some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

a numerical study on the solution of the cauchy problem in elasticity this work deals with the cauchy problem in two dimensional linear elasticity the equations of the problem are discretized through a standard fem approach and the resulting ill conditioned discrete problem is solved within the frame of the tikhonov approach the choice of the required regularization parameter is accomplished through the generalized cross validation criterion on this basis a numerical experimentation has been performed and the calculated solutions have been used to highlight the sensitivity to the amount of known data the noise always present in the data the regularity of boundary conditions and the choice of the regularization parameter the aim of the numerical study is to implicitly device some guidelines to be used in the solution of this kind of problems

a fast and efficient cross validation method for smoothing parameter choice in spline regression abstract the spline smoothing approach to nonparametric regression is considered with particular reference to the problem of choosing how much to smooth both computational and statistical aspects of the method of cross validation are discussed an approximate cross validation method is proposed that requires a small linear amount of computer time a simulation study indicates that the method has very good statistical properties some mathematical justification and motivation are also provided

finite element solution of two dimensional inverse elastic problems using spatial smoothing a finite element method fem is presented for calculating the surface tractions on a body from internal measurements of displacement at discrete sensor locations the solution algorithm employs a sensitivity analysis which minimizes the difference between the calculated and measured displacements at each sensor location spatial regularization is one technique employed to stabilize the minimization process by imposing various degrees of smoothness on the solution it also allows the problem to be solved with fewer sensors than traction boundary nodes as an alternative to spatial regularization a method based on keynodes is introduced which assumes that each component of the boundary traction distribution can be described by a polynomial of specified order the methods are applied to several two dimensional examples including a rolling contact problem the effects of parameters such as the number of sensors the location of the sensors and the error in the sensor displacements are discussed

an iterative method for the cauchy problem in linear elasticity with fading regularization effect in this paper an iterative method for solving the cauchy problem in linear elasticity is introduced this problem consists in recovering missing data displacements and forces on some parts of a domain boundary from the knowledge of overspecified data displacements and forces on the remaining parts the algorithm reads as a least square fitting of the given data with a regularization term whose effect fades as the iterations go on so the algorithm converges to the solution of the cauchy problem numerical simulations using the finite element method highlight the algorithm s efficiency accuracy robustness to noisy data as well as its ability to deblur noisy data

an interlaced extended kalman filter an estimation algorithm for a class of discrete time nonlinear systems is proposed the system structure we deal with is partitionable into in subsystems each affine w r t the corresponding part of the state vector the algorithm consists of a bank of m interlaced kalman filters and each of them estimates a part of the state considering the remaining parts as known time varying parameters whose values are evaluated by the other filters at the previous step the procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably increasing the noise covariance matrices comparisons through numerical simulations with the extended kalman filter and its modified versions proposed in the literature illustrate the good trade off provided by the algorithm between the reduction of the computational load and the estimation accuracy

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract in this paper yet another iterative procedure namely the minimal error method mem for solving stably the cauchy problem in linear elasticity is introduced and investigated furthermore this method is compared with another two iterative algorithms i e the conjugate gradient cgm and landweber fridman methods lfm previously proposed by marin et al marin l hao d n lesnic d 2002b conjugate gradient boundary element method for the cauchy problem in elasticity quarterly journal of mechanics and applied mathematics 55 227 247 and marin and lesnic marin l lesnic d 2005 boundary element landweber method for the cauchy problem in linear elasticity ima journal of applied mathematics 18 817 825 respectively in the case of two dimensional homogeneous isotropic linear elasticity the inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions the numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method bem for two dimensional homogeneous isotropic linear elastic materials

a practical stopping rule for iterative signal restoration iterative signal restoration is a common and simple approach for recovering degraded signals unfortunately iterative algorithms often converge slowly and the convergence point is usually not the best restoration because of noise amplification therefore a stopping rule should be imposed to maximize the effectiveness of the iterative algorithm we demonstrate the value of randomized generalized cross validation rgcv as a stopping rule for linear iterative restoration algorithms with simple initial conditions and show that it can be used on relatively small data sets with confidence we also illustrate the performance of the rgcv criterion for various degrees of blurring and noise

random dynamic load identification based on error analysis and weighted total least squares method abstract in most cases random dynamic load identification problems in structural dynamics are in general ill posed a common approach to treat these problems is to reformulate these problems into some well posed problems by some numerical regularization methods in a previous paper by the authors a random dynamic load identification model was built and a weighted regularization approach based on the proper orthogonal decomposition pod was proposed to identify the random dynamic loads in this paper the upper bound of relative load identification error in frequency domain is derived the selection condition and the specific form of the weighting matrix is also proposed and validated analytically and experimentally in order to improve the accuracy of random dynamic load identification a weighted total least squares method is proposed to reduce the impact of these errors to further validate the feasibility and effectiveness of the proposed method the comparative study of the proposed method and other methods are conducted with the experiment the experimental results demonstrated that the weighted total least squares method is more effective than other methods for random dynamic load identification

smoothing and matching of 3 d space curves we present a new approach to the problem of matching 3 d curves the approach has a low algorithmic complexity in the number of models and can operate in the presence of noise and partial occlusions our method builds upon the seminal work of kishon et al 1990 where curves are first smoothed using b splines with matching based on hashing using curvature and torsion measures however we introduce two enhancements we present experimental results using synthetic data and also using characteristic curves extracted from 3 d medical images an earlier version of this article was presented at the 2nd european conference on computer vision in italy

non iterative regularized mfs solution of inverse boundary value problems in linear elasticity numerical reconstruction of the missing boundary data in 2d and 3d linear elasticity noisy over prescribed data are available on the remaining accessible boundary the inverse problem is solved by a meshless method i e the method of fundamental solutions mfs stabilization is achieved via several svd based regularization methods the regularization parameter is selected via the criteria used for inverse problems the numerical reconstruction of the missing dirichlet and neumann data on an inaccessible part of the boundary in the case of two and three dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part is investigated this inverse problem is solved using the method of fundamental solutions mfs whilst its stabilization is achieved through several singular value decomposition svd based regularization methods such as the tikhonov regularization method 48 the damped svd and the truncated svd 18 the regularization parameter is selected according to the discrepancy principle 40 generalized cross validation criterion 14 and hansen s l curve method 20

neural crack identification in steady state elastodynamics an inverse crack identification problem with harmonic excitation in linear elastodynamics is treated here by means of back propagation neural network methods and boundary element techniques the problem concerns the determination of the existence and the characteristics of a hidden crack within an elastic structure by means of measurements of the structural response on the accessible boundary for given external time periodic loadings the direct problem is solved by a boundary element formulation in the frequency domain which leads to a system of linear equations with frequency dependent matrices thus for a given frequency certain similarities with linear elastostatics exist feed forward multilayer neural networks trained by back propagation are used to learn the inverse input output relation of the structural system then the inverse problem is solved by a simple application of the neural network recalling production ability

soft and hard classification by reproducing kernel hilbert space methods reproducing kernel hilbert space rkhs methods provide a unified context for solving a wide variety of statistical modelling and function estimation problems we consider two such problems we are given a training set yi ti i 1 n where yi is the response for the ith subject and ti is a vector of attributes for this subject the value of yi is a label that indicates which category it came from for the first problem we wish to build a model from the training set that assigns to each t in an attribute domain of interest an estimate of the probability pj t that a future subject with attribute vector t is in category j the second problem is in some sense less ambitious it is to build a model that assigns to each t a label which classifies a future subject with that t into one of the categories or possibly none of the above the approach to the first of these two problems discussed here is a special case of what is known as penalized likelihood estimation the approach to the second problem is known as the support vector machine we also note some alternate but closely related approaches to the second problem these approaches are all obtained as solutions to optimization problems in rkhs many other problems in particular the solution of ill posed inverse problems can be obtained as solutions to optimization problems in rkhs and are mentioned in passing we caution the reader that although a large literature exists in all of these topics in this inaugural article we are selectively highlighting work of the author former students and other collaborators

a fast monte carlo cross validation procedure for large least squares problems with noisy data we propose a fast monte carlo algorithm for calculating reliable estimates of the trace of the influence matrixa involved in regularization of linear equations or data smoothing problems where is the regularization or smoothing parameter this general algorithm is simply as follows i generaten pseudo random valuesw 1 w n from the standard normal distribution wheren is the number of data points and letw w 1 w n t ii compute the residual vectorw a w iii take the normalized inner product w t w a w w t w as an approximation to 1 n tr i a we show both by theoretical bounds and by numerical simulations on some typical problems that the expected relative precision of these estimates is very good whenn is large enough and that they can be used in practice for the minimization with respect to of the well known generalized cross validation gcv function this permits the use of the gcv method for choosing in any particular large scale application with only a similar amount of work as the standard residual method numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency

determination of an optimal regularization factor in system identification with tikhonov regularization for linear elastic continua this paper presents a geometric mean scheme gms to determine an optimal regularization factor for tikhonov regularization technique in the system identification problems of linear elastic continua the characteristics of non linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses it is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution based on this observation the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses the validity of the gms is demonstrated through two numerical examples with measurement errors and modelling errors copyright 2001 john wiley sons ltd

numerical solution of cauchy problems in linear elasticity in axisymmetric situations abstract an iterative method for solving axisymmetric cauchy problems in linear elasticity is presented this kind of problem consists in recovering missing displacements and forces data on one part of a domain boundary from the knowledge of overspecified displacements and forces data on another part of this boundary numerical simulations using the finite element method highlight the algorithm s efficiency accuracy and robustness to noisy data as well as its ability to deblur noisy data an application of the inverse technique to the identification of a friction coefficient is also presented

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

inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection the problem of determining an unknown pair gamma lambda mu of lame parameters inside a body by its dirichlet to neumann data map lambda gamma is considered using explicit exact solutions for the case of constant gamma it is seen that the deflection h between gamma h and gamma is uniquely determined by the first order approximation of lambda gamma h at gamma

time domain galerkin method for dynamic load identification summary this paper proposes a new method called time domain galerkin method tdgm for investigating the structural dynamic load identification problems firstly the shape functions are adopted to approximate three parameters such as the dynamic load kernel function response and measured structural response secondly defining a residual function could be expressed as the difference of the measured response and the computational response thirdly select an appropriate weighting function to multiply the defined residual function and make integral operation with respect to time to be zero finally when the shape functions are chosen as the weighting function it establishes the forward model called tdgm furthermore the regularization method could have effectiveness in solving the ill posed matrix of load reconstruction and obtaining the accurate identified results of the dynamic load compared with the traditional green kernel function method gkfm tdgm can effectively overcome the influences of noise and improve the accuracy of the dynamic load identification three numerical examples are provided to demonstrate the correctness and advantages of tdgm copyright 2015 john wiley sons ltd

generalized cross validation applied to conjugate gradient for discrete ill posed problems in this paper we propose a new method to apply the generalized cross validation gcv as a stopping rule for the conjugate gradient cg in general to apply gcv to an iterative method one must estimate the trace of the so called influence matrix which appears in the denominator of the gcv function in the case of cg unlike what happens with stationary iterative methods the regularized solution has a nonlinear dependence on the noise which affects the data of the problem this fact is often pointed out as a cause of poor performance of gcv to overcome this drawback our proposal linearizes the dependence by computing the derivatives through iterative formulas we compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1d and 2d test problems

automated smoothing of image and other regularly spaced data this paper is primarily motivated by the problem of automatically removing unwanted noise from high dimensional remote sensing imagery the initial step involves the transformation of the data to a space of intrinsically lower dimensionality and the smoothing of images in the new space different images require different amounts of smoothing the signal assumed to be mostly smooth with relatively few discontinuities is estimated from the data using the method of generalized cross validation it is shown how the generalized cross validated thin plate smoothing spline with observations on a regular grid in d dimensions is easily approximated and computed in the fourier domain space domain approximations are also investigated the technique is applied to some remote sensing data

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

a new spatial regularization scheme for the identification of the geometric shape of an inclusion in a finite body this paper presents a system identification scheme to determine the geometric shape of an inclusion in a finite body the proposed algorithm is based on the minimization of the least squared errors between the measured displacement field and calculated displacement field by the finite element model the domain parameterization technique is adopted to manipulate the shape variation of an inclusion to stabilize the optimization process a new regularization function defined by the length of the boundary curve of an inclusion is added to the error function a variable regularization factor scheme is proposed for a consistent regularization effect the modified newton method with the active set method is adopted for optimization copyright 1999 john wiley sons ltd

shape sensing of 3d frame structures using an inverse finite element method abstract a robust and efficient computational method for reconstructing the elastodynamic structural response of truss beam and frame structures using measured surface strain data is presented known as shape sensing this inverse problem has important implications for real time actuation and control of smart structures and for monitoring of structural integrity the present formulation based on the inverse finite element method ifem uses a least squares variational principle involving section strains also known as strain measures of timoshenko theory for stretching torsion bending and transverse shear the present ifem methodology is based on strain displacement relations only without invoking force equilibrium consequently both static and time varying displacement fields can be reconstructed without the knowledge of material properties applied loading or damping characteristics two finite elements capable of modeling frame structures are derived using interdependent interpolations in which interior degrees of freedom are condensed out at the element level in addition relationships between the order of kinematic element interpolations and the number of required strain gauges are established several example problems involving cantilevered beams and three dimensional frame structures undergoing static and dynamic response are discussed to simulate experimentally measured strains and to establish reference displacements high fidelity msc nastran finite element analyses are performed furthermore numerically simulated measurement errors based on gaussian distribution are also considered in order to verify the stability and robustness of the methodology the ifem solution accuracy is examined with respect to various levels of discretization and the number of strain gauges

a strategy to identify exciting forces acting on structures the strategy for identifying the external forces acting on structures sketched by turco comput method appl mech eng 1998 165 291 306 is extended to the dynamic case starting from the measurement of the stress or strain history in a suitable number of points of the structure the exciting forces are reconstructed this inverse problem is tackled in the frequency domain by using the standard tools of computational mechanics along with a classical approach to treat ill conditioned problems some numerical results relating to pin jointed trusses are discussed in order to validate the proposed strategy copyright 2005 john wiley sons ltd

numerical solution to a two dimensional inverse heat conduction problem description de la methode de programmation dynamique pour la resolution de problemes inverses de conduction thermique application a un probleme impliquant la determination de 2 evolutions dans le temps inconnues de flux thermique sur les 2 faces d une plaque execution de differentes experiences numeriques pour etudier les effets du bruit et des parametres de ponderation

inverse boundary traction reconstruction with the bem abstract a boundary integral formulation is presented for the solution of the inverse elastostatics problem iesp of reconstructing missing boundary tractions in two dimensional structural members traction reconstruction may involve the determination of the location of the traction distribution in addition to its extent and amplitude the missing boundary tractions are rebuilt from measured quantities such as displacements strains or stresses these quantities may be obtained from sensors located at some internal or boundary points of the object the proposed formulation starts with an initial guess for the magnitude extent and location of the missing boundary tractions and proceeds towards the final traction distribution in a sequence of iterative steps the inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured quantities at each sensor location and the corresponding computed quantities for the assumed boundary traction distribution the constraints that the missing traction distribution lies within a certain portion of the boundary of the object are imposed this is done using the step retraction and inverse penalty function approach in which the objective function is augmented by the constraint equations using a penalty parameter the unknown traction distribution and its location are defined in terms of load and geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach a series of numerical examples involving the reconstruction of linear parabolic and trigonometric boundary tractions respectively are solved using the present approach the effect of gaussian errors in the sensors is also studied good reconstruction of the missing boundary tractions is obtained for the examples studied the advantages of the present boundary element formulation over the corresponding finite element formulations are also outlined

some new properties of the structured singular value j c doyle et al 1982 have shown that a necessary and sufficient condition for robust stability or robust performance in the h sup infinity frame work may be formulated as a bound on the structured singular value mu of a specific matrix m which includes information on the system model the controller the model uncertainty and the performance specifications often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices t which are of direct interest to the engineer e g sensitivity or complementary sensitivity the present paper shows how to derive bounds on sigma t from bounds on mu m

a boundary element formulation for the inverse elastostatics problem iesp of flaw detection a boundary integral formulation is presented for the detection of flaws in planar structural members from the displacement measurements given at some boundary locations and the applied loading such inverse problems usually start with an initial guess for the flaw location and size and proceed towards the final configuration in a sequence of iterative steps a finite element formulation will require a remeshing of the object corresponding to the revised flaw configuration in each iteration making the procedure computationally expensive and cumbersome no such remeshing is required for the boundary element approach the inverse problem is written as an optimization problem with the objective function being the sum of the squares of the differences between the measured displacements and the computed displacements for the assumed flaw configuration the geometric condition that the flaw lies within the domain of the object is imposed using the internal penalty function approach in which the objective function is augmented by the constraint using a penalty parameter a first order regularization procedure is also implemented to modify the objective function in order to minimize the numerical fluctuations that may be caused in the numerical procedure due to errors in the experimental measurements for displacements the flaw configuration is defined in terms of geometric parameters and the sensitivities with respect to these parameters are obtained in the boundary element framework using the implicit differentiation approach a series of numerical examples involving the detection of circular and elliptical flaws of various sizes and orientations are solved using the present approach good predictions of the flaw shape and location are obtained

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

non cartesian mri reconstruction with automatic regularization via monte carlo sure magnetic resonance image mri reconstruction from undersampled k space data requires regularization to reduce noise and aliasing artifacts proper application of regularization however requires appropriate selection of associated regularization parameters in this work we develop a data driven regularization parameter adjustment scheme that minimizes an estimate based on the principle of stein s unbiased risk estimate sure of a suitable weighted squared error measure in k space to compute this sure type estimate we propose a monte carlo scheme that extends our previous approach to inverse problems e g mri reconstruction involving complex valued images our approach depends only on the output of a given reconstruction algorithm and does not require knowledge of its internal workings so it is capable of tackling a wide variety of reconstruction algorithms and nonquadratic regularizers including total variation and those based on the l1 norm experiments with simulated and real mr data indicate that the proposed approach is capable of providing near mean squared error optimal regularization parameters for single coil undersampled non cartesian mri reconstruction

experimental study of random dynamic loads identification based on weighted regularization method abstract in the identification of random dynamic loads acting on the structure it is necessary to take into account the effect of measuring error of responses and estimation error of frequency response functions frfs for this purpose a dynamic load identification model was built in which the relative error bound of the identified loads was estimated to mitigate error propagation and ill posed problem during the process of identification a weighted regularization approach based on the proper orthogonal decomposition pod was proposed where the regularization parameter was selected by the gcv method to verify the feasibility and effectiveness of this method experiments of loads identification were performed on a plate structure which was randomly excited by three electromagnetic exciters the results show a good agreement between the identified random dynamic loads and the actually exerted loads

identification of geometric shapes and material properties of inclusions in two dimensional finite bodies by boundary parameterization a system identification scheme based on the boundary element method bem is proposed to determine geometric shape and elastic material properties of an inclusion in a finite body the proposed algorithm is based on the minimization of least squared errors between measured displacement and calculated displacement by a boundary element model a regularization function that consists of the geometric term and material property term is added to the error function to overcome ill posedness of inverse problems to deal with the shape variation of an inclusion during the optimization process the boundary parameterization technique is applied to the bem the recursive quadratic programming rqp technique with line search is used for optimization

optimal space varying regularization in iterative image restoration it has been shown that space variant regularization in image restoration provides better results than space invariant regularization however the optimal choice of the regularization parameter is usually unknown a priori in previous work the generalized cross validation gcv criterion was shown to provide accurate estimates of the optimal regularization parameter the author introduces a modified form of the gcv criterion that incorporates space variant regularization and data error terms furthermore he presents an efficient method for estimating the gcv criterion for the space variant case using iterative image restoration techniques this method performs nearly as well as the exact criterion for the image restoration problem in addition he proposes a wiener filter interpretation for choosing the local weighting of the regularization this interpretation suggests the use of a multistage estimation procedure to estimate the optimal choice of the local regularization weights experiments confirm the value of the modified gcv estimation criterion as well as the multistage procedure for estimating the local regularization weights

a boundary element based inverse problem in estimating transient boundary conditions with conjugate gradient method a boundary element method bem based inverse algorithm utilizing the iterative regularization method i e the conjugate gradient method cgm is used to solve the inverse heat conduction problem ihcp of estimating the unknown transient boundary temperatures in a multi dimensional domain with arbitrary geometry the results obtained by the cgm are compared with that obtained by the standard regularization method rm the error estimation based on the statistical analysis is derived from the formulation of the rm a 99 per cent confidence bound is thus obtained finally the effects of the measurement errors to the inverse solutions are discussed results show that the advantages of applying the cgm in the inverse calculations lie in that i the major difficulties in choosing a suitable quadratic norm determining a proper regularization order and determining the optimal smoothing or regularization coefficient in the rm are avoided and ii it is less sensitive to the measurement errors i e more accurate solutions are obtained 1998 john wiley sons ltd

an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem abstract in this paper we establish new density results for the equilibrium equations based on the denseness result of the elastic potential functions the cauchy problem for the equilibrium equations is investigated for this ill posed problem we construct a regularizing solution using the single layer potential function the well posedness of the regularizing solution as well as the convergence property is rigorously analyzed the advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented the method combines minimum norm solution with morozov discrepancy principle to solve an inverse problem convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method the numerical convergence accuracy and stability of the method with respect to the discretisation about the integral equations on pseudo boundary and the distance between the pseudo boundary and the real boundary of the solution domain and decreasing the amount of noise added into the input data respectively are also analysed with some examples

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

asymptotic optimality of the fast randomized versions of gcv and c_l in ridge regression and regularization ridge regression is a well known technique to estimate the coefficients of a linear model the method of regularization is a similar approach commonly used to solve underdetermined linear equations with discrete noisy data when applying such a technique the choice of the smoothing or regularization parameter h is crucial generalized cross validation gcv and mallows c l are two popular methods for estimating a good value for h from the data their asymptotic properties such as consistency and asymptotic optimality have been largely studied craven and wahba 1979 golub heath and wahba 1979 speckman 1985 very interesting convergence results for the actual random parameter given by gcv and c l have been shown by li 1985 1986 recently girard 1987 1989 has proposed fast randomized versions of gcv and c l the purpose of this paper is to show that the above convergence results also hold for these new methods

backfitting in smoothing spline anova a computational scheme for fitting smoothing spline anova models to large data sets with a near tensor product design is proposed such data sets are common in spatial temporal analyses the proposed scheme uses the backfitting algorithm to take advantage of the tensor product design to save both computational memory and time several ways to further speed up the backfitting algorithm such as collapsing component functions and successive over relaxation are discussed an iterative imputation procedure is used to handle the cases of near tensor product designs an application to a global historical surface air temperature data set which motivated this work is used to illustrate the scheme proposed

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

a boundary element solution of an inverse elasticity problem and applications to determining residual stress and contact stress abstract a boundary element solution is developed for an inverse elasticity problem in this inverse problem boundary conditions are incompletely specified strains can be determined experimentally in practice at a number of internal points and used as input to completely resolve these boundary conditions in this paper these strains including random errors are numerically simulated this inverse problem finds applications in the evaluation of residual stress and contact stress in contrast to previous studies the construction of the sensitivity matrix is embedded in the boundary element formulation thereby avoiding the solution of a series of forward problems further the effects of prescribed non zero boundary conditions and body forces are included in the relation between measured strains and the primary traction unknowns unfortunately the inverse problem is still ill posed physical constraints are introduced to stabilize the solution as a result the algorithm presented here has reasonable tolerance to error in the measurement of strains numerical examples are given to validate the inverse algorithm in these examples the input strains are numerically simulated and stable and accurate solutions are obtained with up to 5 random error in the input

a decoupled approach to filter design for stochastic systems abstractthis paper presents a new theorem to guarantee the almost sure exponential stability for a class of stochastic triangular systems by studying only the stability of each diagonal subsystems this result allows to solve the filtering problem of the stochastic systems with multiplicative noises by using the almost sure exponential stability concept two kinds of observers are treated the full order and reduced order cases

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

efficient algorithms for robust generalized cross validation spline smoothing generalized cross validation gcv is a widely used parameter selection criterion for spline smoothing but it can give poor results if the sample size n is not sufficiently large an effective way to overcome this is to use the more stable criterion called robust gcv rgcv the main computational effort for the evaluation of the gcv score is the trace of the smoothing matrix tra while the rgcv score requires both tra and tra 2 since 1985 there has been an efficient o n algorithm to compute tra this paper develops two pairs of new o n algorithms to compute tra and tra 2 which allow the rgcv score to be calculated efficiently the algorithms involve the differentiation of certain matrix functionals using banded cholesky decomposition

ck surface approximation from surface patches we study the problem of constructing a smooth approximant from a finite set of patches given on a surface defined by an equation x3 f x1 x2 as an approximant of f a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed error results and numerical results are given

linearized reduced order filtering a reduced order version of extended kalman filtering is presented in which both the filtering equation and the associated riccati equation have been reduced in dimension to allow for real time processing the procedure for designing the reduced order filter is similar to that for designing the extended kalman filter the same approximations being applied one technique useful for limiting the computational burden in a linearized filter design problems is presented and illustrated by an example the primary limitation of the result is that the nonlinearity must be in terms of the vector to be estimated

fading regularization mfs algorithm for inverse boundary value problems in two dimensional linear elasticity abstract we investigate the numerical reconstruction of the missing displacements dirichlet data and tractions neumann data on an inaccessible part of the boundary in the case of two dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part this inverse problem is solved using the fading regularization method originally proposed by cimetiere et al 2000 2001 for the laplace equation in conjunction with a meshless method namely the method of fundamental solutions mfs the stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to morozov s discrepancy principle morozov 1966

natural spline functions their associated eigenvalue problem we consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots we show that they are o left frac i 2m n right i 1 ldots n m wherem is the order of the spline andn the number of knots this result is of particular interest to prove optimality properties of the generalized cross validation method and had been conjectured by craven and wahba in a recent paper

load distribution modelling for pin jointed trusses by an inverse approach the paper considers the load distribution modelling for pin jointed trusses when an approximate value of the normal stress on a suitable number of elements is known this inverse problem is formulated in the least square sense directly relating the stress on the elements with the external loads and successively solved using an iterative scheme which uses a standard decomposition method for the stiffness matrix and the conjugate direction strategy in order to obtain a regularized solution of this problem and so cut out the errors contained in the known term of the equation system the generalized cross validation criterion is used in a form suitable for iterative methods some numerical results allow an analysis of the algorithm s performance

investigation of regularization parameters and error estimating in inverse elasticity problems the method of tarantola1 based on bayesian statistical theory for solving general inverse problems is applied to inverse elasticity problems and is compared to the spatial regularization technique presented in schnur and zabaras 2 it is shown that when normal gaussian distributions are assumed and the error in the data is uncorrelated the bayesian statistical theory takes a form similar to the deterministic regularization method presented earlier in schnur and zabaras 2 as such the statistical theory can be used to provide a statistical interpretation of regularization and to estimate error in the solution of the inverse problem examples are presented to demonstrate the effect of the regularization parameters and the error in the initial data on the solution

on computing the forces from the noisy displacement data of an elastic body this study is concerned with the accurate computation of the unknown forces applied on the boundary of an elastic body using its measured displacement data with noise vision based minimally intrusive force sensing using elastically deformable grasping tools is the motivation for undertaking this problem since this problem involves incomplete and inconsistent displacement force of ail elastic body it leads to an ill posed problem known as cauchy s problem in elasticity vision based displacement measurement necessitates large displacements of the elastic body for reasonable accuracy therefore we use geometrically non linear modelling of the elastic body which was not considered by others who attempted to solve cauchy s elasticity problem before we present two methods to solve the problem the first method uses the pseudo inverse of an over constrained system of equations this method is shown to be not effective when the noise in the measured displacement data is high we attribute this to the appearance of spurious forces at regions where there should not be any forces the second method focuses on minimizing the spurious forces by varying the measured displacements within the known accuracy of the measurement technique both continuum and frame elements are used in the finite element modelling of the elastic bodies considered in the numerical examples the performance of the two methods is compared using seven numerical examples all of which show that the second method estimates the forces with an error that is not more than the noise in the measured displacements an experiment was also conducted to demonstrate the effectiveness of the second method in accurately estimating the applied forces

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

a relaxation method of an alternating iterative algorithm for the cauchy problem in linear isotropic elasticity we propose two algorithms involving the relaxation of either the given dirichlet data boundary displacements or the prescribed neumann data boundary tractions on the over specified boundary in the case of the alternating iterative algorithm of kozlov et al 16 applied to cauchy problems in linear elasticity a convergence proof of these relaxation methods is given along with a stopping criterion the numerical results obtained using these procedures in conjunction with the boundary element method bem show the numerical stability convergence consistency and computational efficiency of the proposed method

a boundary element strategy for elastostatic inverse problems involving uncertain boundary conditions in this paper a convenient strategy is developed to find solutions for a class of uncertain boundary value problems by the boundary element method bem such problems are ill posed but ill conditioning of the associated algebraic systems of equations can be controlled to a large extent and useful boundary data can be obtained despite ill conditioning interior data of not only sufficient quantity but also of good quality at good locations contribute to good solutions our strategy permits the condition number of the algebraic systems as a function of interior data locations to be monitored such that measured data from displacement sensors and or strain sensors at locations found to be good ones for the problem at hand can be used the present strategy is based upon the concept of a green s function library through partitioning of the bem algebraic system algebraic systems are solved using least squares via singular value decomposition svd the library idea takes advantage of modern data storage and retrieval technology and permits the process of repeated trials in order to determine good data sensor locations to be done quickly and efficiently several numerical examples are given to demonstrate the strategy some examples examine the consequences of errors in measured data copyright 1999 john wiley sons ltd

numerical solution of nonlinear inverse heat conduction problem with a radiation boundary condition a finite difference solution for the transient nonlinear heat conduction equation in a finite slab with a radiation boundary condition is proposed an implicit finite difference approximation is used which enables accurate estimation of the surface temperature as well as prevention of oscillation of computed values at the surfaces a two time level implicit method is used while taylor s forward projection method is employed for taking account of the nonlinearities an iterative method is described which predicts unknown surface conditions an example is illustrated which is typical of those that arise in practical applications the results demonstrate that the method is remarkable in its stability to predict surface conditions without debilitation

relaxation procedures for an iterative mfs algorithm for the stable reconstruction of elastic fields from cauchy data in two dimensional isotropic linear elasticity we investigate two numerical procedures for the cauchy problem in linear elasticity involving the relaxation of either the given boundary displacements dirichlet data or the prescribed boundary tractions neumann data on the over specified boundary in the alternating iterative algorithm of kozlov et al 1991 the two mixed direct well posed problems associated with each iteration are solved using the method of fundamental solutions mfs in conjunction with the tikhonov regularization method while the optimal value of the regularization parameter is chosen via the generalized cross validation gcv criterion an efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase is also presented the mfs based iterative algorithms with relaxation are tested for cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence stability accuracy and computational efficiency of the proposed method

l sub 2 and l sub 1 beamformers recursive implementation and performance analysis studies array beamformers as optimal waveform estimators the authors apply an inverse problem formulation presenting an integrated design to quadratic l sub 2 and least absolute value l sub 1 beamformers the general solution of the l sub 2 beamformers is parameterized by a regularizing parameter that weights the confidence placed by the designer on prior knowledge versus the quality of the measurements this regularizing parameter is used to establish an equivalence between alternative l sub 2 beamformers the authors then develop time recursive implementations of the l sub 2 and l sub 1 beamformers the performance of these beamformers is studied next the authors show that 1 in the presence of correlated arrivals the mmse beamformer uses constructively the correlation between incoming signals in reconstructing the estimated field while rejecting the uncorrelated returns and 2 the l sub 1 beamformer has the ability to adjust itself to unexpected noise conditions because it is considerably more robust than the l sub 2 beamformers to unmodeled impulsive noise or to the occurrence of malfunctioning sensors the analysis is confirmed by simulated studies

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

an inverse method for determining elastic material properties and a material interface the problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

a reduced order observer for non linear discrete time systems abstract in this note a simple and useful reduced order observer for a large class of non linear discrete time systems is presented neither detectability of the linear part nor resolution of the sylvester constraint is required from the stability analysis we show that the proposed state observer has strong tracking properties under very general conditions and generalizes results in m boutayeb et al ieee trans automat control 42 4 1997 581 586 high performances from the design and implementation point of view will be shown through three numerical examples of physical processes largely used in industrial applications

smoothing spline anova for exponential families with application to the wisconsin epidemiological study of diabetic retinopathy the 1994 neyman memorial lecture let y i i 1 n be independent observations with the density of y i of the form h y i f i exp y i f i b f i c y i where b and c are given functions and b is twice continuously differentiable and bounded away from 0 let i t f t i where t t 1 t d t 1 t d t the t are measurable spaces of rather general form and f is an unknown function on t with some assumed smoothness properties given y i t i i 1 n it is desired to estimate f t for t in some region of interest contained in t we develop the fitting of smoothing spline anova models to this data of the form f t c f t f t t the components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric anova the estimate of f is obtained as the minimizer in an appropriate function space of l y f j f j f where l y f is the negative log likelihood of y y 1 y n given f the j j are quadratic penalty functionals and the anova decomposition is terminated in some manner there are five major parts required to turn this program into a practical data analysis tool 1 methods for deciding which terms in the anova decomposition to include model selection 2 methods for choosing good values of the smoothing parameters 3 methods for making confidence statements concerning the estimate 4 numerical algorithms for the calculations and finally 5 public software in this paper we carry out this program relying on earlier work and filling in important gaps the overall scheme is applied to bernoulli data from the wisconsin epidemiologic study of diabetic retinopathy to model the risk of progression of diabetic retinopathy as a function of glycosylated hemoglobin duration of diabetes and body mass index it is believed that the results have wide practical application to the analysis of data from large epidemiologic studies

bayesian inference of models and hyperparameters for robust optical flow estimation selecting optimal models and hyperparameters is crucial for accurate optical flow estimation this paper provides a solution to the problem in a generic bayesian framework the method is based on a conditional model linking the image intensity function the unknown velocity field hyperparameters and the prior and likelihood motion models inference is performed on each of the three levels of this so defined hierarchical model by maximization of marginalized a posteriori probability distribution functions in particular the first level is used to achieve motion estimation in a classical a posteriori scheme by marginalizing out the motion variable the second level enables to infer regularization coefficients and hyperparameters of non gaussian m estimators commonly used in robust statistics the last level of the hierarchy is used for selection of the likelihood and prior motion models conditioned to the image data the method is evaluated on image sequences of fluid flows and from the middlebury database experiments prove that applying the proposed inference strategy yields better results than manually tuning smoothing parameters or discontinuity preserving cost functions of the state of the art methods

boundary element solution for the cauchy problem in linear elasticity using singular value decomposition abstract in this paper the singular value decomposition svd truncated at an optimal number is analysed for obtaining approximate solutions to ill conditioned linear algebraic systems of equations which arise from the boundary element method bem discretisation of an ill posed boundary value problem in linear elasticity the regularisation parameter namely the optimal truncation number is chosen according to the discrepancy principle the numerical results obtained confirm that the svd bem produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data

regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and sure based methods regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values we focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive possibly complex gaussian noise generalized cross validation gcv and weighted mean squared error mse approaches based on stein s unbiased risk estimate sure need the jacobian matrix of the nonlinear reconstruction operator representative of the iterative algorithm with respect to the data we derive the desired jacobian matrix for two types of nonlinear iterative algorithms a fast variant of the standard iterative reweighted least squares method and the contemporary split bregman algorithm both of which can accommodate a wide variety of analysis and synthesis type regularizers the proposed approach iteratively computes two weighted sure type measures predicted sure and projected sure which require knowledge of noise variance 2 and gcv which does not need 2 for these algorithms we apply the methods to image restoration and to magnetic resonance image mri reconstruction using total variation and an analysis type l1 regularization we demonstrate through simulations and experiments with real data that minimizing predicted sure and projected sure consistently lead to near mse optimal reconstructions we also observe that minimizing gcv yields reconstruction results that are near mse optimal for image restoration and slightly suboptimal for mri theoretical derivations in this paper related to jacobian matrix evaluations can be extended in principle to other types of regularizers and reconstruction algorithms

an effective algorithm for reconstructing boundary conditions in elastic solids this paper discusses the reconstruction of boundary conditions for linear elastic solids when there are not enough known boundary conditions but the stress tensor is known in a sufficient number of internal points starting with a refined boundary elements discretization of the elastic solid an iterative strategy based on the conjugate gradient method is presented this iterative strategy uses a monte carlo simulation of the generalized cross validation criterion to select the optimal solution the proposed algorithm possesses some advantages with respect to direct approaches indeed it furnishes almost as accurate results but at a low computational cost some numerical results allow an analysis of the algorithm s performance

an efficient method for calculating smoothing splines using orthogonal transformations a procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m 1 using an orthogonal factorization is presented the procedure substantially overcomes the problem of ill conditioning encountered by a recently developed method which employs a cholesky factorization but still requires only orderm 2 n operations and ordermn storage

plant and controller reduction problems for closed loop performance model reduction problems which consider preserving closed loop performance h sub 2 h sub infinity and mu in the presence of reduction error are developed these are formulated as weighted multiplicative error problems for plant reduction and weighted additive error problems for controller reduction with the weight function incorporating explicitly such control information as the desired sensitivity operator bound the setpoint disturbance spectrum and the plant uncertainties these problems are efficiently solved using the frequency weighted balanced realization technique the benefits of these reduction problems are illustrated with examples taken from the control of a binary distillation column

a least squares variational method for full field reconstruction of elastic deformations in shear deformable plates and shells abstract a variational principle is formulated for the inverse problem of full field reconstruction of three dimensional plate shell deformations from experimentally measured surface strains the formulation is based upon the minimization of a least squares functional that uses the complete set of strain measures consistent with linear first order shear deformation theory the formulation which accommodates for transverse shear deformation is applicable for the analysis of thin and moderately thick plate and shell structures the main benefit of the variational principle is that it is well suited for c 0 continuous displacement finite element discretizations thus enabling the development of robust algorithms for application to complex civil and aeronautical structures the methodology is especially aimed at the next generation of aerospace vehicles for use in real time structural health monitoring systems