a parameter identification problem of mixed type related to the manufacture of car windshields we study the identification of a parameter in a fourth order elliptic partial differential equation that models the optimal design of car windshields to be manufactured by the sagging process considered as a second order equation for the unknown parameter the problem is of mixed type i e changing between elliptic and hyperbolic numerical routines for directly solving this equation are not available in this paper we both theoretically and numerically show that the inverse problem can instead be solved in a stable way by means of a derivative free iterative regularization method the course of the iteration nevertheless depends markedly on the mixed type of the second order equation

a projection regularized newton method for nonlinear ill posed problems and its application to parameter identification problems with finite element discretization this paper is concerned with nonlinear ill posed operator equations f a y e g parameter identification problems and their approximate solution by a newton type method that is regularized by projecting the linearized equation in each newton step onto a finite dimensional space e g a finite element space and by stopping the newton iteration at an appropriate index we prove convergence as the iteration index n goes to infinity in the noise free case and convergence as the data noise level delta goes to zero in the case of noisy data as well as convergence rates under certain additional conditions

a runge kutta type modified landweber method for nonlinear ill posed operator equations in this paper we investigate the convergence behavior of a runge kutta type modified landweber method for nonlinear ill posed operator equations in order to improve the stability and convergence of the landweber iteration a 2 stage gauss type runge kutta method is applied to the continuous analogy of the modified landweber method to give a new modified landweber method called r k type modified landweber method under some appropriate conditions we prove the convergence of the proposed method we conclude with a numerical example confirming the theoretical results including comparisons to the modified landweber iteration

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

a comparison of the landweber method and the gauss newton method for an inverse parabolic boundary value problem we consider an inverse boundary value problem for the heat equation that consists of the identification of the interior boundary curve of an annuli from given temperature and heat flux measurements on the exterior boundary in order to solve the problem in a stable way we apply the landweber method where we treat both the direct and the adjoint problem by means of the boundary integral equation method numerical examples including a comparison to the gauss newton method are presented

further convergence results on the general iteratively regularized gauss newton methods under the discrepancy principle we consider the general iteratively regularized gauss newton methods xk 1 x0 g k f xk f xk f xk f xk y f xk xk x0 for solving nonlinear inverse problems f x y using the only available noise y of y satisfying y y with a given small noise level 0 in order to produce reasonable approximation to the sought solution we terminate the iteration by the discrepancy principle under much weaker conditions we derive some further convergence results which improve the existing ones and thus expand the applied range

an iterative multi level algorithm for solving nonlinear ill posed problems the convergence analysis of landweber s iteration for the solution of nonlinear ill posed problem has been developed recently by hanke neubauer and scherzer in concrete applications sufficient conditions for convergence of the landweber iterates developed there although quite natural turned out to be complicated to verify analytically however in numerical realizations when discretizations are considered sufficient conditions for local convergence can usually easily be stated this paper is motivated by these observations initially a discretization is fixed and a discrete landweber iteration is implemented until an appropriate stopping criterion becomes active the output is used as an initial guess for a finer discretization an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces the numerical performance of this multi level algorithm is compared with landweber s iteration

an analysis of lavrentiev regularization method and newton type process for nonlinear ill posed problems abstract in this paper we consider the lavrentiev regularization method and a modified newton method for obtaining stable approximate solution to nonlinear ill posed operator equations f x y where f d f x x is a nonlinear monotone operator or f x 0 is nonnegative selfadjoint operator defined on a real hilbert space x we assume that only a noisy data y x with y y are available further we assume that frechet derivative f of f satisfies center type lipschitz condition a priori choice of regularization parameter is presented we proved that under a general source condition on x 0 x the error x x n between the regularized approximation x n x 0 x 0 and the solution x is of optimal order in the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem

convergence analysis of a block iterative version of the loping landweber kaczmarz iteration abstract the loping landweber kaczmarz iteration is a numerically efficient regularization method for solving ill posed problems in this article we investigate a block iterative version that allows variable block sizes and variable grouping of the sub equations during the iterations we show that the proposed method converges for exact data and is a strongly convergent regularization method for noisy data

an algorithm for solving large capacitated warehouse location problems abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon lagrangean relaxation of a mixed integer formulation of the problem feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the lagrangean relaxation by incorporating the lower bound and the reduction tests into a tree search procedure we are able to solve problems involving up to 500 potential warehouse locations and 1000 customers

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

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

applications of the modified discrepancy principle to tikhonov regularization of nonlinear ill posed problems in this paper we consider the finite dimensional approximations of tikhonov regularization for nonlinear ill posed problems with approximately given right hand sides we propose an a posteriori parameter choice strategy which is a modified form of morozov s discrepancy principle to choose the regularization parameter under certain assumptions on the nonlinear operator we obtain the convergence and rates of convergence for tikhonov regularized solutions this paper extends the results which were developed by plato and vainikko in 1990 for solving linear ill posed equations to nonlinear problems

inexact newton landweber iteration in banach spaces with nonsmooth convex penalty terms by making use of tools from convex analysis we formulate an inexact newton landweber iteration method for solving nonlinear inverse problems in banach spaces the method consists of two components an outer newton iteration and an inner scheme the inner scheme provides increments by applying landweber iteration with nonsmooth uniformly convex penalty terms to local linearized equations the outer iteration is then terminated by the discrepancy principle detailed convergence analysis is present under standard conditions on the nonlinearity finally numerical simulations are reported to test the performance of the method the pdf has been replaced

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

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

a dynamical system method for solving nonlinear ill posed problems this paper investigates the dynamical systems for solving nonlinear ill posed problems and uses lyapunov theory which is constructed in this paper to prove the stability of the solution of the dynamical system problems finally numerical experiments are presented with pc matlab

an implicit landweber method for nonlinear ill posed operator equations in this paper we are interested in the solution of nonlinear inverse problems of the form f x y we propose an implicit landweber method which is similar to the third order midpoint newton method in form and consider the convergence behavior of the implicit landweber method using the discrepancy principle as a stopping criterion we obtain a regularization method for ill posed problems we conclude with numerical examples confirming the theoretical results including comparisons with the classical landweber iteration and presented modified landweber methods

on the iteratively regularized gauss newton method for solving nonlinear ill posed problems the iteratively regularized gauss newton method is applied to compute the stable solutions to nonlinear ill posed problems f x y when the data y is given approximately by y with y y in this method the iterative sequence x k is defined successively by x k 1 x k k i f x k f x k 1 f x k f x k y k x k x 0 where x 0 x 0 is an initial guess of the exact solution x and k is a given decreasing sequence of positive numbers admitting suitable properties when x k is used to approximate x the stopping index should be designated properly in this paper an a posteriori stopping rule is suggested to choose the stopping index of iteration and with the integer k determined by this rule it is proved that x k x c inf x k x k k 0 1 with a constant c independent of where x k denotes the iterative solution corresponding to the noise free case as a consequence of this result the convergence of x k is obtained and moreover the rate of convergence is derived when x 0 x satisfies a suitable source wise representation the results of this paper suggest that the iteratively regularized gauss newton method combined with our stopping rule defines a regularization method of optimal order for each 0 v 1 numerical examples for parameter estimation of a differential equation are given to test the theoretical results

on the discrepancy principle for some newton type methods for solving nonlinear inverse problems we consider the computation of stable approximations to the exact solution x dagger of nonlinear ill posed inverse problems f x y with nonlinear operators f x y between two hilbert spaces x and y by the newton type methods x k 1 delta x 0 g alpha k left f x k delta f x k delta right f x k delta left f x k delta y delta f x k delta x k delta x 0 right of y satisfying y delta y le delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle in which the stopping index k delta is determined as the first integer such that f x k delta delta y delta le tau delta f x k delta y delta quad 0 le k k delta converges to x dagger as delta rightarrow 0 and establish various order optimal convergence rate results it is remarkable that we even can show the order optimality under merely the lipschitz condition on the frechet derivative f of f if x 0 x dagger is smooth enough

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

a search tree algorithm for plant location problems abstract this paper deals with the classical plant location problem by introducing a new algorithm based on a ranking procedure of all the possible combinations of plant locations the ranking is obtained by introducing a properly defined search tree and by exploiting its structural properties so that the algorithm suitably combines a search procedure on the tree with a ranking method the number of subproblems to be examined is greatly reduced by the use of a stopping rule and a lower bound the lower bound is given by the value of the objective function of a dual feasible solution for the transportation problem associated to each combination computational results for a set of real size problems with different data structures are finally discussed

a general convergence analysis of some newton type methods for nonlinear inverse problems we consider the methods x n 1 delta x n delta g alpha n f x n delta f x n delta f x n delta f x n delta y delta for solving nonlinear ill posed inverse problems f x y using the only available noise data y delta satisfying y delta y leq delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle f x n delta delta y delta leq tau delta 1 under certain conditions on alpha n and f we prove for a large class of spectral filter functions g alpha the convergence of x n delta delta to a true solution as delta rightarrow0 moreover we derive the order optimal rates of convergence when certain holder source conditions hold numerical examples are given to test the theoretical results

a new total variational regularization method for nonlinear inverse problems in fluorescence molecular tomography abstract fluorescence molecular tomography fmt is an imaging way of in vivo optical imaging which is widely used in biomedical photon imaging for its higher sensitivity for years the propagation process of photons in tissue is simulated based on the equation of radiative transfer in this work we analyze an inverse problem in fmt which uses the reading of body surface detector to reconstruct the optical parameter distribution we propose a novel total variational regularization method to solve the absorption parameter inversion problem in fmt the proof of the proposed method is also provided finally numerical experiments demonstrate the feasibility of the proposed method

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

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

regularization in model updating this paper presents the theory of sensitivity based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters model updating together with the additional regularization criterion is an optimization with two objective functions and must be linearized to obtain the solution structured solutions are obtained based on the generalized singular value decomposition gsvd and specific features of the parameter and response paths as the regularization parameter varies are explored the four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families these concepts are demonstrated on a simulated discrete example and on an experimental case study copyright 2007 john wiley sons ltd

local identification of nonparametric and semiparametric models in parametric nonlinear structural models a classical sufficient condition for local identification like fisher 1966 and rothenberg 1971 is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix we derive an analogous result for the nonparametric nonlinear structural models establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification importantly we show that additional conditions are often needed in nonlinear nonparametric models to avoid nonlinearities overwhelming linear effects we give restrictions on a neighborhood of the true value that are sufficient for local identification we apply these results to obtain new primitive identification conditions in several important models including nonseparable quantile instrumental variable iv models and semiparametric consumption based asset pricing models

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

admittivity imaging from multi frequency micro electrical impedance tomography abstract the aim of this paper is to propose an optimal control optimization algorithm for reconstructing admittivity distributions i e both conductivity and permittivity from multi frequency micro electrical impedance tomography a convergent and stable optimal control scheme is shown to be obtainable from multi frequency data this opens a door for convergence analysis of optimal control type approaches in imaging from internal data the results of this paper have potential applicability in cancer imaging cell culturing and differentiation food sciences and biotechnology

regularizing newton kaczmarz methods for nonlinear ill posed problems we introduce a class of stabilizing newton kaczmarz methods for nonlinear ill posed problems and analyze their convergence and regularization behavior as usual for iterative methods for solving nonlinear ill posed problems conditions on the nonlinearity or the derivatives have to be imposed in order to obtain convergence as we shall discuss in general and in some specific examples the nonlinearity conditions obtained for the newton kaczmarz methods are less restrictive than those for previously existing iteration methods and can be verified for several practical applications we also discuss the discretization and efficient numerical solution of the linear problems arising in each step of a newton kaczmarz method and we carry out numerical experiments for two model problems

iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression this paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression we propose a regularized newton type iteration and establish convergence and convergence rate results a particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption we demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model this is illustrated in computed examples with simulated data

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

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

an iteration regularization method with general convex penalty for nonlinear inverse problems in banach spaces abstract this paper investigates the construction analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in banach spaces the method is formulated by combining the homotopy perturbation technique the penalty term is allowed to be nonsmooth to include l 1 norm or total variation like penalty functionals especially suitable for detecting the special features of the sought solutions such as sparsity or piecewise constant by using tools from convex analysis in banach spaces we establish the detailed convergence and stability results numerical simulations for one dimensional and two dimensional parameter identification problems are performed to validate that our approach is competitive in terms of reducing the overall computational time in comparison with the existing landweber iteration with general convex penalty moreover our approach is used to the non gaussian noisy data to illustrate the iterative regularization in banach spaces

on the convergence of a regularizing levenberg marquardt scheme for nonlinear ill posed problems in this note we study the convergence of the levenberg marquardt regularization scheme for nonlinear ill posed problems we consider the case that the initial error satisfies a source condition our main result shows that if the regularization parameter does not grow too fast not faster than a geometric sequence then the scheme converges with optimal convergence rates our analysis is based on our recent work on the convergence of the exponential euler regularization scheme hochbruck et al in inverse probl 25 7 075009 2009

