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

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

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

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

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

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

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

finite frequency model reduction of two dimensional digital filters this technical note is concerned with the model reduction problem of two dimensional 2 d digital filters over finite frequency ranges the 2 d digital filter is described by the fornasini marchesini local state space fm lss model with the aid of the generalized kalman yakubovich popov gkyp lemma for 2 d systems sufficient conditions for the finite frequency model reduction problem are derived compared with full frequency methods the proposed finite frequency method can get a better approximation performance over finite frequency ranges an example is given to demonstrate the effectiveness of the proposed method

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

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

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

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

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

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

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

generalized kalman yakubovich popov lemma for 2 d fm lss model this paper is concerned with the generalized kalman yakubovich popov kyp lemma for 2 d fornasini marchesini local state space fm lss systems by carefully analyzing the feature of the states in 2 d fm lss models a linear matrix inequality lmi characterization for a rectangular finite frequency region is constructed and then by combining this characterization with s procedure a generalized kyp lemma is proposed for 2 d fm lss models as applications of the developed kyp lemma new conditions are further derived for designing controllers guaranteeing finite frequency positive realness of the closed loop systems where both state feedback and dynamic output feedback control laws are considered finally two illustrative examples clearly show the effectiveness of the proposed results the main contribution of the paper is trifold 1 a novel characterization for a finite frequency region has been developed 2 a generalized kyp lemma has been proposed including the existing bounded realness and positive realness results as special cases and 3 systematic methods for finite frequency positive real control of 2 d systems have been presented

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

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

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

prime and primitive ideals of a class of iterated skew polynomial rings hodges and levasseur described the primitive spectra of quantized coordinate rings of sl3 14 and then of sln 15 these results established a close connection between primitive ideals torus action and poisson geometry the proofs relied on explicit computations involving generators and relations subsequently joseph generalized the hodges levasseur program to semisimple algebraic quantum groups 17 hodges et al then expanded joseph s work to include certain multiparameter deformations 16 these papers rely less on concrete calculations and more on deeper more conceptual techniques it is a natural and important question then as to how the preceding theory might apply to other algebras particularly other algebras arising in the study of quantum groups goodearl and letzter established parallel results for certain iterated skew polynomial rings 13 with application to quantum weyl algebras and to some quantum coordinate rings

adaptive h control in finite frequency domain for uncertain linear systems this paper is concerned with the problem of adaptive h controller design in finite frequency domains for uncertain linear systems the uncertainties are assumed to be time invariant unknown but bounded which appear affinely in the matrices of system models an adaptive mechanism is introduced to construct a novel finite frequency h controller with time varying gains by using lyapunov theory and parseval s theorem the controller design conditions are given in terms of a set of linear matrix inequalities lmis it is shown that the proposed finite frequency controller with time varying gains can achieve better h performance than the traditional ones with fixed gains finally a numerical example of the f 18 aircraft model is given to illustrate the presented theoretical results

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

finite frequency model reduction of takagi sugeno fuzzy systems this paper considers the model reduction problem for continuous time takagi sugeno t s fuzzy systems different from existing full frequency methods a finite frequency model reduction method is proposed in this paper the proposed method can get a better approximation performance when input signals belong to a finite frequency domain to this end a finite frequency h infty performance index is first defined then a sufficient finite frequency performance analysis condition is derived by the aid of parseval s theorem and quadratic functions based on this condition and projection lemma three model reduction algorithms for t s fuzzy systems with input signals in low frequency middle frequency and high frequency domain are obtained respectively finally an example is given to illustrate the effectiveness of the proposed method

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

reduced order aggregated models for bilinear time invariant dynamical systems a necessary and sufficient condition for a bilinear time invariant dynamical system to admit an aggregated reduced order bilinear model is given

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

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

families of moment matching based low order approximations for linear systems the problem of finding a framework for linear moment matching based approximation of linear systems is discussed it is shown that it is possible to define five equivalent notions of moments i e one from a complex s domain point of view two using krylov projections and two from a time domain perspective based on these notions classes of parameterized reduced order models that achieve moment matching are obtained we analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models we find the subclasses of models of orders lower than the number of matched moments i e we find the sets of parameters that allow for pole zero cancellations to occur furthermore we compute the subclass of minimal order model s the minimal order is equal to half of the number of matched moments i e generically the largest number of possible pole zero cancellations is half of the number of matched moments finally we present classes of reduced order models that match larger numbers of moments based on interconnections between reduced order models

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

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

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

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

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

unbiased minimum variance fault and state estimation for linear discrete time varying two dimensional systems the fault and state estimation problem is addressed for a class of linear discrete time varying two dimensional systems subject to state and measurement noises two estimators are proposed to compute the estimation of the system state and or fault recursively both of which are unbiased with minimum variance through formulating the estimation problem as the solvability problem of the corresponding matrix equations of estimator gains and system constraint the necessary and sufficient condition of the existence and the solution for the proposed estimators are given an example is used to demonstrate the effectiveness of the proposed estimators