aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

ramsey properties of nonlinear diophantine equations abstract we present general sufficient and necessary conditions for the partition regularity of diophantine equations which extend the classic rado s theorem by covering large classes of nonlinear equations the goal is to contribute to an overall theory of ramsey properties of nonlinear diophantine equations that encompasses the known results in this area under a unified framework sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters n grounding on combinatorial properties of positive density sets and ip sets necessary conditions are proved by a new technique in nonstandard analysis based on the use of the relation of u equivalence for the hypernatural numbers n

albanese maps and off diagonal long time asymptotics for the heat kernel we discuss long time asymptotic behaviors of the heat kernel on a non compact riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient a local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted laplacians our ideas and techniques are motivated partly by analogy with floque bloch theory on periodic schrodinger operators for the asymptotic expansion we make careful use of the classical laplace method in the course of a discussion we observe that the notion of albanese maps associated with the abelian group action is closely related to the asymptotics a similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph the results obtained in the present paper refine our previous ones 4 in the asymptotics the euclidean distance associated with the standard realization of the lattice graph which we call the albanese distance plays a crucial role

convergence criteria of iterative methods based on landweber iteration for solving nonlinear problems abstract landweber iteration xk 1 xk f xk f xk y for the solution of a nonlinear operator equation f x0 y0 can be viewed as a fixed point iteration with fixed point operator x f x f x y especially for nonlinear ill posed problems it seems impossible to verify that this fixed point operator is of contractive type which is a typical assumption for proving weak convergence of fixed point iteration schemes however for specific examples of nonlinear ill posed problems it is possible to verify conditions of quasi contractive type weak convergence of landweber iteration can be proven by application of general results for fixed point iterations based on quasi contractive type conditions in a recent paper by hanke et al a condition on the operator f has been investigated which guarantees convergence of the landweber s method a geometrical interpretation of this condition is given and is compared with well known conditions in the theory of fixed point iterations

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

convergence of the modified mann s iteration method for asymptotically strict pseudo contractions abstract let c be a closed convex subset of a real hilbert space h and assume that t is an asymptotically strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the modified mann s algorithm generates a sequence x n via the formula x n 1 n x n 1 n t n x n n 0 it is proved that if the control sequence n is chosen so that n 1 for some 0 1 then x n converges weakly to a fixed point of t we also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence

some new results in multiplicative and additive ramsey theory there are several notions of largeness that make sense in any semigroup and others such as the various kinds of density that make sense in sucien tly well behaved semigroups including n and n it is known that sets with positive multiplicative density must contain arbitrarily large geoarithmetic progressions that is sets of the form fr j a id i j 2 f0 1 kgg we establish some combined additive and multiplicative ramsey theoretic consequences of known algebraic results in the semigroups n and n derive some new algebraic results and derive consequences of them involving geoarithmetic progressions for example we show that in any nite partition of n there must be for each k sets of the form fb a id j i j 2 f0 1 kgg together with d the arithmetic progression fa id i 2 f0 1 kgg and the geometric progression fbd j j 2 f0 1 kgg in one cell of the partition

weak convergence of symmetric diffusion processes in this paper we show the convergence of forms in the sense of mosco associated with the part form on relatively compact open set of dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular dirichlet space setting we also get the same assertion under dirichlet space in infinite dimensional setting as a result of this we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by modified pseudo metric used in k th sturm

weak and strong convergence theorems for strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t is a strict pseudo contraction on c with a fixed point for some 0 1 given an initial guess x 0 c and given also a real sequence n in 0 1 the mann s algorithm generates a sequence x n by the formula x n 1 n x n 1 n t x n n 0 it is proved that if the control sequence n is chosen so that n 1 and n 0 n 1 n then x n converges weakly to a fixed point of t however this convergence is in general not strong we then modify mann s algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence this result extends a recent result of nakajo and takahashi k nakajo w takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 from nonexpansive mappings to strict pseudo contractions

a simple and efficient method for modeling constant mean curvature surfaces a constant mean curvature cmc surface is a critical point of surface area with respect to variations that preserve the volume bounded by the surface we present a simple and elegant method for constructing a triangle mesh approximation to a cmc surface by minimizing the discretized surface area subject to a fixed volume and other constraints such as fixed boundary curves the key idea is an optimization procedure in which the surface area functional is replaced by a least squares energy functional that is easier to minimize and whose critical points are uniformly parameterized surfaces or more generally surfaces parameterized in accord with an arbitrary density function a good initial solution estimate is not required in fact the input mesh quality may be almost arbitrarily poor the user specified volume is taken to be a shape control parameter and since the volume constraint need not be satisfied precisely it is simply treated as an additional equation in the least squares system the nonlinear

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

a hybrid algorithm for pseudo contractive mappings abstract in this paper we suggest and analyze a hybrid algorithm for pseudo contractive mappings in hilbert spaces further we prove the strong convergence of the proposed iterative algorithm for lipschitz pseudo contractive mappings in hilbert spaces

