共查询到20条相似文献,搜索用时 0 毫秒
1.
Johnathan M. Bardsley 《BIT Numerical Mathematics》2008,48(4):651-664
Image data is often collected by a charge coupled device (CCD) camera. CCD camera noise is known to be well-modeled by a Poisson
distribution. If this is taken into account, the negative-log of the Poisson likelihood is the resulting data-fidelity function.
We derive, via a Taylor series argument, a weighted least squares approximation of the negative-log of the Poisson likelihood
function. The image deblurring algorithm of interest is then applied to the problem of minimizing this weighted least squares
function subject to a nonnegativity constraint. Our objective in this paper is the development of stopping rules for this
algorithm. We present three stopping rules and then test them on data generated using two different true images and an accurate
CCD camera noise model. The results indicate that each of the three stopping rules is effective.
AMS subject classification (2000) 65F20, 65F30 相似文献
2.
For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study. 相似文献
3.
J. Thomas King 《Numerische Mathematik》1992,61(1):311-334
Summary In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.Dedicated to Jim Bramble on the occasion of his sixtieth birthday 相似文献
4.
The truncated singular value decomposition is a popular method for the solution of linear ill-posed problems. The method requires the choice of a truncation index, which affects the quality of the computed approximate solution. This paper proposes that an L-curve, which is determined by how well the given data (right-hand side) can be approximated by a linear combination of the first (few) left singular vectors (or functions), be used as an aid for determining the truncation index. 相似文献
5.
Lothar Reichel Andriy Shyshkov 《Journal of Computational and Applied Mathematics》2010,233(5):1314-1325
Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle. 相似文献
6.
A mollification method for ill-posed problems 总被引:3,自引:0,他引:3
Summary.
A mollification method for a class of ill-posed
problems is suggested. The idea of the method is very simple and
natural: if the data are given inexactly then we try to find a
sequence of ``mollification operators" which map the improper
data into well-posedness classes of the problem (mollify the
improper data). Within these mollified data our problem becomes
well-posed. And when these facts are in hand we try to obtain
error estimates and optimal or ``quasi-optimal" mollification
parameters. The
method is working not only for problems in Hilbert spaces, but
also for problems in Banach spaces. Applications of the
method to concrete problems, like numerical differentiation,
parabolic equations backwards in time, the Cauchy problem for the
Laplace equation,
one- and multidimensional non-characteristic Cauchy problems for
parabolic equations (in infinite or finite domains),... give us
very sharp stability estimates of H\"older continuous type. In
these cases the method is optimal in the sense that it gives the
same order of H\"older continuous dependence on the data as for
the regularized problems. Furthermore, the method may be
implemented numerically using fast Fourier transforms. For the
first time a uniform stability estimate of H\"older continuous
type of the solution of the heat equation backwards in time in the
space for all could
be established by our mollification method. A new simple sharp
pointwise estimate of H\"older type for the weak solution of a
non-characteristic Cauchy problem for parabolic equations in a
finite domain is established.
Received June 25, 1993 / Revised version received February
18, 1994 相似文献
7.
The computation of an approximate solution of linear discrete ill-posed problems with contaminated data is delicate due to
the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution
to errors in the data by replacing the given ill-posed problem by a nearby problem, whose solution is less sensitive to perturbation.
This regularization method requires that a suitable value of the regularization parameter be chosen. Recently, Brezinski et al.
(Numer Algorithms 49, 2008) described new approaches to estimate the error in approximate solutions of linear systems of equations and applied these
estimates to determine a suitable value of the regularization parameter in Tikhonov regularization when the approximate solution
is computed with the aid of the singular value decomposition. This paper discusses applications of these and related error
estimates to the solution of large-scale ill-posed problems when approximate solutions are computed by Tikhonov regularization
based on partial Lanczos bidiagonalization of the matrix. The connection between partial Lanczos bidiagonalization and Gauss
quadrature is utilized to determine inexpensive bounds for a family of error estimates.
In memory of Gene H. Golub.
This work was supported by MIUR under the PRIN grant no. 2006017542-003 and by the University of Cagliari. 相似文献
8.
Robert Deville Julian P. Revalski 《Proceedings of the American Mathematical Society》2000,128(4):1117-1124
We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is -porous.
9.
For the well-known Fay-Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate. 相似文献
10.
Ronaldo Dias Clécio S. Ferreira Nancy L. Garcia 《Statistical Inference for Stochastic Processes》2008,11(1):11-34
Let be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ2. The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure
to be consistent as and .
相似文献
11.
In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting. 相似文献
12.
Pham Ky Anh 《Ukrainian Mathematical Journal》1994,46(7):1049-1055
The method of coordinated approximation suggested by Yu. Gaponenko [1] for the spaceL
2 (0, 1) is generalized to the case of abstract Hilbert spaces. In particular, we show that, forL
2 (0, 1), some assumptions imposed on exact solutions can be weakened.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 956–961, July, 1994. 相似文献
13.
The representation of linear operator equations in terms of wavelet bases yields a multilevel framework, which can be exploited
for iterative solution. This paper describes cascadic multilevel methods that employ conjugate gradient-type methods on each
level. The iterations are on each level terminated by a stopping rule based on the discrepancy principle. 相似文献
14.
Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals.We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on Lp-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions. 相似文献
15.
16.
Thorsten Hohage 《Numerical Functional Analysis & Optimization》2013,34(3-4):439-464
Linear and nonlinear inverse problems which are exponentially ill-posed arise in heat conduction, satellite gradiometry, potential theory and scattering theory. For these problems logarithmic source conditions have natural interpretations whereas standard Hölder-type source conditions are far too restrictive. This paper provides a systematic study of convergence rates of regularization methods under logarithmic source conditions including the case that the operator is given onlyapproximately. We also extend previous convergence results for the iteratively regularized Gauß-Newton method to operator approximations. 相似文献
17.
18.
Julio E. Guacaneme 《Integral Equations and Operator Theory》1988,11(4):610-613
Schock has given a nearly optimal a posteriori parameter choice strategy for Tikhonov regularization. We show that a modification of his argument leads, in fact, to an optimal convergence rate. 相似文献
19.
Summary In this paper we study the noncharacteristic Cauchy problem, ut–(a(x)ux)x=0, x(0, l), t.(0, T], u(0, t)=(t), ux(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.
Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto ut= (a(x)ux)x, x(0, l), t(0, T), u(0, t)=(t), ux(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.相似文献
20.
The problem of detecting singularities (discontinuities of the first kind) of a noisy function in L 2 is considered. A wide class of regularizing algorithms that can detect discontinuities is constructed. New estimates of accuracy of determining the location of discontinuities are obtained and their optimality in terms of order with respect to the error level δ is proved for some classes of functions with isolated singularities. New upper bounds for the singularity separation threshold are obtained. 相似文献