combined sequence of markov parameters and moments in linear systems the combined sequence of markov parameters and moments in linear systems is discussed this sequence is shown to be generated by a linear feedback shift register application to the minimal degree partial realization from a mixture of any number of markov parameters and moments is considered central in this particular realization problem is the extension of a toeplitz partial behavior matrix this is accomplished on the basis of the algebraic theory of iohvidov indexes for toeplitz matrices

high order nonlinear initial value problems countably determined we give a numerical approximation of the solution of a high order nonlinear initial value problem by making use of certain properties of an adequate schauder basis

convergence rates for regularized solutions of integral equations from discrete noisy data given data yi kg u ei where the e s are random errors the u s are known g is an unknown function in a reproducing kernel space with kernel r and k is a known integral operator it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points u become dense in the interval of interest these rates are shown to depend on the eigenvalue asymptotics of krk where r is the integral operator with kernel r the theory is applied to abel s equation and the estimation of particle size densities in stereology rates of convergence of regularized histogram estimates of the particle size density are given

a fourier approach to nonparametric deconvolution of a density estimate we consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f when a represents the convolution of h and f our approach is based on truncated fourier inversion in which the truncation point plays the role of a smoothing parameter we derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data strikingly when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves to first order as though the true f were known

fast inversion of triangular toeplitz matrices an algorithm requiring 0 n log n operations is proposed for the inversion of triangular toeplitz matrices this algorithm is faster than existing techniques even for relatively small values of n and can easily be written in fortran

petrov galerkin methods for linear volterra integro differential equations in this paper we study a class of petrov galerkin solutions that have global optimal convergence rates for linear volterra integro differential equations these solutions also possess certain local and global superconvergence asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by richardson extrapolation and defect corrections several postprocessing techniques are introduced to enhance these solutions as by products these higher order numerical approximations can be used to generate a posteriori error estimators representative numerical results are also provided

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

the numerical treatment of volterra integro differential equations with unbounded delay abstract initial value problems for ordinary and partial volterra integro differential equations with infinite and finite delay arise in the mathematical modelling of various physical and biological phenomena in this paper we give a survey of some recent developments in the numerical treatment of such functional equations with the focus being on equations with infinite delay in addition we will discuss a number of numerical aspects that are not yet understood well

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

pade techniques for model reduction in linear system theory a survey abstract techniques of pade approximation and continued fractions have been used often in model reduction problems an extensive bibliography on this topic is given the ideas are explained for the simple situation of a scalar function where no singular blocks appear in the pade table extensions to the matrix case and the multivariable case are not explained in detail

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

regularization of some linear ill posed problems with discretized random noisy data for linear statistical ill posed problems in hilbert spaces we introduce an adaptive procedure to recover the unknown so lution from indirect discrete and noisy data this procedure is shown to be order optimal for a large class of problems smooth ness of the solution is measured in terms of general source condi tions the concept of operator monotone functions turns out to be an important tool for the analysis

radial basis function networks in the numerical solution of linear integro differential equations the purpose of this paper is to present a new method to solve the linear integro differential equations using radial basis function rbf networks the approximate solution is represented by means of rbfs whose coefficients are computed by training a rbf network and is written as a sum of two parts the first part employs a growing rbf network whose parameters are adjusted to minimize an appropriate error function and are constructed so as not to effect the supplementary conditions of the equations the second part only satisfies the supplementary conditions numerical examples are given to demonstrate the proposed idea and method

high order methods for the numerical solution of volterra integro differential equations abstract if a first order volterra integro differential equation is solved by collocation in the space of continuous polynomial splines of degree m 1 with collocation occuring at the gauss legendre points then the resulting approximation u converges at its knots like o h 2m while its derivative u exhibits only o h m convergence this paper deals with the question of how to choose the collocation points so that both u and u converge like o h q with q maximal

convergence rates of general regularization methods for statistical inverse problems and applications previously the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut off and tikhonov type estimators spectral cut off estimators achieve minimax rates for a broad range of smoothness classes and operators but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes in this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods spectral methods including the aforementioned estimators as well as many iterative methods such as nu methods and the landweber iteration the latter estimators converge at the same rate as spectral cut off but require only matrix vector products our results are applied to various problems in particular we obtain precise convergence rates for satellite gradiometry l 2 boosting and errors in variable problems

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

linear volterra integro differential equation and schauder bases in this paper we present a numerical method to approximate the solution of the linear volterra integro differential equation by making use of the expression of a function of the banach space c 0 1 in terms of a basis schauder we are able to define a sequence of functions which approximate in the uniform sense the solution of such equation likewise we study the error committed in each approximation some advantages that this method possess are that it is not necessary to solve linear equation systems and moreover the involved integrals are immediate

a modified procedure for the realization of the transfer function matrix from a mixture of markov parameters and moments a procedure for the realization of a transfer function matrix using a mixture of markov parameters and moments is given the procedure results in the determination of a b and c and does not involve the determination of a 1and a k 1 etc as needed in existing techniques

inversion of toeplitz operators levinson equations and gohberg krein factorization a simple and unified approach for the rational case the problem of the inversion of the t oeplitz operator r associated with the operator valued function 0 defined on the unit circle is known to involve the associated levinson system of equations and the gohberg krein factorization of 0 a simplified and self contained approach making clear the connections betireen these three problems is presented in the case nhere is matrix valued and rational the key idea consists in looking at the levinson system of equations associated with 2 rather than that associated with z j as a consequence a new invertibility criterion for toeplitz operators with rational matrix lalued symbols is derived

an algorithm for solving discrete time wiener hopf equations based upon euclid s algorithm an algorithm for solving a discrete time wiener hopf equation is presented based upon euclid s algorithm the discrete time wiener hopf equation is a system of linear inhomogeneous equations with a given toeplitz matrix m a given vector b and an unknown vector lambda such that m lambda b the algorithm is able to find a solution of the discrete time wiener hopf equation for any type of toeplitz matrices except for the all zero matrix while the levinson algorithm and the trench algorithm are not available when at least one of the principal submatrices of the toeplitz matrix m is singular the algorithm gives a solution if one exists even when the toeplitz matrix m is singular while the brent gustavson yun algorithm only states that the toeplitz matrix m is singular the algorithm requires o t 2 arithmetic operations for t unknowns in the sense that the number of multiplications or divisions is directly proportional to t 2 like the levinson and trench algorithms furthermore a faster algorithm is also presented based upon the half greatest common divisor algorithm and hence it requires o t log 2 t arithmetic operations like the brent gustavson yun algorithm

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

a simple hankel interpretation of the berlekamp massey algorithm abstract a simple interpretation of the berlekamp massey algorithm in the light of the hankel matrix is presented the salient result is that the jump of the linear feedback shift register lfsr length is derived almost trivially from the so called iohvidov index of the hankel matrix prior to making any reference to the berlekamp massey algorithm itself next the hankel system of equations that yields the updated connection polynomial is solved via the natural lu factorization of the hankel matrix which itself leads to the berlekamp massey algorithm in a simple and transparent manner

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

about some numerical approaches for mixed integral equations abstract in this paper we tackle on mixed volterra fredholm integral equations as in linear as in non linear cases to the aim to obtain numerical solutions of these models the authors propose in the linear case the direct collocation method using a p order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by schauder tensor bases the advantages of both methods are outlined and their convergence is studied numerical results confirm the theoretical statements

direct and iterative methods for the numerical solution of mixed integral equations abstract in this paper we analyze and compare two classical methods to solve volterra fredholm integral equations the first is a collocation method the second one is a fixed point method both of them are proposed on a particular class of approximating functions precisely the first method is based on a linear spline class approximation and the second one on schauder linear basis we analyze some problems of convergence and we propose some remarks about the peculiarities and adaptability of both methods numerical results complete the work

time point relaxation methods for volterra integro differential equations abstract time point relaxation methods based on direct quadrature methods and on runge kutta methods for the numerical solution of volterra integro differential systems are proposed the convergence of the discrete time iterations is analyzed

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

an algebraic approach to the partial realization problem abstract the partial realization problem is treated algebraically using polynomial models the main idea is the kernels of certain projections of the hankel map which give rise to orthogonal direct sum decomposition of the state space minimality of partial realizations is based on minimal elements of these subspaces which arise as a natural consequence of the structural properties

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

nonlinear initial value problems and schauder bases abstract in this work we obtain an approximation of the solution of a nonlinear initial value problem by means of certain schauder basis associated in a natural way with that differential problem

biorthogonal systems for solving volterra integral equation systems of the second kind with the aid of biorthogonal systems in adequate banach spaces the problem of approximating the solution of a system of nonlinear volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically