of each geometry mapping g i and the inverse of the mapping h obtained by projection into the thb spline space

global convergence of general derivative free trust region algorithms to first and second order critical points in this paper we prove global convergence for first and second order stationary points of a class of derivative free trust region methods for unconstrained optimization these methods are based on the sequential minimization of quadratic or linear models built from evaluating the objective function at sample sets the derivative free models are required to satisfy taylor type bounds but apart from that the analysis is independent of the sampling techniques a number of new issues are addressed including global convergence when acceptance of iterates is based on simple decrease of the objective function trust region radius maintenance at the criticality step and global convergence for second order critical points

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

orbit optimization by radial basis function interpolation in trust regions we present a new derivative free algorithm orbit for unconstrained local optimization of computationally expensive functions a trust region framework using interpolating radial basis function rbf models is employed the rbf models considered often allow orbit to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques approximation guarantees are obtained by ensuring that a subset of the interpolation points is sufficiently poised for linear interpolation the rbf property of conditional positive definiteness yields a natural method for adding additional points we present numerical results on test problems to motivate the use of orbit when only a relatively small number of expensive function evaluations are available results on two very different application problems calibration of a watershed model and optimization of a pde based bioremediation plan are also encouraging and support orbit s effectiveness on blackbox functions for which no special mathematical structure is known or available

balancing sets of vectors for n 0 d or 0 n identical to d mod 2 let k n d denote the minimal cardinality of a family v of or 1 vectors of dimension n such that for any or 1 vector w of dimension n there is a v in v such that mod v w mod

finding local optima of high dimensional functions using direct search methods this paper focuses on a subclass of box constrained non linear optimization problems we are particularly concerned with settings where gradient information is unreliable or too costly to calculate and the function evaluations themselves are very costly this encourages the use of derivative free optimization methods and especially a subclass of these referred to as direct search methods the thrust of our investigation is twofold first we implement and evaluate a number of traditional direct search methods according to the premise that they should be suitable as local optimizers when used in a metaheuristic framework second we introduce a new direct search method based on scatter search designed to remedy the lack of a good derivative free method for solving problems of high dimensions our new direct search method has convergence properties comparable to those of existing methods in addition to being able to solve larger problems more effectively

sobolev seminorm of quadratic functions with applications to derivative free optimization this paper studies the h 1 sobolev seminorm of quadratic functions the research is motivated by the least norm interpolation that is widely used in derivative free optimization we express the h 1 seminorm of a quadratic function explicitly in terms of the hessian and the gradient when the underlying domain is a ball the seminorm gives new insights into least norm interpolation it clarifies the analytical and geometrical meaning of the objective function in least norm interpolation we employ the seminorm to study the extended symmetric broyden update proposed by powell numerical results show that the new thoery helps improve the performance of the update apart from the theoretical results we propose a new method of comparing derivative free solvers which is more convincing than merely counting the numbers of function evaluations

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

a digital requantizer with shaped requantization noise that remains well behaved after nonlinear distortion a major problem in oversampling digital to analog converters and fractional n frequency synthesizers which are ubiquitous in modern communication systems is that the noise they introduce contains spurious tones the spurious tones are the result of digitally generated quantized signals passing through nonlinear analog components this paper presents a new method of digital requantization called successive requantization special cases of which avoids the spurious tone generation problem sufficient conditions are derived that ensure certain statistical properties of the quantization noise including the absence of spurious tones after nonlinear distortion a practical example is presented and shown to satisfy these conditions

parallel radial basis function methods for the global optimization of expensive functions abstract we introduce a master worker framework for parallel global optimization of computationally expensive functions using response surface models in particular we parallelize two radial basis function rbf methods for global optimization namely the rbf method by gutmann gutmann h m 2001a a radial basis function method for global optimization journal of global optimization 19 3 201 227 gutmann rbf and the rbf method by regis and shoemaker regis r g shoemaker c a 2005 constrained global optimization of expensive black box functions using radial basis functions journal of global optimization 31 153 171 cors rbf we modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel we compare the performance of the two parallel rbf methods with a parallel multistart derivative based algorithm a parallel multistart derivative free trust region algorithm and a parallel evolutionary algorithm on eleven test problems and on a 6 dimensional groundwater bioremediation application the results indicate that the two parallel rbf algorithms are generally better than the other three alternatives on most of the test problems moreover the two parallel rbf algorithms have comparable performances on the test problems considered finally we report good speedups for both parallel rbf algorithms when using a small number of processors

characterization of su q l 1 equivariant spectral triples for the odd dimensional quantum spheres the quantum group suq l 1 has a canonical action on the odd dimensional sphere s 2l 1 q all odd spectral triples acting on the l2 space of s 2l 1 q and equivariant under this action have been characterized this characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial k homology class these generalize the results of chakraborty pal for suq 2 ams subject classification no 58b34 46l87 19k33

objective derivative free methods for constrained optimization we propose feasible descent methods for constrained minimization that do not make explicit use of the derivative of the objective function the methods iteratively sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective function value is not sampled the search arcs are obtained by projecting search direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone of first order feasible variations at the current iterate we show that these methods have desirable convergence properties under certain regularity assumptions on the constraints in the case of linear constraints the projections are redundant and the regularity assumptions hold automatically numerical experience with the methods in the linearly constrained case is reported

global convergence of radial basis function trust region derivative free algorithms we analyze globally convergent derivative free trust region algorithms relying on radial basis function interpolation models our results extend the recent work of conn scheinberg and vicente siam j optim 20 2009 pp 387 415 to fully linear models that have a nonlinear term we characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first order critical points for the orbit algorithm of wild regis and shoemaker siam j sci comput 30 2008 pp 3197 3219 using orbit we present numerical results for different types of radial basis functions on a series of test problems we also demonstrate the use of orbit in finding local minima on a computationally expensive environmental engineering problem

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

on the adaptive selection of the parameter in stabilized finite element approximations a systematic approach is developed for the selection of the stabilization parameter for stabilized finite element approximation of the stokes problem whereby the parameter is chosen to minimize a computable upper bound for the error in the approximation the approach is applied in the context of both a single fixed mesh and an adaptive mesh refinement procedure the optimization is carried out by a derivative free optimization algorithm and is based on minimizing a new fully computable error estimator numerical results are presented illustrating the theory and the performance of the estimator together with the optimization algorithm

condor a new parallel constrained extension of powell s uobyqa algorithm experimental results and comparison with the dfo algorithm this paper presents an algorithmic extension of powell s uobyqa algorithm unconstrained optimization by quadratical approximation we start by summarizing the original algorithm of powell and by presenting it in a more comprehensible form thereafter we report comparative numerical results between uobyqa dfo and a parallel constrained extension of uobyqa that will be called in the paper condor constrained non linear direct parallel optimization using trust region method for high computing load function the experimental results are very encouraging and validate the approach they open wide possibilities in the field of noisy and high computing load objective functions optimization from 2 min to several days like for instance industrial shape optimization based on computation fluid dynamic codes or partial differential equations solvers finally we present a new easily comprehensible and fully stand alone implementation in c of the parallel algorithm

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

a numerical approach for shape optimization of fluid flow domains abstract a numerical method for the shape optimization of fluid flow domains is presented and analyzed the procedure is based on a flow solver a mathematical optimization tool and a technique for shape variation which are combined into an integrated procedure the flow solver relies on the discretization of the incompressible navier stokes equations by means of the finite volume method for block structured boundary fitted grids with multi grid acceleration the optimization tool is an implementation of a trust region based derivative free method it is designed to minimize smooth functions whose evaluations are considered expensive and whose derivatives are not available or not desirable to approximate the shape variation is obtained by deforming the computational grid employed by the flow solver for this purpose displacement fields scaled by the design variables are added to the initial grid the displacement vectors are computed once before starting the optimization cycle by using a free form deformation technique applications illustrating the functionality and the properties of the method are presented for some examples of engineering interest such as the minimization of a pressure drop the maximization of a lift force and the optimization of a wall temperature

on the shannon capacity of dc and nyquist free codes the design of codes which are simultaneously dc and nyquist free is considered the shannon capacity of codes that are simultaneously dc and nyquist free is tabulated as a function of the number of allowed running digital sum and alternating digital sum states for this constraint the set of possible rational capacities is shown to be 1 4 1 2 3 4 a rate 3 4 dc and nyquist free code is used to demonstrate the ability to code at the highest rational capacity available

least frobenius norm updating of quadratic models that satisfy interpolation conditions quadratic models of objective functions are highly useful in many optimization algorithms they are updated regularly to include new information about the objective function such as the difference between two gradient vectors we consider the case however when each model interpolates some function values so an update is required when a new function value replaces an old one we let the number of interpolation conditions m say be such that there is freedom in each new quadratic model that is taken up by minimizing the frobenius norm of the second derivative matrix of the change to the model this variational problem is expressed as the solution of an m n 1 m n 1 system of linear equations where n is the number of variables of the objective function further the inverse of the matrix of the system provides the coefficients of quadratic lagrange functions of the current interpolation problem a method is presented for updating all these coefficients in m n 2 operations which allows the model to be updated too an extension to the method is also described that suppresses the constant terms of the lagrange functions these techniques have a useful stability property that is investigated in some numerical experiments

on codes with spectral nulls at rational submultiples of the symbol frequency in digital data transmission respectively storage systems line codes respectively recording codes are used to tailor the spectrum of the encoded sequences to satisfy constraints imposed by the channel transfer characteristics or other system requirements for instance pilot tone insertion requires codes with zero mean and zero spectral density at tone frequencies embedded tracking focus servo signals produce similar needs codes are studied with spectral nulls at frequencies f kf s n where f is the symbol frequency and k n are relatively prime integers with k leq n in other words nulls at rational submultiples of the symbol frequency a necessary and sufficient condition is given for a null at f in the form of a finite discrete fourier transform dft running sum condition a corollary of the result is the algebraic characterization of spectral nulls which can be simultaneously realized specializing to binary sequences we describe canonical mealy type state diagrams directed graphs with edges labeled by binary symbols for each set of realizable spectral nulls using the canonical diagrams we obtain a frequency domain characterization of the spectral null systems obtained by the technique of time domain interleaving

the correlated knowledge gradient for simulation optimization of continuous parameters using gaussian process regression we extend the concept of the correlated knowledge gradient policy for the ranking and selection of a finite set of alternatives to the case of continuous decision variables we propose an approximate knowledge gradient for problems with continuous decision variables in the context of a gaussian process regression model in a bayesian setting along with an algorithm to maximize the approximate knowledge gradient in the problem class considered we use the knowledge gradient for continuous parameters to sequentially choose where to sample an expensive noisy function in order to find the maximum quickly we show that the knowledge gradient for continuous decisions is a generalization of the efficient global optimization algorithm proposed in d r jones m schonlau and w j welch j global optim 13 1998 pp 455 492

constant weight and constant charge binary run length limited codes constant weight and constant charge binary sequences with constrained run length of zeros are introduced the weight and charge distribution for these sequences are found next recurrent and direct formulas for calculating the number of these sequences are obtained by considering these numbers of constant weight and constant charge run length limited rll sequences as coefficients of convergent power series generating functions are derived the fact that generating function for enumerating constant charge rll sequences does not have a closed form is proved subsequently implementation of encoding and decoding procedures using cover s enumerative scheme is shown based on the obtained results some examples such as enumeration of running digital sum rds constrained rll sequences or peak shifts control capability are also provided

trust region methods without using derivatives worst case complexity and the nonsmooth case trust region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts in particular they have been studied and applied for problems without using derivatives the analysis of trust region derivative free methods has focused on global convergence and they have been proven to generate a sequence of iterates converging to stationarity independently of the starting point most of such an analysis is carried out in the smooth case and moreover little is known about the complexity or global rate of these methods in this paper we start by analyzing the worst case complexity of general trust region derivative free methods for smooth functions for the nonsmooth case we propose a smoothing approach for which we prove global convergence and bound the worst case complexity effort for the special case of nonsmooth functions that result from the composition of smooth and nonsmooth convex components we show how to improve the existing resu

a derivative free multidimensional optimal search method using lucas number we consider a unimodal continuous function of n variables over an n dimensional rectangular parallelopiped and explain a search method to find the domain of uncertainty which contains the point that maximizes the function our search method does not require differentiability assumption the function is projected along each dimension an algorithm is developed to eliminate the subset of the domain at each step which does not contain the solution lucas numbers are used in the elimination process

new sequential and parallel derivative free algorithms for unconstrained minimization this paper presents sequential and parallel derivative free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest it is proved that under mild assumptions a sufficient decrease condition holds for a nonsmooth function based on this property the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value if the function is strictly differentiable at its limit points a sub sequence of points generated by the algorithm converges to a first order stationary point nabla f x 0 if the function is convex around its limit points convergence of a subsequence to a point with nonnegative directional derivatives on a set of search directions is ensured preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods

a derivative free nonmonotone line search technique for unconstrained optimization a tolerant derivative free nonmonotone line search technique is proposed and analyzed several consecutive increases in the objective function and also nondescent directions are admitted for unconstrained minimization to exemplify the power of this new line search we describe a direct search algorithm in which the directions are chosen randomly the convergence properties of this random method rely exclusively on the line search technique we present numerical experiments to illustrate the advantages of using a derivative free nonmonotone globalization strategy with approximated gradient type methods and also with the inverse sr1 update that could produce nondescent directions in all cases we use a local variation finite differences approximation to the gradient

a derivative free algorithm for least squares minimization we develop a framework for a class of derivative free algorithms for the least squares minimization problem these algorithms are designed to take advantage of the problem structure by building polynomial interpolation models for each function in the least squares minimization under suitable conditions global convergence of the algorithm is established within a trust region framework promising numerical results indicate the algorithm is both efficient and robust numerical comparisons are made with standard derivative free software packages that do not exploit the special structure of the least squares problem or that use finite differences to approximate the gradients

uobyqa unconstrained optimization by quadratic approximation uobyqa is a new algorithm for general unconstrained optimization calculations that takes account of the curvature of the objective function f say by forming quadratic models by interpolation therefore because no first derivatives are required each model is defined by n 1 n 2 values of f where n is the number of variables and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them a typical iteration of the algorithm generates a new vector of variables widetilde underline x t say either by minimizing the quadratic model subject to a trust region bound or by a procedure that should improve the accuracy of the model then usually f widetilde underline x t is obtained and one of the interpolation points is replaced by widetilde underline x t therefore the paper addresses the initial positions of the interpolation points the adjustment of trust region radii the calculation of widetilde underline x t in the two cases that have been mentioned and the selection of the point to be replaced further uobyqa works with the lagrange functions of the interpolation equations explicitly so their coefficients are updated when an interpolation point is moved the lagrange functions assist the procedure that improves the model and also they provide an estimate of the error of the quadratic approximation to f which allows the algorithm to achieve a fast rate of convergence these features are discussed and a summary of the algorithm is given finally a fortran implementation of uobyqa is applied to several choices of f in order to investigate accuracy robustness in the presence of rounding errors the effects of first derivative discontinuities and the amount of work the numerical results are very promising for n 20 but larger values are problematical because the routine work of an iteration is of fourth order in the number of variables

analysis of generalized pattern searches this paper contains a new convergence analysis for the lewis and torczon generalized pattern search gps class of methods for unconstrained and linearly constrained optimization this analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems specifically even if the objective function is discontinuous or extended valued the methods find a limit point with some minimizing properties simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm but also on the directions it uses and on the smoothness of the objective at the limit point in question the contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm and it gives previous results as corollaries

on codes satisfying mth order running digital sum constraints multi level sequences with a spectral null of order m at frequency f meaning that the power spectral density and its first 2m 1 derivatives vanish at f are characterized by finite state transition diagrams fstds whose edge labels satisfy bounds on the variation of the mth order running digital sum rds necessary and sufficient conditions for fstds with higher order null constraints at dc and at an arbitrary submultiple of the symbol frequency are derived analytical results are given concerning the performance of codes satisfying an mth order rds constraint on partial response channels specific code designs for quaternary channel inputs are presented the euclidean distance properties of this new class of codes aside from their spectral shaping properties are demonstrated

spectral metric spaces on extensions of c algebras we construct spectral triples on c algebraic extensions of unital c algebras by stable ideals satisfying a certain toeplitz type property using given spectral triples on the quotient and ideal our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones using our construction we find new spectral triples on the quantum 2 and 3 spheres giving a new perspective on these algebras in noncommutative geometry

global convergence of radial basis function trust region algorithms for derivative free optimization we analyze globally convergent derivative free trust region algorithms relying on radial basis function interpolation models our results extend the recent work of conn scheinberg and vicente siam j optim 20 2009 pp 387 415 to fully linear models that have a nonlinear term we characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first order critical points for the orbit algorithm of wild regis and shoemaker siam j sci comput 30 2008 pp 3197 3219 using orbit we present numerical results for different types of radial basis functions on a series of test problems we also demonstrate the use of orbit in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

binary transmission codes with higher order spectral zeros at zero frequency corresp a method is presented for designing binary channel codes in such a way that both the power spectral density function and its low order derivatives vanish at zero frequency the performance of the new codes is compared with that of channel codes designed with a constraint on the unbalance of the number of transmitted positive and negative pulses some remarks are made on the error correcting capabilities of these codes

derivative free optimization methods for optimizing stirrer configurations abstract in this paper a numerical approach for the optimization of stirrer configurations is presented the methodology is based on a flow solver and a mathematical optimization tool which are integrated into an automated procedure the flow solver is based on the discretization of the incompressible navier stokes equations by means of a fully conservative finite volume method for block structured boundary fitted grids for allowing a flexible discretization of complex stirrer geometries two derivative free optimization algorithms the dfo and condor are considered they are implementations of trust region based derivative free methods using multivariate polynomial interpolation both are designed to minimize smooth functions whose evaluations are considered to be expensive and whose derivatives are not available or not desirable to approximate an exemplary application for a standard stirrer configuration illustrates the functionality and the properties of the proposed methods it also gives a comparison of the two optimization algorithms

baseband trellis codes with a spectral null at zero a method is described for modifying classical n dimensional trellis codes to provide baseband codes that combine a spectral null at dc with significant coding gain the information rate of the classical code is decreased by one bit and this extra redundancy is used to keep the running digital sum bounded equivalently if the rate is held constant then twice as many signal points are needed causing a power penalty of 6 n db baseband trellis codes are presented for several information rates together with complete spectral plots and performance comparisons a method of constructing baseband codes with multiple spectral nulls is also described

sensitivity of optimal shapes of artificial grafts with respect to flow parameters abstract the difficulties arising in the numerical solution of pde constrained shape optimization problems are manifold key ingredients are the optimization strategy and the shape deformation method furthermore the robustness of the optimal shape with respect to simulation parameters is of great interest in this paper we consider fluid flows described by the incompressible navier stokes equations previous studies on artificial bypass grafts indicated the need for specific constitutive models to account for the non newtonian nature of blood in particular the constitutive model was shown to affect the solution of the shape optimization problem we employ a shape optimization framework that couples a finite element solver with quasi newton type optimizers and a bezier spline shape parametrization to compute derivatives of the optimal shapes with respect to viscosity we transform the entire optimization framework by combining the automatic differentiation tools adifor2 and tapenade we demonstrate the impact of the geometry parametrization and of geometric constraints on the optimization outcome finally we employ the transformed framework to compute the sensitivity of the optimal shape of bypass grafts with respect to kinematic viscosity the resulting sensitivities predict very accurately the influence of viscosity changes on the optimal shape the proposed methodology provides a powerful tool to further investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters

binary convolutional codes with application to magnetic recording calderbank heegard and ozarow 1 have suggested a method of designing codes for channels with intersymbol interference such as the magnetic recording channel these codes are designed to exploit intersymbol interference the standard method is to minimize intersymbol interference by constraining the input to the channel using run length limited sequences calderbank heegard and ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function 1 d n 2 where the channel inputs are constrained to be pm 1 this problem is considered here channel inputs are generated using a nontrivial coset of a binary convolutional code the coset is chosen to limit the zero run length of the output of the channel and so maintain clock synchronization the minimum squared euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code an interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic

self correcting geometry in model based algorithms for derivative free unconstrained optimization several efficient methods for derivative free optimization are based on the construction and maintenance of an interpolation model for the objective function most of these algorithms use special geometry improving iterations where the geometry poisedness of the underlying interpolation set is made better at the cost of one or more function evaluations we show that such geometry improvements cannot be completely eliminated if one wishes to ensure global convergence but we also provide an algorithm where such steps occur only in the final stage of the algorithm where criticality of a putative stationary point is verified global convergence for this method is proved by making use of a self correction mechanism inherent to the combination of trust regions and interpolation models this mechanism also throws some light on the surprisingly good numerical results reported by fasano morales and nocedal optim methods softw 24 2009 pp 145 154 for a method where no care is ever taken to guarantee poisedness of the interpolation set

a performance metric for codes with a high order spectral null at zero frequency sum variance is a well known metric for assessing the performance of dc free codes first order spectral null codes however as we show in this paper it is unsuitable for comparing the magnitude of spectral components of high order spectral null hosn codes at low frequencies in this paper we introduce a new performance metric for evaluating the spectrum compression of arbitrarily hosn codes around zero frequency we call this metric the low frequency spectrum weight lfsw we show that the asymptotic low frequency spectral components of kth order spectral null codes k spl ges 1 are exclusively determined by the order k and the lfsw and that the lfsw equals the zero frequency value in the spectrum of the corresponding sequence of kth order running digital sum values we derive this result for symbol by symbol encoding and then extend it to block hosn codes we then derive a closed form expression for the lfsw of hosn codes constructed through state independent encoding closed form expressions for lfsw of first order zero disparity codes and for the asymptotic lsfw of maxentropic dc free sequences are also given

a note on the shannon capacity of run length limited codes corresp it is proven that 100 percent efficient fixed rate codes for run length limited rll d k and rll charge constrained d k c channels are possible in only two eases namely d k c 0 1 1 and 1 3 3 specifically the binary shannon capacity of rll d k constrained systems is shown to be irrational for all values of d k 0 leq d for rll charge constrained systems with parameters d k c the binary capacity is irrational for all values of d k c 0 leq d except 0 1 1 and 1 3 3 which both have binary capacity 1 2

information rates and power spectra of digital codes the encoding of independent data symbols as a sequence of discrete amplitude real variables with given power spectrum is considered the maximum rate of such an encoding is determined by the achievable entropy of the discrete sequence with the given constraints an upper bound to this entropy is expressed in terms of the rate distortion function for a memoryless finite alphabet source and mean square error distortion measure a class of simple dc free power spectra is considered in detail and a method for constructing markov sources with such spectra is derived it is found that these sequences have greater entropies than most codes with similar spectra that have been suggested earlier and that they often come close to the upper bound when the constraint on the power spectrum is replaced by a constraint on the variance of the sum of the encoded symbols a stronger upper bound to the rate of dc free codes is obtained finally the optimality of the binary biphase code and of the ternary bipolar code is decided

voiculescu theorem sobolev lemma and extensions of smooth algebras we present the analytic foundation of a unified b d f extension functor ext on the category of noncommutative smooth algebras for any frechet operator ideal k combining the techniques devised by arveson and voiculescu we generalize voiculescu s theorem to smooth algebras and frechet operator ideals a key notion involved is r smoothness which is verified for the algebras of smooth functions via a noncommutative sobolev lemma the groups ext are computed for many examples

construction and characteristics of codewords and pulse shapes that satisfy spectral constraints constrained sequence codes and pulse shapes are used to control signal characteristics in digital communication systems this letter outlines a technique to construct codewords and pulse shapes to meet specified spectral constraints based on a block coded representation of the signal it is shown how symbol values can be evaluated to satisfy the most significant terms of a fourier series representation of the desired spectral response several properties of the resulting signals are given and examples are presented to confirm the validity of this approach

the mean power spectral density of markov chain driven signals many signals ocurring in data communications can be described by markovian models expressions for the mean power spectral density function of a signal driven by a stationary discrete parameter finite state markov chain are derived and discussed both for irreducible and reducible chains two examples are presented to illustrate the application of the results

dc constrained codes from hadamard matrices the authors consider the construction of balanced error correcting codes with distance close to half of the block length and bounded running digital sum use of these codes in cascade constructions allows derivation of a number of classes of dc constrained codes of various lengths the mathematical framework underlying the code construction is the theory of incomplete exponential sums essentially the authors consider a class of codes formed by values of legendre symbols of polynomials of bounded degree on the set of residues modulo a prime in particular taking linear polynomials they obtain the hadamard codes applying well known estimates of the exponential sums they compute the code parameters and prove that the proposed codes are in fact dc constrained

codes with a multiple spectral null at zero frequency codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated and necessary and sufficient conditions for the spectral properties are given in particular for the most common balanced alphabets a complete characterization is furnished for the class of symbol by symbol codes with vanishing spectrum and second derivative the fundamental results are extended to block codes and suitable criteria for their design are given specific examples of binary and ternary block codes are presented their performance in terms of low power content at low frequencies compares advantageously with that of some conventional codes

a derivative free trust region algorithm for the optimization of functions smoothed via gaussian convolution using adaptive multiple importance sampling in this paper we consider the optimization of a functional f defined as the convolution of a function f with a gaussian kernel we propose this type of objective function for the optimization of the output of complex computational simulations which often present some form of deterministic noise and need to be smoothed for the results to be meaningful we introduce a derivative free algorithm that computes trial points from the minimization of a regression model of the noisy function f over a trust region the regression model is constructed from function values at sample points that are chosen randomly around iterates and trial points of the algorithm the weights given to the individual sample points in the regression problem are obtained according to an adaptive multiple importance sampling strategy this has two advantages first it makes it possible to reuse all noisy function values collected over the course of the optimization second the resulting regression model converges to the second o

the dirac operator on compact quantum groups for the q deformation gq 0 q 1 of any simply connected simple compact lie group g we construct an equivariant spectral triple which is an isospectral deformation of that defined by the dirac operator d on g our quantum dirac operator dq is a unitary twist of d considered as an element of ug cl g the commutator of dq with a regular function on gq consists of two parts one is a twist of a classical commutator and so is automatically bounded the second is expressed in terms of the commutator of the associator with an extension of d we show that in the case of the drinfeld associator the latter commutator is also bounded

pattern search methods for user provided points application to molecular geometry problems this paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry the goal is to find a configuration of a cluster or a molecule with minimum total energy the minimization problems in this class of molecular geometry problems have no constraints and the objective function is smooth the difficulties arise from the existence of several local minima and especially from the expensive function evaluation total energy and the possible nonavailability of first order derivatives we introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives numerical results for a particular instance of this new class of pattern search methods are presented showing the promise of our approach the new pattern search methods can be used in any other context where there is a user provided scheme to generate points leading to a potential objective function decrease