an extension of a bound for functions in sobolev spaces with applications to m s spline interpolation and smoothing given a function f on a bounded open subset of mathbb r n with a lipschitz continuous boundary we obtain a sobolev bound involving the values of f at finitely many points of overline omega this result improves previous ones due to narcowich et al math comp 74 743 763 2005 and wendland and rieger numer math 101 643 662 2005 we then apply the sobolev bound to derive error estimates for interpolating and smoothing m s splines in the case of smoothing noisy data as well as exact data are considered

semi iterative methods for the approximate solution of ill posed problems we study semi iterative two step interative methods of the form x n 1 t t alpha x n gamma x n 1 beta x n 1 beta x n 1 alpha gamma t y for the approximate solution of ill posed or ill conditioned linear equationstx y in infinite or finite dimensional hilbert spaces we present results on convergence convergence rates the influence of perturbed data and on the comparison of different methods

optimal discretization of inverse problems in hilbert scales regularization and self regularization of projection methods we study the efficiency of the approximate solution of ill posed problems based on discretized noisy observations which we assume to be given beforehand a basic purpose of the paper is the consideration of stochastic noise but deterministic noise is also briefly discussed we restrict ourselves to problems which can be formulated in hilbert scales within this framework we shall quantify the degree of ill posedness provide general conditions on projection schemes to achieve the best possible order of accuracy we pay particular attention on the problem of self regularization vs tikhonov regularization moreover we study the information complexity asymptotically any method which achieves the best possible order of accuracy must use at least such amount of noisy observations

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

on converse and saturation results for tikhonov regularization of linear ill posed problems in this paper we prove some new converse and saturation results for tikhonov regularization of linear ill posed problems tx y where t is a linear operator between two hilbert spaces

new pocs algorithms for regularization of inverse problems abstract ill posed problems described by first kind fredholm equations appear in many interesting practical cases in engineering or mathematical physics such as deconvolution and require regularization techniques to get adequate solutions this paper presents under a projection operators onto convex sets pocs framework to introduce the needed regularization operations a series of non adaptive and adaptive pocs regularization algorithms which offer as main advantages besides including previously proposed methods a lower computational load and the possibility of including any kind of constraints linear and nonlinear a series of simulation examples serve to appreciate the high performance of the proposed techniques in a typical problem recovering the discontinuites of a sequence showing sharp edges

krylov subspace iterative methods for boundary element method based near field acoustic holography the reconstruction of the acoustic field for general surfaces is obtained from the solution of a matrix system that results from a boundary integral equation discretized using boundary element methods the solution to the resultant matrix system is obtained using iterative regularization methods that counteract the effect of noise on the measurements these methods will not require the calculation of the singular value decomposition which can be expensive when the matrix system is considerably large krylov subspace methods are iterative methods that have the phenomena known as semi convergence i e the optimal regularization solution is obtained after a few iterations if the iteration is not stopped the method converges to a solution that generally is totally corrupted by errors on the measurements for these methods the number of iterations play the role of the regularization parameter we will focus our attention to the study of the regularizing properties from the krylov subspace methods like conjugate gradients least squares qr and the recently proposed hybrid method a discussion and comparison of the available stopping rules will be included a vibrating plate is considered as an example to validate our results

mathcal y scale regularization many inverse problems are ill posed in the sense that their solution is unstable with respect to data perturbations and hence regularization methods have to be used for their stable solution two possible drawbacks of standard regularization methods are a saturation i e only suboptimal approximations can be found for smooth solutions this is the case e g for tikhonov regularization b numerical effort e g the large number of iterations of methods like landweber iteration a framework that allows us to overcome both effects at least for certain classes of inverse problems is regularization in hilbert scales where a solution is searched in a scale of spaces hilbert scale but convergence is achieved in the original space regularization methods in hilbert scales can be viewed as modified preconditioned versions of standard methods in order to make the advantages of the hilbert scale approach applicable to a new class of problems we propose to use a scale of spaces over the image space instead this results in a new family of methods which we call mathcal y scale regularization and whose optimal convergence properties are analyzed one of the key steps in the analysis is the formulation of an adequate a posteriori stopping rule and the proof of optimal convergence rates the theoretical results are illustrated in several numerical examples

