on the geometry of the term structure of interest rates we present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates this includes an extension of the frobenius theorem for fr echet spaces in particular this approach puts new light on many of the classical models such as the hull white extended vasicek and cox ingersoll ross short rate models the notion of a finite dimensional realization fdr is central for this analysis we motivate it classify all generic fdrs and provide some new results for the corresponding factor processes such as hypoellipticity of its generators and the existence of smooth densities furthermore we include finite dimensional external factors thus admitting a stochastic volatility structure

approximation of ill posed volterra problems via predictor corrector regularization methods first kind volterra problems arise in numerous applications from inverse problems in mathematical biology to inverse heat conduction problems unfortunately such problems are also ill posed due to lack of continuous dependence of solutions on data consequently numerical methods to solve first kind volterra equations are effective only when regularizing features are built into the algorithms or used to control stepsize classical methods often combine numerical discretization with tikhonov regularization but in doing so the underlying volterra or causal nature of the original problem is often destroyed instead a predictor corrector type of numerical method is proposed which combines at each step local regularization ideas with the use of small intervals of future data the result is a regularized numerical method which retains much of the causal nature of the volterra problem and may be solved in fast sequential steps often improving upon the performance of classical algorithms such as those

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

approximately exact inference for the common odds ratio in several 2 2 tables abstract the conditional maximum likelihood estimator mle of the common odds ratio in a sequence of independent 2 2 tables is known to be superior to the mantel haenszel estimator in terms of asymptotic efficiency and has the further advantage that its exact distribution is known however a long standing barrier to the widespread use of this estimator has been computational intractability in particular the calculation of significance levels confidence sets and power based on the exact distribution requires fast and efficient algorithms an important class of such algorithms form the basis of statxact a software package able to solve various aspects of the exact inference problem for a sequence of several 2 2 tables in real time we provide an alternative methodology by developing several useful saddlepoint approximations to the exact distribution of the conditional mle the approximations are derived from an interesting representation for hypergeometric random variables as a sum of independent

globally convergent iterative numerical schemes for nonlinear variational image smoothing and segmentation on a multiprocessor machine we investigate several iterative numerical schemes for nonlinear variational image smoothing and segmentation implemented in parallel a general iterative framework subsuming these schemes is suggested for which global convergence irrespective of the starting point can be shown we characterize various edge preserving regularization methods from the image processing literature involving auxiliary variables as special cases of this general framework as a by product global convergence can be proven under conditions slightly weaker than these stated in the literature efficient krylov subspace solvers for the linear parts of these schemes have been implemented on a multiprocessor machine the performance of these parallel implementations has been assessed and empirical results concerning convergence rates and speed up factors are reported

acceleration methods for total variation based image denoising in this paper we apply a fixed point method to solve the total variation based image denoising problem an algebraic multigrid method is used to solve the corresponding linear equations krylov subspace acceleration is adopted to improve convergence in the fixed point iteration a good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid numerical experiments demonstrate that this method is efficient and robust even for images with large noise to signal ratios

local regularization for the nonlinear inverse autoconvolution problem we develop a local regularization theory for the nonlinear inverse autoconvolution problem unlike classical regularization techniques such as tikhonov regularization this theory provides regularization methods that preserve the causal nature of the autoconvolution problem allowing for fast sequential numerical solution cal o rn 2 r 2n flops where r ll n for the method discussed in this paper as applied to the nonlinear problem in comparison the cost for tikhonov regularization applied to a general linear problem is cal o n 3 flops we prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both l 2 and continuous data we propose several regularization methods and provide a theoretical basis for their convergence of note is that this class of methods does not require an initial guess of the unknown solution our numerical results confirm the effectiveness of the methods with results comparing favorably to numerical examples found in the literature for the autoconvolution problem e g g fleischer r gorenflo and b hofmann zamm z angew math mech 79 1999 pp 149 159 for examples using tikhonov regularization with total variation constraints and j janno inverse problems 16 2000 pp 333 348 for examples using the method of lavrent ev this especially seems to be true when it comes to the recovery of sharp features in the unknown solution we also show the effectiveness of our method in cases not covered by the theory

on effective methods for implicit piecewise smooth surface recovery this paper considers the problem of reconstructing a piecewise smooth model function from given measured data the data are compared to a field which is given as a possibly nonlinear function of the model a regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities regularization operators related to total variation tv are therefore employed two popular methods are modified tv and huber s function both contain a parameter which must be selected the huber variant provides a more natural approach for selecting its parameter and we use this to propose a scheme for both methods our selected parameter depends both on the resolution and on the model average roughness thus it is determined adaptively its variation from one iteration to the next yields additional information about the progress of the regularization process the modified tv operator has a smoother generating function nonetheless we obtain a huber variant with comparable and occasionally better performance for large problems e g high resolution the resulting reconstruction algorithms can be tediously slow we propose two mechanisms to improve efficiency the first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution the second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function

comparison of different maximum likelihood estimators in a small sample logistic regression with two independent binary variables in order to examine the bias of the estimate of the log odds ratio in a 2 2 contingency table walter computed the entire distribution of the estimated log odds ratio using various small sample sizes this is equivalent to computing the distribution of the estimated parameter b1 in a logistic regression with one independent binary variable in this paper the distributions of the estimated parameters b1 and b2 for two independent binary variables are computed for some small sample logistic regressions using six different estimation methods based on maximum likelihood these estimates are then compared to the true parameter values the best estimation method depends on the frequency of the outcome of interest and on whether the bias or mean square error is considered more important

existence of invariant manifolds for stochastic equations in infinite dimension we provide a frobenius type existence result for finite dimensional invariant submanifolds for stochastic equations in infinite dimension in the spirit of da prato and zabczyk 5 we recapture and make use of the convenient calculus on frechet spaces as developed by kriegl and michor 16 our main result is a weak version of the frobenius theorem on frechet spaces as an application we characterize all finite dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates

invariant manifolds for weak solutions to stochastic equations viability and invariance problems related to a stochastic equation in a hilbert space h are studied finite dimensional invariant c2 submanifolds of h are characterized we derive nagumo type conditions and prove a regularity result any weak solution which is viable in a finite dimensional c2 submanifold is a strong solution these results are related to finding finite dimensional realizations for stochastic equations there has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

dynamic regularization of input estimation problems by explicit block inversion input estimation problems in structural dynamics often lead to solving damped least squares problems explicit block inversion algorithms are developed to invert the associated upper block triangular toeplitz matrix moreover the derived inversion algorithms enable the introduction of a new type of sequential regularization technique for optimal regularization parameters it is demonstrated by a numerical example that the solution using the proposed regularization technique has an upper bound equal to zeroth order tikhonov regularization

estimating functions and approximate conditional likelihood summary the approximate conditional likelihood method proposed by cox reid 1987 is applied to the estimation of a scalar parameter 0 in the presence of nuisance parameters the estimating function of 0 based on the approximate conditional likelihood is shown to be preferable to that based on the profile likelihood a sufficient condition for both approaches to be equivalent is given the role of parameter orthogonality is emphasized several examples including bivariate normal means with known coefficient of variation are presented

bayesian video dejittering by the bv image model line jittering or random horizontal displacement in video images occurs when the synchronization signals are corrupted in video storage media or by electromagnetic interference in wireless video transmission the goal of intrinsic video dejittering is to recover the ideal video directly from the observed jittered and often noisy frames the existing approaches in the literature are mostly based on local or semilocal filtering techniques and autoregressive image models and are complemented by various image processing tools in this paper based on the statistical rationale of bayesian inference we propose the first variational dejittering model based on the bounded variation bv image model which is global clean and self contained and intrinsically combines dejittering with denoising the mathematical properties of the model are studied based on the direct method of calculus of variations we design one effective algorithm and present its computational implementation based on techniques from numeri

adaptive wavelet estimator for nonparametric density deconvolution 1999 institute of mathematical statistics the electronic version of this article is the complete one and can be found online at http projecteuclid org euclid aos 1017939249

a convergent overlapping domain decomposition method for total variation minimization in this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint to our knowledge this is the first successful attempt of addressing such a strategy for the nonlinear nonadditive and nonsmooth problem of total variation minimization we provide several numerical experiments showing the successful application of the algorithm for the restoration of 1d signals and 2d images in interpolation inpainting problems respectively and in a compressed sensing problem for recovering piecewise constant medical type images from partial fourier ensembles

generalized mantel haenszel estimators for k 2 x j tables mickey and elashoff 1985 biometrics 41 623 635 gave an extension of mantel haenszel estimation to log linear models for 2 x j x k tables their extension yields two generalizations of the mantel haenszel odds ratio estimator to k 2 x j tables this paper provides variance and covariance estimators for these generalized mantel haenszel estimators that are dually consistent i e consistent in both large strata and sparse data and presents comparisons of the efficiency of the generalized mantel haenszel estimators

deconvolution based partial volume correction in raclopride pet and monte carlo comparison to mr based method abstract in this work we evaluated three iterative deconvolution algorithms and compared their performance to partial volume pv correction based on structural imaging in brain positron emission tomography pet using a database of monte carlo simulated images we limited our interest to quantitative radioligand pet imaging particularly to 11c raclopride and striatal imaging the studied deconvolution methods included richardson lucy reblurred van cittert and reblurred van cittert with the total variation regularization we studied the bias and variance of the regional estimates of binding potential bp values and the accuracy of regional tacs as a function of the applied image processing the resolution noise tradeoff in parametric bp images was addressed as well the regional bp values and tacs obtained by deconvolution were almost as accurate than those by structural imaging based pv correction gtm method when the ideal volumes of interests vois were used to extract tacs from the images for deconvolution methods the ideal vois were slightly eroded from the exact anatomical voi to limit the bias due to tissue fraction effect which is not corrected for by deconvolution based methods for the gtm method the ideal vois were the exact anatomical vois the bp values and tacs by deconvolution were less affected by segmentation and registration errors than those with the gtm based pv correction the bp estimates and tacs with deconvolution based pv correction were more accurate than bps and tacs derived without pv correction the parametric images obtained by the deconvolution based pv correction showed considerably improved resolution with only slightly increased noise level compared to the case with no pv correction the reblurred van cittert method was the best of the studied deconvolution methods we conclude that the deconvolution is an interesting alternative to structural imaging based pv correction as it leads to quantification results of similar accuracy and it is less prone to registration and segmentation errors than structural imaging based pv correction moreover pv corrected parametric images can be readily computed based on deconvolved dynamic images

regularization for curve representations uniform convergence for discontinuous solutions of ill posed problems this paper is concerned with a new approach for regularizing problems with discontinuous solutions regularization for curve representations the idea of this approach is to represent a discontinous function as a curve with parameterization a t b t a combination with nonlinear tikhonov regularization then yields uniform convergence of the regularized solutions the method is applied to deblurring and denoising problems in signal processing a numerical example for a deblurring problem is presented

noise suppression by removing singularities a method is proposed for suppressing gaussian noise by extracting local singularities the method is based on truncating riemann series decomposition whose components naturally characterize different orders of holder regularity the approach yields a single step filtering technique whose performance is comparable to three step wavelet decomposition and thresholding techniques

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

using qr factorization and svd to solve input estimation problems in structural dynamics input estimation problems in structural mechanics are in general ill posed treatment of these problems requires that the problems are reformulated usually by numerical regularization techniques several methods of solving input estimation problems which take the form of structured block matrix problems are studied in this paper the investigated methods are based on forming the normal equations qr factorization and singular value decomposition a specific block qr factorization algorithm utilizing the structure of the associated upper block triangular toeplitz matrix is also introduced moreover a criterion for choosing the level of regularization based on the data used to construct l curves is suggested numerical examples are given to illustrate various numerical aspects of the different methods and the performance of the advocated criterion

restoration of color images by vector valued bv functions and variational calculus we analyze a variational problem for the recovery of vector valued functions and compute its numerical solution the data of the problem are a small set of complete samples of the vector valued function and some significant incomplete information where the former are missing the incomplete information is assumed as the result of a distortion with values in a lower dimen sional manifold for the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints we show the existence of minimizers in the space of vector valued bounded variation functions for the computa tion of minimizers we provide a stable and efficient method first we approximate the functional by coercive functionals on w 1 2 in terms of convergence then we realize approximations of minimiz ers of the latter functionals by an iterative procedure to solve the pde system of the corresponding euler lagrange equations the numerical implementation comes naturally by finite element dis cretization we apply the algorithm to the restoration of color images from limited color information and gray levels where the colors are missing the numerical experiments show that this scheme is very fast and robust the reconstruction capabilities of the model are shown also from very limited randomly distributed color data several examples are included from the real restoration problem of a mantegna s art frescoes in italy

phaseless inverse scattering problems in three dimensions three dimensional inverse scattering problems in the frequency domain are considered in the case when only the modulus of the scattered field is given while the phase is unknown uniqueness theorems are proved

invariance of closed convex cones for stochastic partial differential equations abstract the goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation spde driven by a wiener process and a poisson random measure and to provide conditions on the parameters of the spde which are necessary and sufficient

convergence of a fixed point iteration method for the osv model in image processing image denoising and texture extraction are important problems in which many new methods recently have been developed one of the most important models is the osv model s osher a sole l vese image decomposition and restoration using total variation minimization and the h 1 norm multiscale model simul a siam interdisciplinary j 1 3 2003 349 370 which is constructed by the total variation and h 1 norm this paper proves the existence of the minimizer of the functional from the osv model and analyzes the convergence of an iterative method for solving the problems our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated euler lagrange equation and the limit of our iterations satisfies the minimizer of the functional from the osv model in numerical experiments we compare the numerical results of our works with those of the rof model l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms phys d 60 1992 259 268

identification of discontinuous parameters in flow equations a problem of identifying possibly discontinuous diffusion coefficients in parabolic equations is considered general theorems on existence and convergence of galerkin approximations are proved in l 1 setting classes of functions of bounded variation are discussed and the variation estimates are obtained a double discretization method with the variations constraints is used in two and three dimensional problems and the numerical experiments are presented

a comparison of several point estimators of the odds ratio in a single 2 x 2 contingency table the relative performance of the unconditioned maximum likelihood estimators umles conditional mles cmles and jewell type estimators of the odds ratio or and its logarithm were investigated in sets of single 2 x 2 contingency tables the tables were generated by complete enumeration of all possible cell frequencies consistent with a single fixed margin the bias mean squared error mse and average absolute error aae were computed for all estimators using the individual table probabilities as weights the results showed that for the or jewell s estimator usually had smaller bias mse and aae than either of the mles while the differences were often slight for mse and aae for bias it was sometimes substantial for the log or the umle usually had the lowest bias and its mse and aae were only slightly greater than those for the other estimators overall we recommend estimation on the log scale using the umle if or is to be estimated jewell s method had strong merit although it is nonsymmetric with respect to the table orientation in view of this the umle may again be favoured in some situations

an empirical evaluation of the use of conditional and unconditional likelihoods for case control data a monte carlo study is made of conditional and unconditional likelihoods in the analysis of case control data where the underlying disease incidence is assumed to follow a logistic model results indicate that the full likelihood may yield estimates of relative risk that are unacceptably biased upwards for strata with fewer than 20 cases and 20 controls the simulations support the need for better approximations to the conditional likelihood with 1 to r matched studies the bias of the full likelihood is excessive for r less than 10 suggesting the general use of the conditional likelihood in matched pair studies the conditional likelihood tends to underestimate the true risk when there are fewer than 50 pairs

an algorithm for image removals and decompositions without inverse matrices partial differential equation pde based methods in image processing have been actively studied in the past few years one of the effective methods is the method based on a total variation introduced by rudin oshera and fatemi rof l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms physica d 60 1992 259 268 this method is a well known edge preserving model and an useful tool for image removals and decompositions unfortunately this method has a nonlinear term in the equation which may yield an inaccurate numerical solution to overcome the nonlinearity a fixed point iteration method has been widely used the nonlinear system based on the total variation is induced from the rof model and the fixed point iteration method to solve the rof model is introduced by dobson and vogel d c dobson c r vogel convergence of an iterative method for total variation denoising siam j numer anal 34 5 1997 1779 1791 however some methods had to compute inverse matrices which led to roundoff error to address this problem we developed an efficient method for solving the rof model we make a sequence like richardson s method by using a fixed point iteration to evade the nonlinear equation this approach does not require the computation of inverse matrices the main idea is to make a direction vector for reducing the error at each iteration step in other words we make the next iteration to reduce the error from the computed error and the direction vector we describe that our method works well in theory in numerical experiments we show the results of the proposed method and compare them with the results by d dobson and c vogel and then we confirm the superiority of our method

high accuracy combination algorithm and a posteriori error estimation for solving the first kind abel integral equations abstract this paper presents high accuracy combination algorithm for solving the first kind abel integral equations to avoid solving ill posed problems we transform the first kind abel integral equation to the second kind volterra integral equation with a continuous kernel expressed by a weakly singular integral by using integration rules the approximation of this kernel can be easily computed then two quadrature algorithms for solving abel integral equations are proposed which possess accuracy order o h 1 0 h 2 moreover a posteriori error estimate for the algorithms is derived both theory and numerical examples show that it is effective and saves storage capacity and computational work

in vivo impedance imaging with total variation regularization we show that electrical impedance tomography eit image reconstruction algorithms with regularization based on the total variation tv functional are suitable for in vivo imaging of physiological data this reconstruction approach helps to preserve discontinuities in reconstructed profiles such as step changes in electrical properties at interorgan boundaries which are typically smoothed by traditional reconstruction algorithms the use of the tv functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation this cannot be efficiently solved with traditional optimization techniques such as the newton method we explore two implementations methods for regularization with the tv functional the lagged diffusivity method and the primal dual interior point method pd ipm first we clarify the implementation details of these algorithms for eit reconstruction next we analyze the performance of these algorithms on noisy simulated data finally we show reconstructed eit images of in vivo data for ventilation and gastric emptying studies in comparison to traditional quadratic regularization tv regulariza tion shows improved ability to reconstruct sharp contrasts

interest rates and information geometry the space of probability distributions on a given sample space possesses natural geometric properties for example in the case of a smooth parametric family of probability distributions on the real line the parameter space has a riemannian structure induced by the embedding of the family into the hilbert space of square integrable functions and is characterized by the fisher rao metric in the non parametric case the relevant geometry is determined by the spherical distance function of bhattacharyya in the context of term structure modelling we show that the derivative of the discount function with respect to the time left until maturity gives rise to a probability density this follows as a consequence of the positivity of interest rates therefore by mapping the density functions associated with a given family of term structures to hilbert space the resulting metrical geometry can be used to analyse the relationship of yield curves to one another we show that the general arbitragefree yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line it follows that the theory of interest rate dynamics can be represented by a class of processes in hilbert space we also derive the dynamics for the central moments associated with the distribution determined by the yield curve

a viscosity method with no spectral radius requirements for the split common fixed point problem abstract this paper is concerned with an algorithmic solution to the split common fixed point problem in hilbert spaces our method can be regarded as a variant of the viscosity approximation method under very classical assumptions we establish a strong convergence theorem with regard to involved operators belonging to the wide class of quasi nonexpansive operators in contrast with other related processes our algorithm does not require any estimate of some spectral radius the technique of analysis developed in this work is new and can be applied to many other fixed point iterations numerical experiments are also performed with regard to an inverse heat problem

existence of affine realizations for stochastic partial differential equations driven by levy processes the goal of this paper is to clarify when a semilinear stochastic partial differential equation driven by levy processes admits an affine realization our results are accompanied by several examples arising in natural sciences and economics

a new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location this work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body beyond the obvious merits of its applications this problem also presents significant mathematical challenges the near incompressibility inherent in the model of linear elasticity in the body gives rise to the locking effect and necessitates a unique treatment of both the direct and inverse problems a general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least squares mols objective functional the mols functional is shown to be convex thus overcoming the nonconvexity of the classical output least squares ols functional and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters generalized derivative formulas for the coefficient to solution map are also given along with a complete convergence analysis

a comparison of tests of homogeneity of odds ratios in k 2 2 tables we conducted a simulation study to determine the performance of nine procedures for testing the homogeneity of odds ratios in k 2 2 contingency tables we recommend tarone s approximate score test based on the mantel haenszel estimator of the common odds ratio for use in practice we also recommend a non iterative statistic developed by gart and based on the modified woolf estimator of the common odds ratio for very large samples in balanced or mildly unbalanced designs we base our recommendation of a statistic on its performance in terms of size and power in comparison with the other statistics considered

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

a posteriori error control for the binary mumford shah model the binary mumford shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision ranging from basic segmentation and labeling to object reconstruction this paper presents robust a posteriori error estimates for a natural error quantity namely the area of the non properly segmented region to this end a suitable strictly convex and non constrained relaxation of the originally non convex functional is investigated and repin s functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the l norm in combination with a suitable cut out argument a fully practical estimate for the area mismatch is derived this estimate is incorporated in an adaptive meshing strategy two different adaptive primal dual finite element schemes and the most frequently used finite difference discretization are investigated and compared numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications

an efficient numerical method for solving abel integral equation in this paper a numerical method for solving abel integral equation is presented we first convert the integral equation to the algebraic equation then we find numerical inversion of laplace transform by power series at last the pade approximants is effectively used to improve the convergence rate and accuracy of the computed series the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to implement

convergence of fixed point iteration for deblurring and denoising problem abstract to solve deblurring and denoising problem we present a proof of the global and linear convergence of fixed point iteration as well as an estimate for the rate of convergence is given we show the equivalence among three different iterative methods half quadratic regularization iteration based on bergman distance weiszfeld s method

laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging in the present paper we consider the problem of laplace deconvolution with noisy discrete non equally spaced observations on a finite time interval which appears in many different contexts we propose a new method for laplace deconvolution which is based on expansions of the convolution kernel the unknown function and the observed signal over laguerre functions basis which acts as a surrogate eigenfunction basis using regression setting the expansion results in a small system of linear equations with the matrix of the system being triangular and toeplitz the number m of the terms in the expansion of the estimator is controlled via complexity penalty the advantage of this methodology is that it leads to very fast computations produces no boundary effects due to extension at zero and cut off at t and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions we emphasize that in the present paper we consider the true observational model with possibly non equispaced observations which are available on a finite interval of length t and account for the bias associated with this model the study is motivated by perfusion imaging using a short injection of contrast agent a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors presence of a tuning parameter a allows to choose the most advantageous time units so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution in addition absence of boundary effects allows for an accurate estimator of the tissue blood flow parameter which is of great interest to medical doctors the methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast efficient accurate usable from a practical point of view and competitive

an alternative approach on the existence of affine realizations for hjm term structure models we propose an alternative approach on the existence of affine realizations for heath jarrow and morton interest rate models it is applicable to a wide class of models and simultaneously it is conceptually rather comprehensible we also supplement some known existence results for particular volatility structures and provide further insights into the geometry of term structure models

the large sample relative efficiency of the mantel haenszel estimator in the fixed strata case the large sample relative efficiency of the mantel haenszel estimator psi mh of an odds ratio is investigated for the case in which the number of strata is fixed and the sample sizes within each stratum increase indefinitely the results show that psi mh is very efficient over a wide range of designs likely to occur in practice however conditions are identified under which the relative efficiency of this widely used estimator can be unusually low

a new perturbative technique for solving integro partial differential equations integro partial differential equations occur in many contexts in mathematical physics typical examples include time dependent diffusion equations containing a parameter e g the temperature that depends on integrals of the unknown distribution function the standard approach to solving the resulting nonlinear partial differential equation involves the use of predictor corrector algorithms which often require many iterations to achieve an acceptable level of convergence in this paper we present an alternative procedure that allows us to separate a family of integro partial differential equations into two related problems namely i a perturbation equation for the temperature and ii a linear partial differential equation for the distribution function we demonstrate that the variation of the temperature can be determined by solving the perturbation equation before solving for the distribution function convergent results for the temperature are obtained by recasting the divergent perturbation expa

on the asymptotic inefficiency of certain noniterative estimators of a common relative risk or odds ratio nurminen 1981 gives conditions for the asymptotic efficiency of general weighted mean estimators his application is a noniterative estimator of a common relative risk for binomial variates this summary estimator and by implication the mantel haenszel summary odds ratio estimator are asserted to be asymptotically efficient we examine conditions for the asymptotic efficiency in the unconditional sample space of the mantelhaenszel estimator for a common odds ratio the estimator of a common relative risk considered by nurminen and tarone s 1981 estimator of a common relative risk for binomial variables contrary to nurminen we find that except for a few degenerate cases these estimators are inefficient whenever the common odds ratio or common relative risk is not equal to 1

methods of estimation in log odds ratio regression models mccullagh s 1984 journal of the royal statistical society series b 46 250 256 approximation to the conditional maximum likelihood estimator in log odds ratio regression models is shown to have negligible asymptotic bias unless the odds ratios are large and the sample sizes in individual 2 x 2 tables are very small in application to two sets of case control data it yields results virtually indistinguishable from those of the conditional analysis a generalization of the mantel haenszel estimator proposed by davis 1985 biometrics 41 487 495 does not approximate the conditional results nearly as well

a first order sequential predictor corrector regularization method for ill posed volterra equations a discrete sequential predictor corrector regularization method for the solution of ill posed volterra equations based on affine continuation into the future interval is presented an infinite dimensional analogue is derived and analyzed stability estimates and convergence in the cases of exact and noisy data are proved a convergence rate result is proved for finitely smoothing kernels if the exact solution is mathcal c 1 alpha smooth numerical examples are presented the results are compared to other standard regularization methods

parameter identification by regularization for surface representation via the moving grid approach we consider the identification of a diffusion parameter in a second order elliptic equation in two dimensions by interior measurements the diffusion parameter is assumed to have discontinuities for its reconstruction we propose regularization algorithms with an adaptive grid the grid is adapted according to a measure of the smoothness of the regularized solution for the numerical computation we compare several iterative methods such as the minimal error method the steepest descent method and an inexact iteratively regularized gauss newton method the computations show that these algorithms can effectively identify the discontinuities

existence of affine realizations for levy term structure models we investigate the existence of affine realizations for term structure models driven by levy processes it turns out that we obtain more severe restrictions on the volatility than in the classical diffusion case without jumps as special cases we study constant direction volatilities and the existence of short rate realizations

an infeasible primal dual algorithm for total bounded variation based inf convolution type image restoration in this paper a primal dual algorithm for total bounded variation tv type image restoration is analyzed and tested analytically it turns out that employing a global boldsymbol l s regularization with 1 s leq 2 in the dual problem results in a local smoothing of the tv regularization term in the primal problem the local smoothing can alternatively be obtained as the infimal convolution of the ell r norm with r 1 s 1 1 and a smooth function in the case r s 2 this results in gauss tv type image restoration the globalized primal dual algorithm introduced in this paper works with generalized derivatives converges locally at a superlinear rate and is stable with respect to noise in the data in addition it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration a comprehensive numerical study ends the paper

likelihood ratio tests with three way tables likelihood ratio lr tests for association and for interaction are examined for three way contingency tables particularly the widely used 2 2 k table mutual information identities are used to characterize the information decomposition and the logical relationship between the omnibus lr test for conditional independence across k strata and its two independent components lr tests for interaction and for uniform association the latter two tests are logically connected to formulating a natural two step test for conditional independence the proposed two step test with reduced nominal levels is suggested instead of the breslow day test and the cochran mantel haenszel test this yields efficient interval estimation for both the interaction parameter and the common odds ratio compared with using the mantel haenszel estimate this allows the development of power analysis for testing general hypotheses of varied interactions using an invariant pythagorean law of relative entropy

new methods of controlled total variation reduction for digital functions we consider the functional of total variation for functions of either one or two discrete variables we introduce a new principle for construction of evolution processes that gradually reduce the total variation of a function the method is derived from analysis of relaxation of the functional along the directions given by the haar wavelets an approach that does not require regularization of either the functional or its singular gradient we further present a rigorous analysis of the analytical properties of the resulting evolution schemes the purely discrete theory grows out of universal observations about the relation between the total variation and the haar functions an additional advantage of the method is that it can be directly implemented as an effective numerical procedure moreover it can be intertwined in a natural way with a nonlinear phase space filtering which further increases control of the deformation of a signal finally we report that the algorithms can be used as remarkable tools for signal processing and image enhancement

hessian based norm regularization for image restoration with biomedical applications we present nonquadratic hessian based regularization methods that can be effectively used for image restoration problems in a variational framework motivated by the great success of the total variation tv functional we extend it to also include second order differential operators specifically we derive second order regularizers that involve matrix norms of the hessian operator the definition of these functionals is based on an alternative interpretation of tv that relies on mixed norms of directional derivatives we show that the resulting regularizers retain some of the most favorable properties of tv i e convexity homogeneity rotation and translation invariance while dealing effectively with the staircase effect we further develop an efficient minimization scheme for the corresponding objective functions the proposed algorithm is of the iteratively reweighted least square type and results from a majorization minimization approach it relies on a problem specific preconditioned conjugate gradient method which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time we validate the overall proposed regularization framework through deblurring experiments under additive gaussian noise on standard and biomedical images

numerical solutions of volterra integral equation with weakly singular kernel using scw method numerical methods for weakly singular volterra integral equations are rarely considered in the literature the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation in this paper we present a numerical solution of weakly singular volterra integral equations including the abels equations by the second chebyshev wavelet scw we give the scw operational matrix of fractional integration and combine with the block pulse functions bpfs to derive the procedure of solving this kind integral equations the proposed method is illustrated with numerical examples the results reveal that the method is accurate and easy to implement

properties and refinements of the fused lasso we consider estimating an unknown signal both blocky and sparse which is corrupted by additive noise we study three interrelated least squares procedures and their asymptotic properties the first procedure is the fused lasso put forward by friedman et al ann appl statist 1 2007 302 332 which we modify into a different estimator called the fused adaptive lasso with better properties the other two estimators we discuss solve least squares problems on sieves one constrains the maximal l 1 norm and the maximal total variation seminorm and the other restricts the number of blocks and the number of nonzero coordinates of the signal we derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso and we derive convergence rates for the sieve estimators explicitly in terms of the constraining parameters

future sequential regularization methods for iii posed volterra equations applications to the inverse heat conduction problem abstract we develop a theoretical context in which to study the future sequential regularization method developed by beck for the inverse heat conduction problem in the process we generalize beck s ideas and view that method as one in a large class of regularization methods in which the solution of an ill posed first kind volterra equation is seen to be the limit of a sequence of solutions of well posed second kind volterra equations such techniques are important because standard regularization methods such as tikhonov regularization tend to transform a naturally sequential volterra problem into a full domain fredholm problem destroying the underlying causal nature of the volterra model and leading to inefficient global approximation strategies in contrast the ideas we present here preserve the original volterra structure of the problem and thus can lead to easily implemented localized approximation strategies theoretical properties of these methods are discussed and proofs of convergence are given

a two step model for image denoising using a duality strategy and surface fitting in order to alleviate the staircase effect or the edge blurring in the course of the image denoising we propose a two step model based on the duality strategy in fact this strategy follows the observation that the dual variable of the restored image can be looked at as the normal vector so we first obtain the dual variable and then reconstruct the image by fitting the dual variable following the augmented lagrangian strategy we propose a projection gradient method for solving this two step model we also give some convergence analyses of the proposed projection gradient method several numerical experiments are tested to compare our proposed model with the rof model and the llt model

an effective method for solving nonlinear equations and its application the linearized partial differential equation from the nonlinear partial differential equation which was proposed by rudin osher and fatemi l i rudin s osher and e fatemi nonlinear total variation based noise removal algorithms for solving image decomposition was introduced by chambolle a chambolle an algorithm for total variation minimization and applications and r acar and c r vogel r acar and c r vogel analysis of bounded variation penalty methods for ill posed problems in this paper we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works

future polynomial regularization of ill posed volterra equations we examine a new discrete method for regularizing ill posed volterra problems unlike many classical regularization techniques such as tikhonov regularization this method preserves the volterra causal nature of the problem allowing the regularized solution to be produced sequentially in near real time as data arrives we analyze the method and for an important class of volterra problems prove that regularized solutions converge to the true solution at the best possible rate with respect to error in the data in fact the future polynomial regularization method discussed here may be applied to quite general operator equations provided that the operator may be discretized by a lower triangular matrix of toeplitz type this enlarges the class of operator equations that may be approximated using the method but it also introduces degenerate situations in which the future polynomial method is no more regularizing than an ordinary discretization method we characterize these degenerate cases and argue that we are unlikely to see them for the problems of interest here in particular such degeneracies cannot occur for the class of volterra problems for which we are able to prove the future polynomial method converges finally we present numerical evidence that this method works well in the recovery of sharp and discontinuous features in the true solution features that can be oversmoothed by classical regularization techniques

an efficient linear solver for nonlinear parameter identification problems in this paper we study some efficient numerical methods for parameter identifications in elliptic systems the proposed numerical methods are conducted iteratively and each iteration involves only solving positive definite linear algebraic systems although the original inverse problems are ill posed and highly nonlinear the positive definite systems can be naturally preconditioned with their corresponding block diagonal matrices numerical experiments are presented to illustrate the efficiency of the proposed algorithms

numerical solution of first kind volterra equations by sequential tikhonov regularization we consider the problem of finding regularized solutions to ill posed volterra integral equations the method we consider is a sequential form of tikhonov regularization that is particularly suited to problems of volterra type we prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation collocation rectangular and midpoint quadrature one obtains convergence of the method at an optimal rate with respect to noise in the data in addition we describe a fast algorithm for the implementation of sequential tikhonov regularization and show that for small values of the regularization parameter the method is only slightly more expensive computationally than the numerical solution of the original unregularized integral equation finally numerical results are presented to show that the performance of sequential tikhonov regularization nearly matches that of standard tikhonov regularization in practice but considerable savings in cost are realized

reconstruction of wavelet coefficients using total variation minimization we propose a model to reconstruct wavelet coefficients using a total variation minimization algorithm the approach is motivated by wavelet signal denoising methods where thresholding small wavelet coefficients leads to pseudo gibbs artifacts by replacing these thresholded coefficients by values minimizing the total variation our method performs a nearly artifact free signal denoising in this paper we detail the algorithm based on a subgradient descent combining a projection on a linear space the convergence of the algorithm is established and numerical experiments are reported

alternative estimators of the common odds ratio for 2 x 2 tables hauck 1984 reported on the comparative efficiencies and biases as evident from simulation studies of three estimators of a common odds ratio or common log odds ratio these three estimators included both the conditional and the unconditional maximum likelihood ml estimators and the estimator based on the mantel haenszel mh 1959 procedure but particular care needs to be taken in the interpretation of seeming biases in the estimation of the common odds ratio reasonably we might look for unbiasedness in the log odds ratio rather than in the odds ratio itself odds ratio estimates are likely to be highly right skewed around the true parameter value while the log odds ratio estimate would tend to show more symmetric behavior if an estimator shows unbiasedness relative to the log odds ratio it will necessarily show positive bias relative to the odds ratio itself this is because in the presence of some variation the arithmetic mean will exceed the geometric mean if the geometric mean is the true odds ratio the arithmetic mean will exceed the true odds ratio another consequence is that if two estimators are unbiased relative to the log odds ratio then the one showing the greater variability in estimation of the log odds ratio will ordinarily show greater bias in the estimation of the odds ratio for log normally distributed estimates which is not exactly the present case the bias in the arithmetic mean due to variability in the logarithmic estimates is known to be the factor exp r 2 2 this kind of thing can be recognized from hauck s results where one can see that the bias in estimates of the odds ratio reflects both the bias and the variance in estimates of the log odds ratio

on the convergence of the lagged diffusivity fixed point method in total variation image restoration in this paper we show that the lagged diffusivity fixed point algorithm introduced by vogel and oman in siam j sci comput 17 1996 pp 227 238 to solve the problem of total variation denoising proposed by rudin osher and fatemi in phys d 60 1992 pp 259 268 is a particular instance of a class of algorithms introduced by voss and eckhardt in computing 25 1980 pp 243 251 whose origins can be traced back to weiszfeld s original work for minimizing a sum of euclidean lengths tohoku math j 43 1937 pp 355 386 there have recently appeared several proofs for the convergence of this algorithm g aubert et al technical report 94 01 informatique signaux et systemes de sophia antipolis 1994 a chambolle and p l lions technical report 9509 ceremade 1995 and d c dobson and c r vogel siam j numer anal 34 1997 pp 1779 1791 here we present a proof of the global and linear convergence using the framework introduced in h voss and u eckhart computing 25 1980 pp 243 251 and give a bound for the convergence rate of the fixed point iteration that agrees with our experimental results these results are also valid for suitable generalizations of the fixed point algorithm

a comparative study of conditional maximum likelihood estimation of a common odds ratio the finite sample properties of various point estimators of a common odds ratio from multiple 2 x 2 tables have been considered in a number of simulation studies however the conditional maximum likelihood estimator has received only limited attention that omission is partially rectified here for cases of relatively small numbers of tables and moderate to large within table sample sizes the conditional maximum likelihood estimator is found to be superior to the unconditional maximum likelihood estimator and equal or superior to the mantel haenszel estimator in both bias and precision

robustness improvement of hyperspectral image unmixing by spatial second order regularization the acquisition of hundreds of images of a scene each at a different wavelength is known as hyperspectral imaging this high amount of data allows the extraction of much more information from hyperspectral images compared with conventional color images the forward looking imaging approach emerged from remote sensing but is still not very widespread in industrial and other practical applications spectral unmixing in particular aims at the determination of the components present in a scene as well as the abundance to which each component contributes this information is valuable for instance when discrimination tasks are to be performed involving not only spectral but also spatial information was found to have the potential to improve the unmixing results several publications use spatial first order regularization closely related to the total variation approach to incorporate this spatial information like in classical image processing this approach favors piecewise constant pixel transitions this is why it was proposed in the literature to use second order regularization instead of first order to approach piecewise linear transitions therefore we introduce hessian based regularization to hyperspectral unmixing and propose an algorithm to calculate the regularized result we use simulated data and images measured in our laboratory to show that both the first and second order approaches share many properties and produce similar results the second order approach however is more robust and thus more accurate in finding the minimum both methods smoothen the images in the case of supervised unmixing i e the component spectra are known beforehand and enhance unsupervised unmixing when the spectra are not known

towards artifact free characterization of surface topography using complex wavelets and total variation minimization a hybrid method based on dual tree complex wavelet transform and total variation minimization is proposed for erasure of undesirable artifacts that arise in the existing wavelet based methods in surface metrology the complex wavelet transform provides approximate shift invariance and good directional selectivity and attempts to solve the weakness of real discrete wavelet methods reconstruct the complex wavelet coefficients using a total variation minimization principle to eliminate the wavelet shape artifacts and the pseudo gibbs artifacts near the discontinuities which are caused when thresholding small wavelet coefficients by replacing these thresholded complex wavelet coefficients by optimal values that minimize the total variation the method performs a close artifact free surface characterization numerical experiments using a series of typical engineering and bioengineering surfaces demonstrate the remarkable potential of the methodology

variational approach for edge preserving regularization using coupled pdes this paper deals with edge preserving regularization for inverse problems in image processing we first present a synthesis of the main results we have obtained in edge preserving regularization by using a variational approach we recall the model involving regularizing functions spl phi and we analyze the geometry driven diffusion process of this model in the three dimensional 3 d case then a half quadratic theorem is used to give a very simple reconstruction algorithm after a critical analysis of this model we propose another functional to minimize for edge preserving reconstruction purposes it results in solving two coupled partial differential equations pdes one processes the intensity the other the edges we study the relationship with similar pde systems in particular with the functional proposed by ambrosio tortorelli 1990 1992 in order to approach the mumford shah 1989 functional developed in the segmentation application experimental results on synthetic and real images are presented

subspace correction methods for total variation and ell_1 minimization this paper is concerned with the numerical minimization of energy functionals in hilbert spaces involving convex constraints coinciding with a seminorm for a subspace the optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces on each subspace an iterative proximity map algorithm is implemented via oblique thresholding which is the main new tool introduced in this work we provide convergence conditions for the algorithm in order to compute minimizers of the target energy analogous results are derived for a parallel variant of the algorithm applications are presented in domain decomposition methods for degenerate elliptic pdes arising in total variation minimization and in accelerated sparse recovery algorithms based on ell 1 minimization we include numerical examples which show efficient solutions to classical problems in signal and image processing

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions

rate of convergence of regularization procedures and finite element approximations for the total variation flow we derive rates of convergence for regularization procedures characterized by a parameter and finite element approximations of the total variation flow which arises from image processing geometric analysis and materials sciences practically useful error estimates which depend on open image in new window only in low polynomial orders are established for the proposed fully discrete finite element approximations as a result scaling laws which relate mesh parameters to the regularization parameter are also obtained numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods