value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

alternate discrete painleve equations from limits of q pvi and d pv abstract we study the special limits of discrete painleve equations belonging to the q p vi and d p v families when the independent variable goes to 1 or 0 respectively we obtain discrete systems which are shown to be either contiguities of solutions of continuous painleve equations usually of p vi but also of the other painleve equations except p i which has no parameters and so no possibility of contiguity or linearisable mappings

on universality of the smoothed eigenvalue density of large random matrices we describe the resolvent approach for the rigorous study of the mesoscopic regime of hermitian matrix spectra we present results reflecting universal behaviour of the smoothed density of the eigenvalue distribution of large random matrices

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

discretising systematically the painleve equations we apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the painleve equations the important point is that in the case of the painleve equations one needs to preserve the integrability we show that this is indeed possible using our systematic procedure applied to the equations obtained from the hamiltonian description of the painleve equations

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

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

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

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

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

tri hamiltonian formulation for certain integrable lattice equations a systematic investigation of integrable differential difference equations with two independent variables admitting multi hamiltonian structure is presented considering the volterra vl toda tl relativistic toda rt belov chaltikian bc and blaszak marciniak both three bm3 and four bm4 coupled lattice equations it is shown that they admit a sequence of operators out of which only three are hamiltonian ones and so they are tri hamiltonian systems only it is observed that the constructed third operator for vl and bc lattice equations is hamiltonian only if the field variable is periodic with even period

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

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

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

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

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

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

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

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

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

special relations for continuous and discrete painleve equations we investigate a relation between solutions of the painleve ii equation corresponding to values of the parameter of pii which cannot be connected through a schlesinger transformation which was first derived by gambier here we present the discrete analogue of this relation relating the discrete pii to the alternate d pii the latter turns out to be a consequence of a quadratic relation existing between two different families of solutions of pv a q discrete analogue of the latter relation is also presented

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

miura transformations for discrete painleve equations coming from the affine e8 weyl group we derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine weyl group e8 1 by deautonomisation we obtain two hitherto unknown systems one of which turns out to be a linearisable one and we show that both these systems arise from the deautonomisation of a non qrt quispel roberts thompson mapping in order to unambiguously prove the integrability of these nonautonomous systems we introduce a series of miura transformations which allows us to prove that one of these systems is indeed a discrete painleve equation related to the affine weyl group e7 1 and to cast it in canonical form a similar sequence of miura transformations allows us to effectively linearise the second system we obtain an interesting off shoot of our calculations is that the series of miura transformations when applied at the autonomous limit allows one to transform a non qrt invariant into a qrt one

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

on a transcendental equation related to painleve iii and its discrete forms we examine the canonical forms of painlev equations and argue that the equation for piii in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard piii our argument is based on the fact that the value of this parameter cannot be modified through auto b cklund transformations we investigate the possible discrete forms of this equation and produce two of them one is of a difference type where the independent variable enters linearly while the second one is of q type where the independent variable enters in a multiplicative way the properties of these discrete equations are also studied

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

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

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

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

on laplace darboux type sequences of generalized weingarten surfaces two novel classes of integrable surfaces generalized weingarten surfaces of class 1 and 2 which may be defined in a coordinate independent manner or equivalently via an extension of the classical lelieuvre formulas are introduced the classes 1 and 2 generalize bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the ernst weyl equation descriptive of gravitational and neutrino fields in general relativity the latter connection is exploited to derive laplace darboux type transformations for class 2 via the associated loewner konopelchenko rogers representation the laplace darboux type transformations are shown to have remarkable geometric properties in terms of sphere congruences connections with the pohlmeyer lund regge vortex model and an inhomogeneous heisenberg spin equation are recorded a backlund transformation for a particular painleve iii equation is derived in a purely geometric manner and a geometric interpretation of a

casorati determinant solutions for the discrete painleve iii equation the discrete painleve iii equation is investigated based on the bilinear formalism it is shown that it admits the solutions expressed by the casorati determinant whose entries are given by the discrete bessel functions moreover based on the observation that these discrete bessel functions are transformed to the q bessel functions by a simple variable transformation we present a q difference analog of the painleve iii equation

new 2 1 dimensional nonisospectral toda lattice hierarchy we give a new 2 1 dimensional nonisospectral generalization of the toda lattice hierarchy reductions yield a variety of new integrable hierarchies along with their underlying linear problems including new 1 1 dimensional differential delay hierarchies nonisospectral and isospectral new ordinary differential delay hierarchies and new discrete painleve hierarchies we also show that a reduction in components yields our previously obtained 2 1 dimensional nonisospectral volterra lattice hierarchy

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

continuous symmetries of the lattice potential kdv equation in this paper we present a set of results on the integration and on the symmetries of the lattice potential korteweg de vries lpkdv equation using its associated spectral problem we construct the soliton solutions and the lax technique enables us to provide infinite sequences of generalized symmetries finally using a discrete symmetry of the lpkdv equation we construct a large class of non autonomous generalized symmetries

growth and integrability in discrete systems we introduce a new discrete integrability criterion inspired from the recent findings of ablowitz and collaborators this criterion is based on the study of the growth of some characteristic of the solutions of a mapping using nevanlinna theory since the practical implementation of the latter does not always lead to a clear cut answer we complement the growth criterion by the singularity confinement property this combination turns out to be particularly efficient its application allows us to recover the known forms of the discrete painleve equations and to show that no new ones may exist within a given parametrization

from the continuous pv to discrete painleve equations we study the discrete transformations that are associated with the auto backlund of the continuous pv equation we show that several two parameter discrete painleve equations can be obtained as contiguity relations of pv among them we find the asymmetric d pii equation which is a well known form of discrete piii the relation between the ternary pi previously obtained through the discrete dressing approach and pv is also established a new discrete painleve equation is also derived

a discrete garnier type system from symmetry reduction on the lattice a symmetry reduction of the lattice modified boussinesq system is studied the full group of lie point symmetries of the relevant system is retrieved and certain group invariant solutions are considered by using an accessional generalized symmetry it is demonstrated that the symmetry reduction leads to a coupled set of second order nonlinear non autonomous ordinary difference equations involving six free parameters generalizing to higher order some of the known discrete analogues of the painleve vi equation the corresponding isomonodromic deformation problem is constructed through the symmetry reduction as well