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

inexact newton regularization using conjugate gradients as inner iteration in our papers inverse problems 15 1999 pp 309 327 and numer math 88 2001 pp 347 365 we proposed algorithm tt reginn an inexact newton iteration for the stable solution of nonlinear ill posed problems tt reginn consists of two components the outer iteration which is a newton iteration stopped by the discrepancy principle and an inner iteration which computes the newton correction by solving the linearized system the convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration especially landweber iteration nu methods and tikhonov phillips regularization in the present paper we prove convergence rates for tt reginn when the conjugate gradient method which is nonlinear serves as inner iteration thereby we add to a convergence analysis of hanke who had previously investigated tt reginn furnished with the conjugate gradient method numer funct anal optim 18 1997 pp 971 993 by numerical experiments we illustrate that the conjugate gradient method outperforms the nu method as inner iteration

recovery of multiple coefficients in a reaction diffusion equation abstract this paper considers the inverse problem of recovering both the unknown spatially dependent conductivity a x and the potential q x in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time t we show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients we also allow a more general setting in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity f of the form q x f u

a new iteratively total variational regularization for nonlinear inverse problems superior to the other well known iterations such as landweber iteration total variational regularization is a well established method for identifying the piecewise smooth parameter and processing the image therefore we propose a new iteratively total variational regularization based on total variational regularization and homotopy perturbation technique in this paper taking advantage of bregman distance we construct a nested iteration to inverse parameters meanwhile we prove that the error of the new method decreases monotonically and the new iterative method with the noisy data is regular according to the discrepancy principle in the last numerical section compared with landweber type total variational regularization and runge kutta type regularization numerical results of this new method indicate that this new regularization is time saving for the same accuracy and more robust to the noisy data

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

a homotopy method for well log constraint waveform inversion the primary difficulty posed by the method of waveform inversion for the acoustic wave equation is the presence of local minima of the objective function waveform inversion fails to converge to the global minimum unless the wavefield contains very low frequencies or the starting model is very close to the true model constraints can improve the convergence of the method however if the starting model is far from the correct model normal iterative methods will still get trapped in local minima we designed a homotopy method to improve the robustness of waveform inversion it makes natural use of constraints such as well logs in addition to further condition the inverse problem we incorporate standard tikhonov regularization we demonstrate through a synthetic example that our method is more likely to find a global minimum than normal iterative methods

numerical approximation of a tikhonov type regularizer by a discretized frozen steepest descent method abstract we present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill posed operator equation f x y the proposed method is a modified form of the method considered by argyros et al 2014 the balancing principle considered by pereverzev and schock 2005 is used for choosing the regularization parameter the error estimate is derived under a general source condition and is of optimal order the provided numerical example proves the efficiency of the proposed method

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

lagrangean dual ascent algorithms for computing bounds in capacitated plant location problems abstract given a mixed integer programming problem if one dualizes all linking constraints and the constraints containing only integer variables the optimum of the lagrangean dual is equal to the optimum of the lp relaxation if one strengthens this dual by adding valid inequalities such that the problem still decomposes into a continuous and a pure integer problem and is relatively easy to solve due to some special structure one may be able to devise dual ascent steps for the lagrangean dual this is investigated for the capacitated plant location problem whose strengthened lagrangean relaxed problem decomposes into a transportation problem and a knapsack problem computational experiments are reported based on some very simple ascent steps

on a regularized levenberg marquardt method for solving nonlinear inverse problems we consider a regularized levenberg marquardt method for solving nonlinear ill posed inverse problems we use the discrepancy principle to terminate the iteration under certain conditions we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source wise representations

a note on using the differential transformation method for the integro differential equations in this paper we show that the generalization of the differential transformation method dtm to integro differential equation is not the same thing that has been proposed in p darania a ebadian a method for the numerical solution of the integro differential equation appl math comput 188 2007 567 668 we extend the integral part by the method of jang et al 2 for the integro differential equations we give a counterexample for the method expressed by darania and ebadian 1 and some examples for comparing our modified method by method s of darania and ebadian 1

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

bouligand landweber iteration for a non smooth ill posed problem this work is concerned with the iterative regularization of a non smooth nonlinear ill posed problem where the forward mapping is merely directionally but not g teaux differentiable using a bouligand subderivative of the forward mapping a modified landweber method can be applied however the standard analysis is not applicable since the bouligand subderivative mapping is not continuous unless the forward mapping is g teaux differentiable we therefore provide a novel convergence analysis of the modified landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition these conditions are verified for an inverse source problem for an elliptic pde with a non smooth lipschitz continuous nonlinearity showing that the corresponding bouligand landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle this is illustrated with a numerical example

on an a posteriori parameter choice strategy for tikhonov regularization of nonlinear ill posed problems in the study of the choice of the regularization parameter for tikhonov regularization of nonlinear ill posed problems scherzer engl and kunisch proposed an a posteriori strategy in 1993 to prove the optimality of the strategy they imposed many very restrictive conditions on the problem under consideration their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid in this paper we give a further study on this strategy and show that tikhonov regularization is order optimal for each 0 nu le 2 with the regularization parameter chosen according to this strategy under some simple and easy checking assumptions this paper weakens the conditions needed in the existing results and provides a theoretical guidance to numerical experiments

a convergence analysis of nonlinear implicit iterative method for nonlinear ill posed problems abstract implicit iterative method acquires good effect in solving linear ill posed problems we have ever applied the idea of implicit iterative method to solve nonlinear ill posed problems under the restriction that is appropriate large we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence numerical results show that the implicit iterative method for nonlinear ill posed problems is efficient in this paper we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when k are determined by hanke criterion

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

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

a homotopy method for nonlinear inverse problems we develop a homotopy method for nonlinear inverse problems where the forward problems are governed by some forms of differential equations a tikhonov style regularization approach yields an optimization problem ordinary iterative methods may fail to solve this problem due to their locally convergent properties then the fixed point homotopy method is introduced to solving the normal equation of the optimization problem and a new and globally convergent algorithm is constructed which is highly effective in the aspects of speed of computation ability of noise suppression and wide region of convergence as a practical application the method is used to solve the inverse problem of 2 d acoustic wave equation we demonstrate the merits and effectiveness of our algorithm on two realistic model problems

a two stage method for nonlinear inverse problems in this work we are interested in the solution of nonlinear inverse problems of the form f x y we consider a two stage method which is third order convergent for well posed problems combining the method with levenberg marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion we obtain a regularization method for ill posed problems numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method

on convergence rates of inexact newton regularizations reginn is an algorithm of inexact newton type for the regularization of nonlinear ill posed problems inverse problems 15 1999 pp 309 327 in the present article convergence is shown under weak smoothness assumptions source conditions moreover convergence rates are established some computational illustrations support the theoretical results

a discrete scheme of landweber iteration for solving nonlinear ill posed problems in this paper we consider the finite dimensional approximation of landweber iteration for nonlinear ill posed problems and propose an a posteriori stopping rule to choose the termination index of the iteration under certain conditions we obtain convergence a pseudo optimality estimate and rates of convergence a numerical example is given to test the theoretical assertions

second order asymptotical regularization methods for inverse problems in partial differential equations abstract we develop second order asymptotical regularization soar methods for solving inverse source problems in elliptic partial differential equations with both dirichlet and neumann boundary data we show the convergence results of soar with the fixed damping parameter as well as with a dynamic damping parameter which is a continuous analog of nesterov s acceleration method moreover by using morozov s discrepancy principle together with a newly developed total energy discrepancy principle we prove that the approximate solution of soar weakly converges to an exact source function as the measurement noise goes to zero a damped symplectic scheme combined with the finite element method is developed for the numerical implementation of soar which yields a novel iterative regularization scheme for solving inverse source problems several numerical examples are given to show the accuracy and the acceleration effect of soar a comparison with the state of the art methods is also provided

mimo nonlinear ultrasonic tomography by propagation and backpropagation method this paper develops a fast ultrasonic tomographic imaging method in a multiple input multiple output mimo configuration using the propagation and backpropagation pbp method by this method ultrasonic excitation signals from multiple sources are transmitted simultaneously to probe the objects immersed in the medium the scattering signals are recorded by multiple receivers utilizing the nonlinear ultrasonic wave propagation equation and the received time domain scattered signals the objects are to be reconstructed iteratively in three steps first the propagation step calculates the predicted acoustic potential data at the receivers using an initial guess second the difference signal between the predicted value and the measured data is calculated third the backpropagation step computes updated acoustical potential data by backpropagating the difference signal to the same medium computationally unlike the conventional pbp method for tomographic imaging where each source takes turns to excite the acoustical field until all the sources are used the developed mimo pbp method achieves faster image reconstruction by utilizing multiple source simultaneous excitation furthermore we develop an orthogonal waveform signaling method using a waveform delay scheme to reduce the impact of speckle patterns in the reconstructed images by numerical experiments we demonstrate that the proposed mimo pbp tomographic imaging method results in faster convergence and achieves superior imaging quality

numerical solution of an inverse medium scattering problem for maxwell s equations at fixed frequency consider a time harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in r 3 in this paper well posedness of the variational problem for the direct scattering is examined an energy estimate for the scattered field is obtained on which the born approximation is based a regularized recursive linearization method for the inverse medium scattering which reconstructs the scatterer of an inhomogeneous medium from the boundary measurements of the scattered field is developed the algorithm requires only single frequency data using an initial guess from the born approximation each update is obtained via continuation on the spatial frequency of a two parameter family of plane waves by solving one direct problem and one adjoint problem of the maxwell equation

runge kutta type total variation regularization for nonlinear inverse problems in this paper we construct a runge kutta type total variation regularization for the nonlinear ill posed problems this method resembles runge kutta type iteration with the bregman distance as a regularization functional in the presence of noise with noise level we prove this method to be convergent under appropriate stopping rule furthermore some numerical experiments are presented to verify this method to be more suitable for problems with discontinuous solutions rk type tv regularization for the nonlinear ill posed problems is constructed this method introduced bregman distance as regularization functional under appropriate stopping rule this method with noise is convergent numerical tests testify the method for problems with discontinuous solutions

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

an iteratively regularized gauss newton halley method for solving nonlinear ill posed problems in this paper we analyze a second order method for regularizing nonlinear inverse problems which comes from adapting halley s method to the ill posed situation the method is related to the iteratively regularized gauss newton method and enhanced by the use of second derivatives of the forward operator for parameter identification problems in pdes these second derivatives often require not much additional computational cost as compared to first order derivatives we prove convergence and convergence rates under certain conditions on the forward operator in hilbert spaces numerical tests for a problem of identifying a distributed coefficient in an elliptic pde illustrate the performance of the method

iterative regularization of a parameter identification problem occurring in polymer crystallization this paper is devoted to the mathematical analysis and regularization of an identification problem related to nonisothermal crystallization of polymers which can be modeled by an initial boundary value problem for a coupled system of parabolic and hyperbolic partial differential equations the identification problem consists of estimating a material function of temperature which appears as a nonlinearity in the equations existence and uniqueness of a solution of the direct problem is shown as well as its stability with respect to the parameter furthermore we develop algorithms for the application of various iterative regularization methods to this particular problem their use is justified by verifying the fr echet differentiability of the parameter to output map which is needed for their realization the numerical performance of the iterative methods is compared with respect to speed of convergence stability and efficiency

iteratively regularized newton type methods for general data misfit functionals and applications to poisson data we study newton type methods for inverse problems described by nonlinear operator equations f u g in banach spaces where the newton equations f prime u n u n 1 u n g f u n are regularized variationally using a general data misfit functional and a convex regularization term this generalizes the well known iteratively regularized gauss newton method irgnm we prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a lepski type a posteriori stopping rule our analysis includes previous order optimal convergence rate results for the irgnm as special cases the main focus of this paper is on inverse problems with poisson data where the natural data misfit functional is given by the kullback leibler divergence two examples of such problems are discussed in detail an inverse obstacle scattering problem with amplitude data of the far field pattern and a phase retrieval problem the performance of the proposed method for these problems is illustrated in numerical examples

a convergence analysis of regularization by discretization in preimage space in this paper we investigate the regularizing properties of discretization in preimage space for linear and nonlinear ill posed operator equations with noisy data we propose to choose the discretization level that acts as a regularization parameter in this context by a discrepancy principle while general convergence has been shown not to hold see 17 we provide convergence results under appropriate conditions on the exact solution