a regularization algorithm for decoding perceptual temporal profiles from fmri data in several biomedical fields researchers are faced with regression problems that can be stated as statistical learning problems one example is given by decoding brain states from functional magnetic resonance imaging fmri data recently it has been shown that the general statistical learning problem can be restated as a linear inverse problem hence new algorithms were proposed to solve this inverse problem in the context of reproducing kernel hilbert spaces in this paper we detail one iterative learning algorithm belonging to this class called method and test its effectiveness in a between subjects regression framework specifically our goal was to predict the perceived pain intensity based on fmri signals during an experimental model of acute prolonged noxious stimulation we found that using a linear kernel the psychophysical time profile was well reconstructed while pain intensity was in some cases significantly over underestimated no substantial differences in terms of accuracy were found between the proposed approach and one of the state of the art learning methods the support vector machines nonetheless adopting the method yielded a significant reduction in computational time an advantage that became more evident when a relevant feature selection procedure was implemented the method can be easily extended and included in typical approaches for binary or multiple classification problems and therefore it seems well suited to build effective brain activity estimators

nonparametric trending regression with cross sectional dependence panel data whose series length t is large but whose cross section size n need not be are assumed to have common time trend of unknown form the model includes additive unknown individual specific components and allows for spatial or other cross sectional dependence and or heteroscedasticity a simple smoothed nonparametric trend estimate is shown to be dominated by an estimate which exploits availability of cross sectional data asymptotically optimal bandwidth choices are justified for both estimates feasible optimal bandwidths and feasible optimal trend estimates are asymptotically justified finite sample performance of the latter being examined in a monte carlo study potential extensions are discussed

on the effect of support estimation and of a new model in 2 d inverse scattering problems starting from the concept of degree of non linearity of the relationship amongst unknown parameters and scattered fields we discuss how knowledge of the geometrical features of the targets favorably affects the solution of inverse scattering problems the same concept is also exploited to optimize the contrast source extended born cs eb model resulting in the optimized cs eb o cs eb inversion approach which remarkably reduces false solutions occurrence then a two steps approach is proposed as a reliable and effective inversion strategy the first step consists in retrieving the shape of the targets using the linear sampling method while in the second step the estimated shapes and the o cs eb are exploited to reconstruct the permittivity profile the approach has been tested against experimental data showing that very accurate reconstructions are achieved from monochromatic data

multiscale analysis in sobolev spaces on bounded domains we study a multiscale scheme for the approximation of sobolev functions on bounded domains our method employs scattered data sites and compactly supported radial basis functions of varying support radii at scattered data sites the actual multiscale approximation is constructed by a sequence of residual corrections where different support radii are employed to accommodate different scales convergence theorems for the scheme are proven and it is shown that the condition numbers of the linear systems at each level are independent of the level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points on a bounded domain

a multi grid method with a priori and a posteriori level choice for the regularization of nonlinear ill posed problems in this paper we study a multi grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill posed problems where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers we provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill posed problem i e a stable solution method that converges to an exact solution of the underlying infinite dimensional problem as the data noise level goes to zero with optimal rates under additional regularity conditions

regularization of inverse problems by two point gradient methods in banach spaces in this paper we propose and analyze a two point gradient method for solving inverse problems in banach spaces which is based on the landweber iteration and an extrapolation strategy the method allows to use non smooth penalty terms including the l 1 like and the total variation like penalty functionals which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications the design of the method involves the choices of the step sizes and the combination parameters which are carefully discussed numerical simulations are presented to illustrate the effectiveness of the proposed method

towards a general convergence theory for inexact newton regularizations we develop a general convergence analysis for a class of inexact newton type regularizations for stably solving nonlinear ill posed problems each of the methods under consideration consists of two components the outer newton iteration and an inner regularization scheme which applied to the linearized system provides the update in this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including landweber and steepest decent iterations iterated tikhonov method and method of conjugate gradients

extension of sampling inequalities to sobolev semi norms of fractional order and derivative data this paper devoted to sampling inequalities provides some extensions of previous results by arcangeli et al numer math 107 2 181 211 2007 j approx theory 161 198 212 2009 given a function u in a suitable sobolev space defined on a domain in mathbb r n sampling inequalities typically yield bounds of integer order sobolev semi norms of u in terms of a higher order sobolev semi norm of u the fill distance d between overline omega and a discrete set a subset overline omega and the values of u on a the extensions established in this paper allow us to bound fractional order semi norms and to incorporate if available values of partial derivatives on the discrete set both the cases of a bounded domain and omega mathbb r n are considered

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

sirt and cg type methods for the iterative solution of sparse linear least squares problems abstract sirt and cg type methods have been successfully employed for the approximate solution of least squares problems arising in tomography in this paper we study and compare their convergence and regularization properties it is pointed out that sirt methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system whereas cg type methods do not for a large class of model problems it is shown that virtually the same solutions as obtained by sirt methods can be obtained by applying a cg type method to a properly rescaled system but with an amount of work proportional to the square root of the amount of work with sirt

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

preconditioning landweber iteration in hilbert scales in this paper we investigate convergence of landweber iteration in hilbert scales for linear and nonlinear inverse problems as opposed to the usual application of hilbert scales in the framework of regularization methods we focus here on the case s 0 which for tikhonov regularization corresponds to regularization in a weaker norm in this case the hilbert scale operator l 2 s appearing in the iteration acts as a preconditioner which significantly reduces the number of iterations needed to match an appropriate stopping criterion additionally we carry out our analysis under significantly relaxed conditions i e we only require inlinemediaobject not available see fulltext instead of inlinemediaobject not available see fulltext which is the usual condition for regularization in hilbert scales the assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones

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

a hybrid quantitative method for inverse scattering of multiple dielectric objects a hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method lsm the hybrid method is composed of three steps combining the capabilities of the lsm the multiple signal classification music algorithm and the generalized multipole technique gmt initially a new measure of the noise level of the multistatic response msr matrix is proposed subsequently the lsm is applied using the tikhonov regularization then the scattered field is back propagated into the investigation domain using the gmt where in order to ensure the stability a music based approach is exploited to determine the optimum reconstructing multipole sources the case of limited scattered field data is also addressed by introducing a modification of the papoulis gerchberg algorithm pga finally the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization the proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner the effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data

asymptotically unbiased inference for a dynamic panel model with fixed effects when both n and t are large we consider a dynamic panel ar 1 model with fixed effects when both n and t are large under the t fixed n large asymptotic approximation the ordinary least squares ols or gaussian maximum likelihood estimator mle is known to be inconsistent due to the well known incidental parameter problem we consider an alternative asymptotic approximation where n and t grow at the same rate it is shown that although ols or the mle is asymptotically biased a relatively simple fix to ols or the mle results in an asymptotically unbiased estimator under the assumption of gaussian innovations the bias corrected mle is shown to be asymptotically efficient by a hajek type convolution theorem

the time series and cross section asymptotics of dynamic panel data estimators in this paper we derive the asymptotic properties of within groups wg gmm and liml estimators for an autoregressive model with random effects when both t and n tend to infinity gmm and liml are consistent and asymptotically equivalent to the wg estimator when t n 0 the fixed t results for gmm and liml remain valid but wg although consistent has an asymptotic bias in its asymptotic distribution when t n tends to a positive constant the wg gmm and liml estimators exhibit negative asymptotic biases of order 1 t 1 n and 1 2n t respectively in addition the crude gmm estimator that neglects the autocorrelation in first differenced errors is inconsistent as t n c 0 despite being consistent for fixed t finally we discuss the properties of a random effects pseudo mle with unrestricted initial conditions when both t and n tend to infinity

accelerated landweber iterations for the solution of ill posed equations in this paper the potentials of so calledlinear semiiterative methods are considered for the approximate solution of linear ill posed problems and ill conditioned matrix equations several efficient two step methods are presented most of which have been introduced earlier in the literature stipulating certain conditions concerning the smoothness of the solution a notion of optimal speed of convergence may be formulated various direct and converse results are derived to illustrate the properties of this concept if the problem s right hand side data are contaminated by noise semiiterative methods may be used asregularization methods assuming optimal rate of convergence of the iteration for the unperturbed problem the regularized approximations will be of order optimal accuracy to derive these results specific properties of polynomials are used in connection with the basic theory of solving ill posed problems rather recent results onfast decreasing polynomials are applied to answer an open question of brakhage numerical examples are given including a comparison to the method of conjugate gradients

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

on landweber iteration for nonlinear ill posed problems in hilbert scales in this paper we derive convergence rates results for landweber iteration in hilbert scales in terms of the iteration index k for exact data and in terms of the noise level delta for perturbed data these results improve the one obtained recently for landweber iteration for nonlinear ill posed problems in hilbert spaces for numerical computations we have to approximate the nonlinear operator and the infinite dimensional spaces by finite dimensional ones we also give a convergence analysis for this finite dimensional approximation the conditions needed to obtain the rates are illustrated for a nonlinear hammerstein integral equation numerical results are presented confirming the theoretical ones

inference on trending panel data abstract semiparametric panel data modelling and statistical inference with fractional stochastic trends nonparametrically time trending individual effects and general cross sectional correlation and heteroscedasticity in innovations are developed the fractional stochastic trends allow for a wide range of nonstationarity indexed by a memory parameter nesting the familiar i 1 case and allowing for parametric short memory the individual effects can nonparametrically vary simultaneously across time and across units the cross sectional covariance matrix is also nonparametric the main focus is on estimation of the time series parameters two methods are considered both of which entail an only approximate differencing out of the individual effects leaving an error which has to be taken account of in our theory in both cases we obtain standard asymptotics with a central limit theorem over a wide range of possible parameter values unlike the nonstandard asymptotics for autoregressive parameter estimates at a unit root for statistical inference consistent estimation of the limiting covariance matrix of the parameter estimates requires consistent estimation of a functional of the cross sectional covariance matrix we examine efficiency loss due to cross sectional correlation in a spatial model example a monte carlo study of finite sample performance is included

inexact newton regularization methods in hilbert scales we consider a class of inexact newton regularization methods for solving nonlinear inverse problems in hilbert scales under certain conditions we obtain the order optimal convergence rate result

sampling inequalities for sparse grids sampling inequalities play an important role in deriving error estimates for various reconstruction processes they provide quantitative estimates on a sobolev norm of a function defined on a bounded domain in terms of a discrete norm of the function s sampled values and a smoothness term which vanishes if the sampling points become dense the density measure which is typically used to express these estimates is the mesh norm or hausdorff distance of the discrete points to the bounded domain such a density measure intrinsically suffers from the curse of dimension the curse of dimension can be circumvented at least to a certain extend by considering additional structures here we will focus on bounded mixed regularity in this situation sparse grid constructions have been proven to overcome the curse of dimension to a certain extend in this paper we will concentrate on a special construction for such sparse grids namely smolyak s method and provide sampling inequalities for mixed regularity functions on such sparse grids in terms of the number of points in the sparse grid finally we will give some applications of these sampling inequalities

an analysis of a multi level projected steepest descent iteration for nonlinear inverse problems in banach spaces subject to stability constraints we consider nonlinear inverse problems described by operator equations in banach spaces assuming conditional stability of the inverse problem that is assuming that stability holds on a compact convex subset of the domain of the operator we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data we proceed with developing a multi level algorithm based on a nested family of compact convex subsets on which stability holds and the stability constants are ordered growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion after a finite number of steps

on the order optimality of the regularization via inexact newton iterations inexact newton regularization methods have been proposed by hanke and rieder for solving nonlinear ill posed inverse problems every such a method consists of two components an outer newton iteration and an inner scheme providing increments by regularizing local linearized equations the method is terminated by a discrepancy principle in this paper we consider the inexact newton regularization methods with the inner scheme defined by landweber iteration the implicit iteration the asymptotic regularization and tikhonov regularization under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of rieder we in fact obtain a more general order optimality result by considering these inexact newton methods in hilbert scales

analysis of profile functions for general linear regularization methods the stable approximate solution of ill posed linear operator equations in hilbert spaces requires regularization tight bounds for the noise free part of the regularization error are constitutive for bounding the overall error norm bounds of the noise free part which decrease to zero along with the regularization parameter are called profile functions and are the subject of our analysis the interplay between properties of the regularization and certain smoothness properties of solution sets which we shall describe in terms of sourcewise representations is crucial for the decay of associated profile functions on the one hand we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness on the other hand if smoothness fits the regularization then decay rates are easily obtained if smoothness does not fit then we will measure this in terms of some distance function tight bounds for these allow us to obtain profile functions finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition in many parts the analysis is done on geometric basis extending classical concepts of linear regularization theory in hilbert spaces we emphasize intrinsic features of linear ill posed problems which are frequently hidden in the classical analysis of such problems

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

the meshless kernel based method of lines for parabolic equations using the heat equation as a simple example we give a rigorous theoretical analysis of the method of lines implemented as a meshless method based on spatial trial spaces spanned by translates of positive definite kernels the technique can be generalized to other parabolic problems and some numerical illustrations are given

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

dilational interpolatory inequalities they also wish to acknowledge the support of the radon institute of computational and applied mathematics where the initial draft of this paper was finalized

a posteriori parameter choice strategies for some newton type methods for the regularization of nonlinearill posed problems this paper treats a class of newton type methods for the approximate solution of nonlinear ill posed operator equations that use so called filter functions for regularizing the linearized equation in each newton step for noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution as the noise level goes to zero under certain smoothness conditions on the nonlinear operator appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates moreover we present an application of the newton type methods under consideration to a parameter identification problem together with some numerical results

dimension reduction method for ode fluid models we develop a new dimension reduction method for large size systems of ordinary differential equations odes obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow the method is also applicable to other large size classical particle systems with negligibly small variations of particle concentration we propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution to illustrate the computational advantages of the proposed reduction method we use it to solve a system of smoothed particle hydrodynamic odes describing single phase and two phase layered poiseuille flows driven by uniform and periodic in space body forces for the single phase poiseuille flow driven by the uniform force the coarse solution was obtained with the zero order deconvolution for the single phase flow driven by the periodic body force and for the two phase flows the higher order the first and second order deconvolutions were necessary to obtain a sufficiently accurate solution

optimal a posteriori parameter choice for tikhonov regularization for solving nonlinear ill posed problems the authors propose an a posteriors strategy for choosing the regularization parameter in tikhonov regularization for solving nonlinear ill posed problems and show that under certain conditions the convergence rate obtained with this strategy is optimal as a by product a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems the authors compare the parameter choice strategy with morozov s discrepancy principle finally numerical results are presented

bias in dynamic panel estimation with fixed effects incidental trends and cross section dependence abstract explicit asymptotic bias formulae are given for dynamic panel regression estimators as the cross section sample size n the results extend earlier work by nickell 1981 biases in dynamic models with fixed effects econometrica 49 1417 1426 and later authors in several directions that are relevant for practical work including models with unit roots deterministic trends predetermined and exogenous regressors and errors that may be cross sectionally dependent the asymptotic bias is found to be so large when incidental linear trends are fitted and the time series sample size is small that it changes the sign of the autoregressive coefficient another finding of interest is that when there is cross section error dependence the probability limit of the dynamic panel regression estimator is a random variable rather than a constant which helps to explain the substantial variability observed in dynamic panel estimates when there is cross section dependence even in situations where n is very large some proposals for bias correction are suggested and finite sample performance is analyzed in simulations

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

the instability of some gradient methods for ill posed problems for the solution of linear ill posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature it is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a priori parameter choice regularization methods upon them

deconvolution with unknown error distribution we consider the problem of estimating a density fx using a sample y1 yn from fy fx fe where fe is an unknown density we assume that an additional sample e1 em from fe is observed estimators of fx and its derivatives are constructed by using nonparametric estimators of fy and fe and by applying a spectral cut off in the fourier domain we derive the rate of convergence of the estimators in case of a known and unknown error density fe where it is assumed that fx satisfies a polynomial logarithmic or general source condition it is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density if the density fx belongs to a sobolev space h mathcal p and fe is ordinary smooth or supersmooth

the linear sampling method as a way to quantitative inverse scattering the linear sampling method lsm is a simple and effective approach to image the shape of unknown targets via the solution of a linear inverse problem in this paper we show that the lsm can also be exploited to devise a novel effective approximation of the scattering phenomenon that leads to a new noniterative linear inversion method for the estimation of the target s electric contrast since the introduced approximation relies on the broad applicability of the lsm the proposed inversion method is suitable to tackle inverse scattering problems involving nonweak scatterers as such it represents an innovative yet effective way to tackle quantitative imaging examples with numerical and experimental data are given to show the performance of the approach in particular results obtained with fresnel data sets show that the proposed method is capable of successfully imaging targets which have been so far processed using nonlinear iterative schemes and taking advantage of frequency diversity

extended domain eigenfunction method edem a study of ill posedness and regularization the extended domain eigenfunction method edem proposed for solving elliptic boundary value problems on annular like domains requires an inversion process the procedure thus represents an ill posed problem whose numerical solution involves an ill conditioned system of equations in this paper the ill posed nature of edem is studied and numerical solutions based on regularization schemes are considered it is shown that the edem solution methodology lends itself naturally to a formulation in terms of the well known iterative landweber method and the more general and faster converging semi iterative regularization schemes theoretical details and numerical results of the regularization schemes are presented for the case of the two dimensional laplace operator on annular domains

an inverse scattering method based on contour deformations by means of a level set method using frequency hopping technique a shape reconstruction method for electromagnetic tomography based on contour deformations by means of a level set method is proposed the level set method adds quite valuable features to the inverse procedure such as natural regularization of the contours and automatic breaking and merging capability the frequency hopping technique turns out to be a key point of the inverse scattering procedure the numerical results clearly indicate that the use of this inversion algorithm yields to accurate reconstructions of one or several objects from one initial guess object even with noise contaminated data and limited coverage of the fields

r k type landweber method for nonlinear ill posed problems in this paper we propose the r k type landweber iteration and investigate the convergence of the method for nonlinear ill posed problem f x y where f h h is a nonlinear operator between hilbert space h moreover for perturbed data with noise level d we prove that the convergence rate is o d 2 3 under appropriate conditions finally the numerical performance of this r k type landweber iteration for a nonlinear convolution equation is compared with the landweber iteration

on steepest descent kaczmarz methods for regularizing systems of nonlinear ill posed equations we investigate modified steepest descent methods coupled with a loping kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill posed operator equations we show that the proposed method is a convergent regularization method numerical tests are presented for a linear problem related to photoacoustic tomography and a nonlinear problem related to the testing of semiconductor devices

experimental microwave validation of level set reconstruction algorithm the capability of the level set method is illustrated for shape and location reconstruction using real measurement data in the range of 3 8 ghz an experimental setup is utilized to measure the s parameters using two vivaldi antennas revolving around metallic pipes of different cross sections a two dimensional reconstruction of the pipe s cross section is retrieved using the level set algorithm combined with the method of moments and the frequency hopping technique the results show good agreement between the reconstructed cross section and the true configuration this indicates to the strength of the level set method and its ability to cope with the noise in real measurements

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

semi iterative regularization in hilbert scales in this paper we investigate the regularizing properties of semi iterative regularization methods in hilbert scales for linear ill posed problems and perturbed data it is well known that standard landweber iteration can be remarkably accelerated by polynomial acceleration methods leading to optimal speed of convergence which can be obtained by several efficient two step methods e g the nu methods by brakhage it was observed earlier that a similar speed of convergence i e similar iteration numbers yielding optimal convergence rates can be obtained if landweber iteration is performed in hilbert scales we show that a combination of both ideas allows for a further acceleration yielding optimal convergence rates with only the square root of iterations as compared to the nu methods or landweber iteration in hilbert scales the theoretical results are illustrated by several examples and numerical tests including a comparison to the method of conjugate gradients

on the discrepancy principle for iterative and parametric methods to solve linear ill posed equations for the numerical solution of non necessarily well posed linear equations in banach spaces we consider a class of iterative methods which contains well known methods like the richardson iteration if the associated resolvent operator fulfils a condition with respect to a sector it is the purpose of this paper to show that for given noisy right hand side the discrepancy principle being a stopping rule for the iteration methods belonging to the mentioned class defines a regularization method and convergence rates are proved under additional smoothness conditions on the initial error this extends similar results obtained for positive semidefinite problems in hilbert spaces then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract cauchy problem and under a weaker resolvent condition the iterated method of lavrentiev a modified discrepancy principle is formulated for them and finally numerical illustrations are presented

convergence rate analysis of a derivative free landweber iteration for parameter identification in certain elliptic pdes we consider the nonlinear inverse problem of identifying a parameter from knowledge of the physical state in an elliptic partial differential equation for a derivative free landweber method convergence rates are proven under a weak source condition not involving the standard frechet derivative of the nonlinear parameter to output map this source condition is discussed both for the estimation of state and space dependent parameters in higher dimensions finally numerical results are presented

an accelerated alternating procedure for the cauchy problem for the helmholtz equation in this paper we study the cauchy problem for the helmholtz equation this problem appears in various applications and is severely ill posed the modified alternating procedure has been proposed by

estimation of dynamic models with error components abstract observations on n cross section units at t time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit different assumptions about the initial conditions are a initial state fixed b initial state random c the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed and d the unobserved individual effect independent of the unobserved dynamic process with initial value random asymptotic properties of the maximum likelihood and covariance estimators are obtained when t and when n the relationship between the pseudo and conditional maximum likelihood estimators is clarified a simple consistent estimator that is independent of the initial conditions and the way in which t or n is also suggested

the fast fourier transform method and ill conditioned matrices we study the problem of finding numerical solutions of the linear algebraic equation a x b where a denotes an nxn ill conditioned coefficient matrix it is well known that gaussian elimination methods coupled with pivoting strategies are ineffective in this setting due to round off error we propose a new and simple application of the fast fourier transform fft method other viable methods such as the qr method qrm or the singular value decomposition method svdm have been proposed in the literature the goal of this paper is to investigate the performance of the proposed method and compare it to other popular methods the comparison is illustrated by computer simulation results using matlab

computational scales of sobolev norms with application to preconditioning this paper provides a framework for developing computationally efficient multilevel preconditioners and representations for sobolev norms specifically given a hilbert space v and a nested sequence of subspaces v 1 c v 2 c c v we construct operators which are spectrally equivalent to those of the form a k k q k q k 1 here k k 1 2 are positive numbers and q k is the orthogonal projector onto v k with q 0 0 we first present abstract results which show when a is spectrally equivalent to a similarly constructed operator a defined in terms of an approximation q k of q k for k 1 2 we show that these results lead to efficient preconditioners for discretizations of differential and pscudo differential operators of positive and negative order these results extend to sums of operators for example singularly perturbed problems such as i e can be preconditioned uniformly independently of the parameter e we also show how to precondition an operator which results from tikhonov regularization of a problem with noisy data finally we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition

efficient inference on fractionally integrated panel data models with fixed effects a dynamic panel data model is considered that contains possibly stochastic individual com ponents and a common fractional stochastic time trend we propose four dierent ways of coping with the individual eects so as to estimate the fractional parameter like models with autoregressive dynamics ours nests a unit root but unlike the nonstandard asymptotics in the autoregressive case estimates of the fractional parameter can be asymptotically normal establishing this property is made di cult due to bias caused by the individual eects or by the consequences of eliminating them and requires the number of time series observations t to increase while the cross sectional size n can either remain xed or increase with t the biases in the central limit theorem are asymptotically negligible only under stringent conditions on the growth of n relative to t but these can be relaxed by bias correction for three of the estimates the biases depend only on the fractional parameter in hypothesis testing bias correc tion of the estimates is readily carried out we evaluate the biases numerically for a range of t and parameter values develop and justify feasible bias corrected estimates and brie y discuss simpli ed but less eective corrections a monte carlo study of nite sample performance is included

pointwise rational approximation and iterative methods for iii posed problems we study the connection between the pointwise approximation of the zero function by rational functions and iterative methods for the approximate solution of ill posed linear equations results are presented on convergence stability and saturation phenomena

a predictor corrector iterated tikhonov regularization for linear ill posed inverse problems many questions in science and engineering often give rise to linear ill posed inverse problems to enable meaningful approximate solutions of the inverse problems various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations one of the most popular regularization techniques is iterated tikhonov regularization which has attracted considerable attention due to its interesting applications in practical however this regularization often suffers from inaccurate results and low computational efficiency in some situations in this paper we proposed an accelerated predictor corrector iterated tikhonov regularization this method combined the classical iterated tikhonov regularization with modified euler method which could improve computational efficiency without sacrificing numerical accuracy the convergence rate of the accelerated regularization was investigated theoretically when the right hand side was corrupted by noise in particular we derived an error estimate of optimal order for the convergence of the accelerated regularization both one and two dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated tikhonov regularization

efficient microwave imaging algorithm based on hybridization of the linear sampling and level set methods a hybridized algorithm based on the linear sampling method and the level set technique is presented the level set technique has shown the capability of accurate reconstruction of the shapes of unknown targets of different materials and arbitrary shapes but with significant computational time the linear sampling method has shown comparatively rapid estimation of the targets shapes the presented algorithm retains the accuracy of the level set algorithm with significantly reduced computational time due to the hybridization of the level set method and the linear sampling method reconstruction results of fresnel institute experimental data are obtained using the hybridized algorithm

image reconstruction and restoration using the simplified topological e algorithm in order to compute meaningful approximations of the solutions of large scale linear inverse ill posed problems some form of regularization should be employed cimmino and landweber methods are well known iterative regularization methods that can be quite successfully applied for tomographic reconstruction and image restoration problems despite their usually slow convergence the goal of this paper is to explore the performance of a recent extrapolation algorithm when applied to accelerate the convergence of these iterative regularization methods in particular we provide insight and algorithmic details about the simplified topological e algorithm applied to slow converging iterative regularization methods the results of many numerical experiments and comparisons with other methods are also displayed

convergence analysis of the homotopy perturbation method for solving nonlinear ill posed operator equations the homotopy perturbation method is used to construct a new iteration algorithm for solving nonlinear ill posed operator equations numerical tests are given showing that the algorithm is more efficient than the well known landweber method

an optimal order regularization method which does not use additional smoothness assumptions this paper defines an optimal method to reconstruct the solutions of operator equations of the first kind only the case of compact operators is considered the method is in principle a discrepancy method it does not require any additional knowledge about the solution and is optimal for all standard smoothness assumptions in order to analyze the properties of the new regularization method variable hilbert scales are introduced and several well known results for hilbert scales are generalized convergence theorems for classes of optimal and suboptimal methods are derived from a generalization of the interpolation inequality

newton lavrentiev regularization for ill posed operator equations in hilbert scales in this paper we consider the two step method for approximately solving the ill posed operator equation f x f where f d f x x is a nonlinear monotone operator defined on a real hilbert space x in the setting of hilbert scales we derive the error estimates by selecting the regularization parameter a according to the adaptive method considered by pereverzev and schock in 2005 when the available data is f d with f f email protected x is of optimal order

on the regularization of projection methods for solving iii posed problems in this paper we consider a class of regularization methods for a discretized version of an operator equation which includes the case that the problem is ill posed with approximately given right hand side we propose an a priori as well as an a posteriori parameter choice method which is similar to the discrepancy principle of ivanov morozov from results on fractional powers of selfadjoint operators we obtain convergence rates which are in many cases the same for both parameter choices

on the acceleration of kaczmarz s method for inconsistent linear systems abstract let the linear system ax b be rectangular but solvable if a is a large sparse matrix then one possibility to solve the system is to use the iterative method of kaczmarz even in the case when the system is unsolvable this method is applicable if the relaxation parameters are kept fixed in a certain interval to get a meaningful result however one has to apply strong underrelaxation we give an interpretation of the limit in terms of a generalized inverse of a and derive bounds with respect to the least squares solution using a result of buoni and varga we estimate the speed of convergence it is further shown that the eigenvalues of the iteration matrix are nearly real numbers and we propose to use chebyshev acceleration this results in a significant speedup