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

split bregman iteration algorithm for total bounded variation regularization based image deblurring many existing algorithms taking the seminorm in bv for regularization have achieved great success in image processing however this paper considers the total bounded variation regularization based approach to perform image deblurring based on this novel model we introduce an extended split bregman iteration to obtain the optimum solution quickly we also provide the rigorous convergence analysis of the iterative algorithm here compared with the results of the rof method numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm

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

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

intersective polynomials and the polynomial szemeredi theorem abstract let p p 1 p r q n 1 n m be a family of polynomials such that p i z m z i 1 r we say that the family p has the psz property if for any set e z with d e lim sup n m e m n 1 n m 0 there exist infinitely many n z m such that e contains a polynomial progression of the form a a p 1 n a p r n we prove that a polynomial family p p 1 p r has the psz property if and only if the polynomials p 1 p r are jointly intersective meaning that for any k n there exists n z m such that the integers p 1 n p r n are all divisible by k to obtain this result we give a new ergodic proof of the polynomial szemeredi theorem based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds we also obtain as a corollary the following generalization of the polynomial van der waerden theorem if p 1 p r q n are jointly intersective integral polynomials then for any finite partition z i 1 k e i of z there exist i 1 k and a n e i such that a a p 1 n a p r n e i

minimal surfaces and sobolev gradients we treat the problem of computing triangle based piecewise linear approximations to parametric minimal surfaces in bf r 3 more specifically given a triangulation t of the unit square omega and a function f 0 from the nodes of t into bf r 3 we seek a function f from the nodes of t into bf r 3 such that f agrees with f 0 on the boundary of omega and the triangulated surface area corresponding to the image of f is minimal we employ a descent method in which at each iteration the gradient of the surface area functional is computed with respect to an inner product that depends on the current approximation to f test results show that starting with extremely poor initial estimates a few descent iterations produce approximations in the vicinity of the solution we also introduce a new characterization of minimal surfaces that eliminates the potential problem of triangle areas approaching zero in place of the surface area functional we minimize a functional whose critical

a new general iterative scheme for split variational inclusion and fixed point problems of k strict pseudo contraction mappings with convergence analysis in this paper we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k strictly pseudo contractive nonself mappings strong convergence theorem is established under some suitable conditions in a real hilbert space which also solves some variational inequality problems results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors finally some examples to study the rate of convergence and some illustrative numerical examples are presented

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 new algorithm for two finite families of demicontractive mappings and equilibrium problems in this paper we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings the strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real hilbert space our result generalizes several recent results in the current literature

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

a multigrid homotopy method for nonlinear inverse problems abstract in the present contribution we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems in which the forward problems are modeled by some forms of partial differential equations the method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest and then carries out the inversion on the coarsest grid by the homotopy method the strategy may give a rapidly and globally convergent method as a practical application this method is used to solve the nonlinear inverse problem of a nonlinear convection diffusion equation which is the saturation equation within the two phase porous media flow we demonstrate the effectiveness and merits of the multigrid homotopy method on two actual model problems

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

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

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

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

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

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 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

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

limits of random differential equations on manifolds consider a family of random ordinary differential equations on a manifold driven by vector fields of the form sum ky k alpha k z t epsilon omega where y k are vector fields epsilon is a positive number z t epsilon is a 1 over epsilon mathcal l 0 diffusion process taking values in possibly a different manifold alpha k are annihilators of mathrm ker mathcal l 0 under hormander type conditions on mathcal l 0 we prove that as epsilon approaches zero the stochastic processes y t over epsilon epsilon converge weakly and in the wasserstein topologies we describe this limit and give an upper bound for the rate of the convergence

strong convergence of iterative methods by strictly pseudocontractive mappings in banach spaces abstract in this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth banach spaces first we prove that the s iteration process recently introduced by sahu in 14 converges strongly to a unique fixed point of a mapping t where t is strongly pseudocontractive mapping from a nonempty closed and convex subset c of a smooth banach space into itself it is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of i strictly pseudocontractive mappings from c into itself our results extend and improve some very recent theorems in fixed point theory and variational inequality problems particularly the results presented here extend some theorems of reich 1980 1 and yamada 2001 15 to a general class of strictly pseudocontractive mappings in uniformly smooth banach spaces

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

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

iterative methods for the computation of fixed points of demicontractive mappings this paper surveys some of the main convergence properties of the mann type iteration for the demicontractive mappings some variants of the mann iteration that ensure the strong convergence like the cq algorithm and a variant for the asymptotically demicontractive mappings are also considered the usual framework of our study is a real hilbert space and only to a certain extent some particular banach spaces historical aspects are pointed out and some applications for the convex feasibility problem are discussed

variable preconditioning via quasi newton methods for nonlinear problems in hilbert space the aim of this paper is to develop stepwise variable preconditioning for the iterative solution of monotone operator equations in hilbert space and apply it to nonlinear elliptic problems the paper is built up to reflect the common character of preconditioned simple iterations and quasi newton methods the main feature of the results is that the preconditioners are chosen via spectral equivalence the latter can be executed in the corresponding sobolev space in the case of elliptic problems which helps both the construction and convergence analysis of preconditioners this is illustrated by an example of a preconditioner using suitable domain decomposition

preconditioning operators and sobolevgradients for nonlinear elliptic problems a preconditioning framework is presented for the iterative solution of nonlinear elliptic problems based on the preconditioning operator approach various fixed preconditioning operators are used in the iteration which can be interpreted as a weighted sobolev gradient method

an equation error approach for the elasticity imaging inverse problem for predicting tumor location the primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body this nonlinear inverse problem not only represents an important and interesting application it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility due to the locking effect classical finite element methods are not effective for incompressible elasticity equations therefore special treatment is necessary for both the direct and inverse problems to study the inverse problem in an optimization framework we propose an extension of the equation error approach we focus on two cases namely when the material parameter is sufficiently smooth and when it is may be discontinuous for the latter case we extend the total variation regularization method to the elasticity imaging inverse problem we give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem we give sufficient details on the discrete formulas as well as on the implementation issues numerical examples for smooth and discontinuous coefficients are given

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

minimization of the ginzburg landau energy functional by a sobolev gradient trust region method we describe a generalized levenberg marquardt method for computing critical points of the ginzburg landau energy functional which models superconductivity the algorithm is a blend of a newton iteration with a sobolev gradient descent method and is equivalent to a trust region method in which the trust region radius is defined by a sobolev metric numerical test results demonstrate the method to be remarkably effective

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

weak convergence of laws of stochastic processes on riemannian manifolds weak convergence of the laws of discrete time re metrized stochastic processes derived from brownian motions on compact riemannian manifolds with heat kernels uniformly bounded by a constant on each compact set of the time parameter and bounded volumes to a stochastic process is given with a weak condition we also give weak convergence of those of brownian motions themselves on manifolds in the same class several examples are given which cover the cases when the manifolds collapse the cases when the original brownian motions converge to a non local markov process and the cases when the gromov hausdorff limit and the spectral limit by kasue and kumura are different

convergence theorems for strict pseudo contractions in q uniformly smooth banach spaces abstract in this paper we discuss some properties of iterates generated by a strict pseudo contraction or a finite family of strict pseudo contractions in a real q uniformly smooth banach space we prove the weak convergence and strong convergence of pseudo contractive mappings the results presented in this paper are interesting extensions and improvements upon those known ones of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 349 acedo and xu g l acedo h k xu iterative methods for strict pseudo contractions in hilbert spaces nonlinear anal 67 2007 2258 2271 and zhou h zhou convergence theorems for strict pseudo contractions in 2 uniformly smooth banach spaces nonlinear anal 69 2008 3160 3173

convergence and adaptive discretization of the irgnm tikhonov and the irgnm ivanov method under a tangential cone condition in banach space in this paper we consider the iteratively regularized gauss newton method irgnm in its classical tikhonov version as well as two further ivanov type and morozov type versions in these two alternative versions regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit respectively we do so in a general banach space setting and under a tangential cone condition while convergence without source conditions thus without rates has so far only been proven under stronger restrictions on the nonlinearity of the operator and or on the spaces moreover we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators the results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem for the cases of a measure valued and of an l infty source for the latter we also provide numerical results with the ivanov type irgnm

iterative methods for strict pseudo contractions in hilbert spaces abstract let t i i 1 n be n strict pseudo contractions defined on a closed convex subset c of a real hilbert space h consider the problem of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem we will prove the weak convergence of these algorithms moreover by applying additional projections we further prove that these algorithms can be modified to have strong convergence

convergence rates for maximum entropy regularization the authors prove results about stability convergence and convergence rates for the maximum entropy method by interpreting it as a nonlinear regularization method the results are applied to a widely studied problem from physical chemistry namely the reconstruction of fluorescence lifetime distributions from the distribution of the delays between absorption and emission of photons

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

global existence and maximal regularity of solutions of gradient systems abstract in this article we use a galerkin method to prove a maximal regularity result for the following abstract gradient system u t g t e u t f t for a e t 0 t u 0 u 0 this abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients

global convergence for ill posed equations with monotone operators the dynamical systems method consider an operator equation f u 0 in a real hilbert space let us call this equation ill posed if the operator f u is not boundedly invertible and well posed otherwise if f is monotone c2loc h operator then we construct a cauchy problem which has the following properties 1 it has a global solution for an arbitrary initial data 2 this solution tends to a limit as time tends to infinity and 3 the limit is the minimum norm solution to the equation f u 0 an example of applications to linear ill posed operator equation is given

strong convergence theorem for pseudo contractive mappings in hilbert spaces abstract the purpose of this paper is to construct an ishikawa type of hybrid algorithm for pseudo contractive mappings in hilbert spaces our results extend the recent ones announced by yao et al y h yao y c liou g marino a hybrid algorithm for pseudo contractive mappings nonlinear anal 71 2009 4997 5002 and many others

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

a levenberg marquardt method based on sobolev gradients abstract we extend the theory of sobolev gradients to include variable metric methods such as newton s method and the levenberg marquardt method as gradient descent iterations associated with stepwise variable inner products in particular we obtain existence uniqueness and asymptotic convergence results for a gradient flow based on a variable inner product

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

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

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