scattering amplitude and algorithm for solving the inverse scattering problem for a class of non convex obstacles abstract consider a class r1 of non convex smooth obstacles b such that there exist a number of directions sj sj j 1 2 n uniformly distributed on the unit sphere with the following property the rays incident in the directions sj are reflected by the obstacle once according to geometrical optics laws it is also assumed that the obstacles in r1 satisfy a non trapping condition for b r1 the high frequency asymptotics of the scattering amplitude f s s0 k is studied and analytical formulas are derived for determining the surface of the obstacle from the scattering amplitude known at a fixed large wave number k and for the directions s0 and s of the incident and scattered waves in solid angles around the directions sj j 1 n the investigation of the scattering amplitude and the solution to the inverse scattering problem are based on approximate formulas for f derived from the kirchhoff approximation and these formulas are justified for non convex obstacles of class r1

on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

a characterization of dual quermassintegrals and the roots of dual steiner polynomials abstract let m 1 r 0 0 r 1 r m be a tuple of distinct real numbers and n 2 we provide a characterization of those tuples 0 1 m of real numbers such that there exist n dimensional star bodies k l with w r j k l j j 0 m where w r k l denotes the r th dual relative quermassintegral of k and l this may be regarded as an analogue within the dual brunn minkowski theory of shephard s classification of quermassintegrals of two convex bodies it turns out that the characterization of dual quermassintegrals is related to the moment problem and based on this relation we also derive new determinantal inequalities among the dual quermassintegrals moreover this characterization will be the key tool in order to investigate structural properties of the set of roots of dual steiner polynomials of star bodies

a study of boundedness in probabilistic normed spaces abstract it was shown in lafuerza guillen rodriguez lallena and sempi 1999 8 that uniform boundedness in a serstnev pn space v named boundedness in the present setting of a subset a v with respect to the strong topology is equivalent to the fact that the probabilistic radius r a of a is an element of d here we extend the equivalence just mentioned to a larger class of pn spaces namely those pn spaces that are topological vector spaces briefly tv spaces but are not serstnev pn spaces we present a characterization of those pn spaces whether they are tv spaces or not in which the equivalence holds then a characterization of the archimedeanity of triangle functions of type t l is given this work is a partial solution to a problem of comparing the concepts of distributional boundedness d bounded in short and that of boundedness in the sense of associated strong topology

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

a study of boundedness in probabilistic normed spaces probabilistic normed spaces have been redefined by alsina schweizer and sklar we give a detailed analysis of various boundedness notions for linear operators between such spaces and we study the relationship among them and also with the notion of continuity

robust global optimization with polynomials we consider the optimization problems maxz minx k p z x and minx k maxz p z x where the criterion p is a polynomial linear in the variables z the set can be described by lmis and k is a basic closed semi algebraic set the first problem is a robust analogue of the generic sdp problem maxz p z whereas the second problem is a robust analogue of the generic problem minx k p x of minimizing a polynomial over a semi algebraic set we show that the optimal values of both robust optimization problems can be approximated as closely as desired by solving a hierarchy of sdp relaxations we also relate and compare the sdp relaxations associated with the max min and the min max robust optimization problems

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

finding numerical derivatives for unstructured and noisy data by multiscale kernels the recently developed multiscale kernel of r opfer adv comput math 25 2006 pp 357 380 is applied to approximate numerical derivatives the proposed method is truly mesh free and can handle unstructured data with noise in any dimension the method of tikhonov and the method of l curve are employed for regularization no information about the noise level is required an error analysis is provided in a general setting for all dimensions numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods

numerical differentiation from a viewpoint of regularization theory in this paper we discuss the classical ill posed problem of numerical differentiation assuming that the smoothness of the function to be differentiated is unknown using recent results on adaptive regularization of general ill posed problems we propose new rules for the choice of the stepsize in the finite difference methods and for the regularization parameter choice in numerical differentiation regularized by the iterated tikhonov method these methods are shown to be effective for the differentiation of noisy functions and the order optimal convergence results for them are proved

cyclic convex bodies and optimization moment problems abstract we deal with two discrete moment problems first deciding when a fixed element of r d is the vector of d first moments for some discrete probability distribution on a given interval a b feasibility moment problem and second maximizing minimizing a given linear combination of moments on the set of discrete probability distributions on a b whose d first moments are given optimization moment problem these problems are linked with the cyclic body which is the union of all cyclic polytopes on a b the cyclic polytopes have been extensively studied and their combinatorial and geometric properties are noteworthy the cyclic body also has interesting geometric properties we totally determine its facial structure and supporting hyperplanes and we construct an external representation by means of linear inequality systems whose coefficients are symmetric polynomials depending on parameters these tools allow us to solve the mentioned moment problems by using linear semi infinite programming and we obtain a representation of non negative polynomials over a b as well

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

the numerical evaluation of hadamard finite part integrals a quadrature rule is described for the numerical evaluation of hadamard finite part integrals with a double pole singularity within the range of integration the rule is based upon the observation that such an integral is the derivative of a cauchy principal value integral

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

wavelet based filters for accurate computation of derivatives let f x be a smooth function whose derivative of a given order must be computed the signal f x is affected by two kinds of perturbation the perturbation caused by the presence of the machine epsilon m of the computer may be considered to be an extremely high frequency noise of very small amplitude the way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient the second perturbation caused by the presence of noise requires first the signal to be treated in some way it is the purpose of this work to construct a wavelet based band pass filter that deals with the two cited perturbations simultaneously in effect our wavelet acts like a smoothed difference quotient whose stepsize is of the same order as that of the usual difference quotient moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise

a modified method for high order numerical derivatives in this paper we propose a new regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence estimate under an appropriate choice of the regularization parameter is obtained some interesting numerical tests show that the proposed method is effective and stable

calculating singular integrals as an ill posed problem applying numerical quadrature to singular integrals in a straightforward way leads to uncontrolled instability with respect to data errors in this paper we describe how to control this instability

probabilistic norms and convergence of random variables abstract we prove that the probabilistic norms of suitable probabilistic normed spaces induce convergence in probability l p convergence and almost sure convergence

voronoi cells via linear inequality systems abstract the theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of voronoi cells and diagrams which constitute basic tools of computational geometry as shown by voigt and weis in 2010 the voronoi cells of a given set of sites t which provide a tesselation of the space called voronoi diagram when t is finite are solution sets of linear inequality systems indexed by t this paper exploits systematically this fact in order to obtain geometrical information on voronoi cells from sets associated with t convex and conical hulls tangent cones and the characteristic cones of their linear representations the particular cases of t being a curve a closed convex set and a discrete set are analyzed in detail we also include conclusions on voronoi diagrams of arbitrary sets

research on some problems for nonlinear operators abstract in this paper we discuss some problems for nonlinear operators in the menger pn space meanwhile we prove existence theorems for the solution of an operator equation and obtain some new results by means of the theory of topological degree for the z p s space

countable products of probabilistic normed spaces abstract countable products of probabilistic normed spaces are introduced and studied in particular a comparison is made with the analogous constructions for probabilistic metric spaces

on computing first and second order derivative spectra enhancing resolution in spectral response and an ability to differentiate spectral mixing in delineating the endmembers from the spectral response are central to the spectral data analysis first and higher order derivatives analysis of absorbance and reflectance spectral data is commonly used techniques in differentiating the spectral mixing but high sensitivity of derivative to the noise in data is a major problem in the robust estimation of derivative of spectral data an algorithm of robust estimation of first and second order derivative spectra from evenly spaced noisy normal spectral data is proposed the algorithm is formalized in the framework of an inverse problem where based on the fundamental theorem of calculus a matrix equation is formed using a volterra type integral equation of first kind a regularization technique where the balancing principle is used in selecting a posteriori optimal regularization parameter is designed to solve the inverse problem for robust estimation of first order derivative spectra the higher order derivative spectra are obtained while using the algorithm in sequel the algorithm is tested successfully with synthetically generated spectral data contaminated with additive white gaussian noise and also with real absorbance and reflectance spectral data for fresh and sea water respectively robust and efficient algorithm of computing first and second orders derivative spectra is designed the algorithm is based on the premise of inverse problem where a robust solution is achieved via regularization it couples discrepancy and balancing principles to determine regularization parameter and implement stopping rule respectively numerical experiments on noisy data suggest superiority of the method over finite difference and fourier transform based techniques the proposed method is applied on absorbance and reflectance spectral data for fresh and sea water

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

differentiation by integration with jacobi polynomials in this paper the numerical differentiation by integration method based on jacobi polynomials originally introduced by mboup et al 19 20 is revisited in the central case where the used integration window is centered such a method based on jacobi polynomials was introduced through an algebraic approach 19 20 and extends the numerical differentiation by integration method introduced by lanczos 1956 21 the method proposed here rooted in 19 20 is used to estimate the nth email protected n order derivative from noisy data of a smooth function belonging to at least c n 1 q email protected n in 19 20 where the causal and anti causal cases were investigated the mismodelling due to the truncation of the taylor expansion was investigated and improved allowing a small time delay in the derivative estimation here for the central case we show that the bias error is o h q 2 where h is the integration window length for email protected c n q 2 in the noise free case and the corresponding convergence rate is o d q 1 n 1 q where d is the noise level for a well chosen integration window length numerical examples show that this proposed method is stable and effective

discrete probability distributions and moment problem numerical aspects discrete probability distributions with finite range generated by maximum entropy technique and constrained by first moments have been considered numerical aspects as stability of parameter s computation entropy variation spectral properties of involved hankel s matrices have been studied the ill conditioning of hankel s matrices suggests minimization methods like newton s should be unreliable as a high number of moments are kept while methods relying upon relative error estimate between assigned and calculated moments should be possible candidates

on the inverse diffraction problem abstract it is proved that for a convex reflecting scatterer the support function is uniquely determined by the scattering amplitude in particular by the backscattering amplitude the support function determines the position orientation and the shape of the scatterer uniquely stability of the solution is discussed

semidefinite programming for min max problems and games we consider two min max problems 1 minimizing the supremum of finitely many rational functions over a compact basic semi algebraic set and 2 solving a 2 player zero sum polynomial game in randomized strategies with compact basic semi algebraic sets of pure strategies in both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations in the spirit of the moment approach developed in lasserre siam j optim 11 796 817 2001 math program b 112 65 92 2008 this provides a unified approach and a class of algorithms to compute nash equilibria and min max strategies of several static and dynamic games each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation and sometimes even for finite convergence a behavior similar to what was observed in polynomial optimization

peridynamic differential operator and its applications abstract the nonlocal peridynamic theory has been proven extremely robust for predicting damage nucleation and propagation in materials under complex loading conditions its equations of motion originally derived based on the principle of virtual work do not contain any spatial derivatives of the displacement components thus their solution does not require special treatment in the presence of geometric and material discontinuities this study presents an alternative approach to derive the peridynamic equations of motion by recasting navier s displacement equilibrium equations into their nonlocal form by introducing the peridynamic differential operator also this operator permits the nonlocal form of expressions for the determination of the stress and strain components the capability of this differential operator is demonstrated by constructing solutions to ordinary partial differential equations and derivatives of scattered data as well as image compression and recovery without employing any special filtering and regularization techniques

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

the topological structure of a certain menger space in this paper we will reconsider the topological structure of menger probabilistic normed spaces briefly pn spaces under the t norm m we will prove that this topology is compatible with the topology induced by a countable and separating family of semi norms and hence the well known theorems of classical functional analysis such as the principle of uniform boundedness open mapping and closed graph theorems are valid in this context also we will meanwhile obtain a method by which one may construct easily a large class of pn spaces finally using this method we see that a certain subspace of bounded linear operators between pn spaces i e the class of strongly bounded linear operators has a natural pn structure

representability of convex sets by analytical linear inequality systems the solution sets of analytical linear inequality systems posed in the euclidean space form a transition class between the polyhedral convex sets and the closed convex sets which are representable by means of linear continuous systems the constraint systems of many semi infinite programming problems are analytical and their feasible sets retain geometric properties of the polyhedral sets which are useful in the numerical treatment of such kind of optimization problems the euclidean closed n dimensional balls admit analytical representation if and only if n 3 this paper solves in a negative way the analytical representation problem for a wide class of n dimensional convex sets with n 3 which includes quasi polyhedral sets and smooth convex bodies

interpreting potential field anomaly of an isolated source of regular geometry revisited abstracti have developed an improved practical method for interpreting a symmetrical shaped potential field anomaly due to an isolated source body of regular geometric configuration the method uses the first order horizontal derivative of the logarithmically transformed absolute value of the anomaly in estimating the source body parameters such as the location depth of burial and shape factor to tackle noise in data a regularization technique is designed which ensures a robust estimate of the first order derivative of logarithmically transformed data the regularization technique uses an optimal value of regularization parameter that although noise dependent requires no a priori knowledge of the noise level in the data a graphical method is designed to determine an optimal value of the regularization parameter from the position of the local minimum of a specially defined functional with respect to the regularization parameters numerical tests have been conducted on the noise contaminated synthe

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

estimates of derivatives of random functions i abstract approximate formulas for the derivatives of functions of several variables contaminated by noise are given error estimates are derived the formulas are optimal in some sense

numerical evaluation of hypersingular integrals abstract boundary integral equations which employ integrals which exist only if defined in the cauchy principal value sense or as the hadamard finite part are currently used with success to solve many two and three dimensional problems of applied mechanics we will recall definitions and main properties of these integrals examine some numerical approaches for their evaluation and present several new results

optimality conditions and finite convergence of lasserre s hierarchy lasserre s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally this paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of lasserre s hierarchy our main results are i lasserre s hierarchy has finite convergence when the constraint qualification strict complementarity and second order sufficiency conditions hold at every global minimizer under the standard archimedean condition the proof uses a result of marshall on boundary hessian conditions ii these optimality conditions are all satisfied at every local minimizer if a finite set of polynomials which are in the coefficients of input polynomials do not vanish at the input data i e they hold in a zariski open set this implies that under archimedeanness lasserre s hierarchy has finite convergence generically

numerical differentiation with noisy signal the aim of this paper is to present a new numerical method which ables one to filter and compute numerical derivatives of a function whose values are known in some points from experimental measurements inducing noisy data we use a piecewise cubic spline interpolation to generate a function whose fourier coefficients give an approximation of the numerical derivatives we are looking for error and stability analysis of this numerical algorithm are provided numerical results are presented for data smoothing and for the first and second derivatives computed from noisy data they show that this method gives good numerical results comparison with other methods is done

remarks on the gauss quadrature rule for a particular class of finite part integrals the method of finite part hadamard type or hypersingular integrals is already a standard approach in applied numerical methods in engineering and mainly in acoustics fluid dynamics elasticity and fracture mechanics here we study the convergence of the non classical possessing a negative node outside the integration interval as well as a negative weight corresponding to this node gauss legendre quadrature rule for finite part integrals on the interval 0 1 with the weight function 1 y and we prove its validity for integrands possessing a continuous first derivative the case of integrands possessing a higher order derivative is also considered and the rate of convergence is established these results are based on the derivation of a bound for the negative node in the above quadrature rule and further on the theory of convergence of the classical gauss quadrature rule numerical results verifying the results of this paper are also displayed the present results can be generalized to the gauss laguerre similarly non classical quadrature rule for finite part integrals on the interval 0 as well as to the two special but important in engineering applications non classical gaussian quadrature rules for cauchy type principal value integrals and mangler type finite part integrals have been proposed by tsamasphyros and dimou

fourier truncation method for high order numerical derivatives we consider a classical ill posed problem numerical differentiation with a new method we propose fourier truncation method to compute high order numerical derivatives a holder type stability estimate is obtained a numerical implementation is described numerical examples show that the proposed method is effective and stable

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

numerical differentiation by a fourier extension method with super order regularization abstract based on the idea of fourier extension we develop a new method for numerical differentiation the tikhonov regularization method with a super order penalty term is presented to deal with the illposdness of the problem and the regularization parameter can be chosen by a discrepancy principle for various smooth conditions the solution process of the new method is uniform and order optimal error bounds can be obtained numerical experiments are also presented to illustrate the effectiveness of the proposed method

numerical differentiation for the second order derivatives of functions of two variables a regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article we prove the existence and uniqueness of the solution to this problem provide a constructive scheme for the solution which is based on bi harmonic green s function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter the efficiency of the constructive scheme is shown by some numerical examples

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

regularized nonlinear acceleration we describe a convergence acceleration technique for generic optimization problems our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method the weights in this average are computed via a simple linear system whose solution can be updated online this acceleration scheme runs in parallel to the base algorithm providing improved estimates of the solution on the fly while the original optimization method is running numerical experiments are detailed on classical classification problems

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

entanglement criteria for dicke states dicke states are a family of multi qubit quantum states with interesting entanglement properties and they have been observed in many experiments we construct entanglement witnesses for detecting genuine multiparticle entanglement in the vicinity of these states we use the approach of positive partial transpose mixtures to derive the conditions analytically for nearly all cases our criteria are stronger than all conditions previously known

numerical differentiation for two dimensional scattered data in this paper we propose a regularization method for numerical differentiation of two dimensional mildly scattered input data a regularized solution is constructed based on the green s function the existence and uniqueness of the regularized solution are proved and the convergence estimates are provided under a simple choice of regularization parameter numerical results show that our method is quite effective one of the advantages for our proposed method is that the basis functions are independent of input data

three algorithms for hadamard finite part integrals and fractional derivatives abstract three algorithms for the evaluation of the hadamard finite part integral of the form 1 1 f t 1 t 1 where is a positive non integer are described one algorithm is based on a knowledge of the chebyshev series expansion of f on 1 1 the other two on polynomial interpolations to f at the zeros of tn the chebyshev polynomial of the first kind convergence theorems are given for each algorithm and each algorithm is demonstrated numerically

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

a wavelet galerkin method for high order numerical differentiation numerical differentiation is a classical ill posed problem in this paper we propose a wavelet galerkin method for high order numerical differentiation by an appropriate choice of the regularization parameter an order optimal stability estimate of holder type is obtained some numerical examples show that the method is effective and stable

new green s function for stress field and a note of its application in quantum wire structures abstract the elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two dimensional hydrostatic inclusion subjected to plane strain conditions the stresses in such a hydrostatic inclusion can be effectively calculated by employing the green s functions developed by downes and faux which tend to be more efficient than the conventional method based on the green s function for the displacement field in this study downes and faux s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains an explicit green s function solution which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly is obtained in a closed form here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived green s functions in the case of uniform eigenstrain the formulae may be simplified to contour integrals by green s theorem however special care is required when green s theorem is applied for the interior field the proposed green s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non uniform eigenstrain two examples concerning circular inclusions are investigated a linearly distributed eigenstrain is attempted in the first example resulting in a linear interior stress field the second example solves a circular thermal inclusion where both the interior and exterior stress fields are obtained simultaneously

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a scheme for stable numerical differentiation a method for stable numerical differentiation of noisy data is proposed the method requires solving a volterra integral equation of the second kind this equation is solved analytically in the examples considered its solution is computed analytically some numerical results of its application are presented these examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods in particular than variational regularization

the numerical evaluation of a 2 d cauchy principal value integral arising in boundary integral equation methods in this paper we consider the problem of computing 2 d cauchy principal value integrals of the form

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

stability of additive mappings in non archimedean fuzzy normed spaces in this paper we introduce a notion of a non archimedean fuzzy norm and study the stability of the cauchy equation in the context of non archimedean fuzzy spaces in the spirit of hyers ulam rassias gavruta as a corollary the stability of the jensen equation is established we indeed present an interdisciplinary relation between the theory of fuzzy spaces the theory of non archimedean spaces and the theory of functional equations

high order numerical derivatives for one dimensional scattered noisy data based on the smoothing spline approximation in this paper we propose a regularization method for computing high order numerical derivatives from one dimensional noisy data the convergence rates under two different choices of the regularization parameter are obtained numerical examples show that the proposed method is effective and stable

letter to the editor numerical differentiation for high orders by an integration method this paper mainly studies the numerical differentiation by integration method proposed first by lanczos new schemes of the lanczos derivatives are put forward for reconstructing numerical derivatives for high orders from noise data the convergence rate of these proposed methods is o d 4 n 4 as the noise level d 0 numerical examples show that the proposed methods are stable and efficient

justification of geometrical optics for non convex obstacles using new results about the propagation of singularities we shall show that it suffices to approximate the exact solution of the helmholtz equation locally for justification of geometrical optics the methods which we describe allow us to show that the geometrical optics approximation converges to the exact solution of the helmholtz equation for a large class of obstacles apart from some technical assumptions concerning the behaviour of the reflected rays we only need that solutions of the wave equation decay exponentially outside of the obstacle this is true if rays are not trapped cf s in this paper however we shall not treat the most general case but approximate the solution of the helmholtz equation only at points x satisfying the additional conditions a2 and a3 given below a more general case will be considered in a sequel to this paper let b be a bounded open set with c boundary in r3 and q r3 b the solution of au k2u 0 1 1

two regularization strategies for an evolutional type inverse heat source problem this paper investigates an evolutional type inverse problem of determining an unknown heat source function in heat conduction equations when the solution is known in a discrete point set being different from other ordinary inverse source problems which often rely on only one variable the unknown coefficient in this paper depends not only on the space variable x but also on time t two regularization strategies which are called the time semi discrete scheme tsds and the integral reconstruction scheme irs respectively are proposed to deal with such a problem by the tsds the inverse problem is transformed into a sequence of stationary inverse problems and the unknown heat source is reconstructed layer by layer while the irs is to recover the source function from the situation as a whole both theoretical and numerical studies are provided two numerical algorithms on the basis of the landweber iteration are designed and some typical numerical experiments are performed in this paper the numerical results show that the proposed methods are stable and the unknown heat source is recovered very well

taylor type 1 step ahead numerical differentiation rule for first order derivative approximation and znn discretization in order to achieve higher computational precision in approximating the first order derivative and discretize more effectively the continuous time zhang neural network znn a taylor type numerical differentiation rule is proposed and investigated in this paper this rule not only greatly remedies some intrinsic weaknesses of the backward and central numerical differentiation rules but also overcomes the limitation of the lagrange type numerical differentiation rules in znn discretization in addition a formula is proposed to obtain the optimal step length of the taylor type numerical differentiation rule moreover based on the proposed numerical differentiation rule the stability convergence and residual error of the taylor type discrete time znn dtznn are analyzed numerical experimental results further substantiate the efficacy and advantages of the proposed taylor type numerical differentiation rule for first order derivative approximation and znn discretization a formula is proposed to approximate the first order derivative an optimal step length rule for the proposed formula is investigated a taylor type znn model is derived for time varying matrix inversion

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

filtered legendre expansion method for numerical differentiation at the boundary point with application to blood glucose predictions let f 1 1 r be continuously differentiable we consider the question of approximating f 1 from given data of the form t j f t j j 1 m where the points t j are in the interval 1 1 it is well known that the question is ill posed and there is very little literature on the subject known to us we consider a summability operator using legendre expansions together with high order quadrature formulas based on the points t j s to achieve the approximation we also estimate the effect of noise on our approximation the error estimates both with or without noise improve upon those in the existing literature and appear to be unimprovable the results are applied to the problem of short term prediction of blood glucose concentration yielding better results than other comparable methods

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

an algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy in this paper we introduce an algorithm and a computer code for numerical differentiation of discrete functions the algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points the algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas in particular when the derivatives are required with high approximation accuracy moreover the given matlab computer code can be implemented to solve boundary value ordinary and partial differential equations with high numerical accuracy the numerical technique is based on the undetermined coefficient method in conjunction with taylor s expansion to avoid the difficulty of solving a system of linear equations an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions this is done by using an explicit closed formula for the vandermonde matrix inverse moreover the code is designed to give a unified approximation order throughout the given domain a numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code it is found that the method and the code work properly for any degree of derivative and any order of accuracy

several nonlinear operator problems in the menger pn space abstract in this paper we investigate some nonlinear operator problems and the theory of topological degree meanwhile we obtain some new fixed point theorems in the m pn space

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

galerkin method with trigonometric basis on stable numerical differentiation abstract this paper considers the p p 1 2 3 order numerical differentiation on function y in 0 2 they are transformed into corresponding fredholm integral equation of the first kind computational schemes with analytic solution formulas are designed using galerkin method on trigonometric basis convergence and divergence are all analysed in corollaries 5 1 5 2 and a priori error estimate is uniformly obtained in theorem 6 1 7 1 7 2 therefore the algorithm achieves the optimal convergence rate o 2 2 1 1 2 or 1 with periodic sobolev source condition of order 2 p besides we indicate a noise independent a priori parameter choice when the function y possesses the form of k 0 p 1 a k t k k 1 n 1 b k cos k t k 1 n 2 c k sin k t b n 1 c n 2 0 in particular in numerical differentiations for functions above good filtering effect error approaches 0 is displayed with corresponding parameter choice in addition several numerical examples are given to show that even derivatives with discontinuity can be recovered well

a new algorithm for cauchy principal value and hadamard finite part integrals an algorithm for the approximate evaluation of integrals defined by cauchy principal value or by hadamard finite part has been proposed the convergence of the procedure is proved the stability of the algorithm is considered and some numerical examples are given

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur