periodic points of holomorphic maps via lefschetz numbers in this paper we study the set of periods of holomorphic maps on compact manifolds using the periodic lefschetz numbers introduced by dold and llibre which can be computed from the homology class of the map we show that these numbers contain information about the existence of periodic points of a given period and if we assume the map to be transversal then they give us the exact number of such periodic orbits we apply this result to the complex projective space of dimension n and to some special type of hopf surfaces partially characterizing their set of periods in the first case we also show that any holomorphic map of cp n of degree greater than one has infinitely many distinct periodic orbits hence generalizing a theorem of fornaess and sibony we then characterize the set of periods of a holomorphic map on the riemann sphere hence giving an alternative proof of baker s theorem

experiments with implicit upwind methods for the euler equations abstract a number of implicit integration schemes for the one dimensional euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow fastest convergence quadratic for the first order backward euler scheme is obtained with the upwind switching provided by van leer s differentiable split fluxes which easily linearize in time with roe s nondifferentiable split flux differences the iterations may get trapped in a limit cycle this also happens in a second order scheme with split fluxes if the matrix coefficients arising in the implicit time linearization are not properly centered in space the use of second order terms computed from split fluxes degrades the accuracy of the solution especially if these are subjected to a limiter for the sake of monotonicity preservation second order terms computed from the characteristic variables perform best

abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

rate of type ii blowup for a semilinear heat equation a solution u of a cauchy problem for a semilinear heat equation left begin array ll u t delta u u p quad rm in bf r n times 0 t u x 0 u 0 x geq 0 quad rm in bf r n end array right is said to undergo type ii blowup at t t if lim sup t nearrow t t t 1 p 1 u t infty infty let varphi infty be the radially symmetric singular steady state suppose that u 0 in l infty is a radially symmetric function such that u 0 varphi infty and u0 t change sign at most finitely many times we determine the exact blowup rate of type ii blowup solution with initial data u0 in the case of p pl where pl is the lepin exponent

an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

multilevel algorithms for constrained compact fixed point problems in this paper the authors extend the multilevel algorithm of atkinson and brakhage for compact fixed point problems and the projected newton method of bertsekas to create a fast multilevel algorithm for parabolic boundary control problems having bound constraints on the control results are extended from finite dimension on constraint identification this approach permits both adaptive integration in time and inexact evaluation of the cost functional

pseudo transient continuation for nonsmooth nonlinear equations pseudo transient continuation is a newton like iterative method for computing steady state solutions of differential equations in cases where the initial data are far from a steady state the iteration mimics a temporal integration scheme with the time step being increased as steady state is approached the iteration is an inexact newton iteration in the terminal phase in this paper we show how steady state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo transient continuation an example of such a case is a discretized pde with a lipschitz continuous but nondifferentiable constitutive relation as part of the nonlinearity in this case we can approximate a generalized derivative with a difference quotient the existing theory for pseudo transient continuation requires lipschitz continuity of the jacobian newton like methods for nonsmooth equations have been globalized by trust region methods smooth approximations and splitting methods in the past but these approaches are not designed to find steady state solutions of time dependent problems the method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo transient continuation

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

pseudo transient continuation for nonlinear transient elasticity this paper demonstrates how pseudo transient continuation improves the efficiency and robustness of a newton iteration within a non linear transient elasticity simulation pseudo transient continuation improves efficiency by enabling larger time steps than possible with a newton iteration robustness improves because pseudo transient continuation recovers the convergence of newton s method when the initial iterate is not within the region of local convergence we illustrate the benefits of pseudo transient continuation on a non linear transient simulation of a buckling cylinder including a comparison with a line search based newton iteration

homotopy method for solving variational inequalities with bounded box constraints abstract in this paper using the gabriel more smoothing function of the median function a smooth homotopy method for solving nonsmooth equation reformulation of bounded box constrained variational inequality problem vip l u f is given without any monotonicity condition on the defining map f for starting point chosen almost everywhere in r n existence and convergence of the homotopy pathway are proven nevertheless it is also proven that if the starting point is chosen to be an interior point of the box the proposed homotopy method can also serve as an interior point method

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

numerical simulation of two dimensional bingham fluid flow by semismooth newton methods this paper is devoted to the numerical simulation of two dimensional stationary bingham fluid flow by semismooth newton methods we analyze the modeling variational inequality of the second kind considering both dirichlet and stress free boundary conditions a family of tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified by using fenchel s duality optimality systems which characterize the original and regularized solutions are obtained the regularized optimality systems are discretized using a finite element method with cross grid p 1 q 0 elements for the velocity and pressure respectively a semismooth newton algorithm is proposed in order to solve the discretized optimality systems using an additional relaxation a descent direction is constructed from each semismooth newton iteration local superlinear convergence of the method is also proved finally we perform numerical experiments in order to investigate the behavior and efficiency of the method

strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems it is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable a classical result of ky fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions which implies that the eigenvalues are semismooth functions based on a recent result of the authors it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere as an application it is demonstrated how this result can be used to analyze the quadratic convergence of newton s method for solving inverse eigenvalue problems ieps and generalized ieps with multiple eigenvalues

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

fixed point indices of iterated smooth maps in arbitrary dimension abstract let f be a smooth self map of r m when m is an arbitrary natural number we give a complete description of possible sequences of indices of iterations of f at an isolated fixed point answering in affirmative the chow mallet paret and yorke conjecture posed in s n chow j mallet parret j a yorke a periodic point index which is a bifurcation invariant in geometric dynamics rio de janeiro 1981 in lecture notes in math vol 1007 springer berlin 1983 pp 109 131

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

on the computation of an element of clarke generalized jacobian for a vector valued max function abstract in this paper we present an algorithm for calculating an element of clarke generalized jacobian for a vector valued max type function the algorithm reduces the computational cost of an existing algorithm

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

some quadrature based versions of the generalized newton method for solving nonsmooth equations abstract in this paper modifications of a generalized newton method based on some rules of quadrature are studied the methods considered are newton like iterative schemes for numerical solving systems of nonsmooth equations some mild conditions are given that ensure superlinear convergence to a solution moreover a parameterized version of the midpoint version is presented finally results of numerical tests are established

an improved method for solving quasi linear convection diffusion problems on a coarse mesh a method is developed for solving quasi linear convection diffusion problems starting on a coarse mesh where the data and solution dependent coefficients are unresolved the problem is unstable and approximation properties do not hold the newton like iterations of the solver are based on the framework of regularized pseudotransient continuation where the proposed time integrator is a variation on the newmark strategy designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite an algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning the method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization derivation and q linear loc

the blow up solutions of the heat equations in fl1 rn abstract in this paper we give a formal solution of some nonlinear evolution equations by the formal solution we can obtain the blow up solution of the heat equations even in the supercritical case

fixed point indices and periodic points of holomorphic mappings let n be the ball x 1 in the complex vector space mathbb c n let f delta n rightarrow mathbb c n be a holomorphic mapping and let m be a positive integer assume that the origin 0 0 dots 0 is an isolated fixed point of both f and the mth iteration f m of f then for each factor m of m the origin is again an isolated fixed point of f m and the fixed point index mu f m 0 of f m at the origin is well defined and so is the local dold s index invent math 74 3 419 435 1983 at the origin p m f 0 sum tau subset p m 1 tau mu f m tau 0 where p m is the set of all primes dividing m the sum extends over all subsets of p m is the cardinal number of and m tau m prod p in tau p 1 p m f 0 can be interpreted to be the number of periodic points of period m of f overlapped at the origin any holomorphic mapping f 1 delta n rightarrow mathbb c n sufficiently close to f has exactly p m f 0 distinct periodic points of period m near the origin provided that all the fixed points of f 1 m near the origin are simple note that f itself has no periodic point of period m near the origin if m 1 according to shub and sullivan s work topology 13 189 191 1974 a necessary condition so that p m f 0 0 is that the linear part of f at the origin has a periodic point of period m the goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings

numerical methods for reaction diffusion problems with non differentiable kinetics we consider a class of steady state semilinear reaction diffusion problems with non differentiable kinetics the analytical properties of these problems have received considerable attention in the literature we take a first step in analyzing their numerical approximation we present a finite element method and establish error bounds which are optimal for some of the problems in addition we also discuss a finite difference approach numerical experiments for one and two dimensional problems are reported

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

finite termination of a newton type algorithm for a class of affine variational inequality problems abstract many optimization problems can be reformulated as a system of equations one may use the generalized newton method or the smoothing newton method to solve the reformulated equations so that a solution of the original problem can be found such methods have been powerful tools to solve many optimization problems in the literature in this paper we propose a newton type algorithm for solving a class of monotone affine variational inequality problems avips for short in the proposed algorithm the techniques based on both the generalized newton method and the smoothing newton method are used in particular we show that the algorithm can find an exact solution of the avip in a finite number of iterations under an assumption that the solution set of the avip is nonempty preliminary numerical results are reported

global and blow up solutions for nonlinear parabolic equations with robin boundary conditions abstract in this paper we discuss the blow up for classical solutions to the following class of parabolic equations with robin boundary condition b u t g u u f u in 0 t u n u 0 on 0 t u x 0 h x 0 in where is a bounded domain of r n n 2 with smooth boundary by constructing some appropriate auxiliary functions and using a first order differential inequality technique we derive conditions on the data which guarantee the blow up or the global existence of the solution for the blow up solution a lower bound on blow up time is also obtained moreover some examples are presented to illustrate the applications

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

fixed point indices of iterated maps on determine toutes les relations universelles qui existent entre les indices de point fixe i f n n 1 2 des iterees f n d une application f v y ou y est un enr vcy est un sous ensemble ouvert

quasi newton trust region algorithm for non smooth least squares problems1this work was supported by cnpq of brazil and the national natural science foundation of china 1 in this paper we present a quasi newton trust region algorithm for non smooth least squares problems which possess the smooth plus non smooth decomposition this method uses a smooth subproblem with second order information to approximate the locally lipschitzian function which is important to increase the convergence rate of the algorithm and deal with the large residual non smooth least squares problems the global and superlinear convergence of this algorithm are established

an l blow up estimate for a nonlinear heat equation on decrit une borne superieure au taux d explosion des solutions de l equation de la chaleur non lineaire u t u u p ou u 0 p 1

stabilized and inexact adaptive methods for capturing internal layers in quasilinear pde a method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh the solution process is assumed to start on a mesh where the problem is badly resolved and approximation properties of the exact problem and its corresponding finite element solution do not hold a sequence of stabilized and inexact partial solves allows the mesh to be refined to capture internal layers while an approximate solution is built eventually leading to an accurate approximation of both the problem and its solution the innovations in the current work include a closed form definition for the numerical dissipation and inexact scaling parameters on each mesh refinement as well as a convergence result for the residual of the discrete problem numerical experiments demonstrate the method on a range of problems featuring steep internal layers and high solution dependent frequencies of the diffusion coefficients

superlinear convergence of smoothing quasi newton methods for nonsmooth equations abstract we study local convergence of smoothing quasi newton methods for solving a system of nonsmooth nondifferentiable equations in r n the feature of smoothing quasi newton methods is to use a smooth function to approximate the nonsmooth mapping and update the quasi newton matrix at each step convergence results are given under directional derivative consistence property without differentiability we establish a dennis more type superlinear convergence theorem for smoothing quasi newton methods and we prove linear convergence of the smoothing broyden method furthermore we propose a superlinear convergent smoothing newton broyden method without using the generalized jacobian and the semismooth assumption we illustrate the smoothing approach on box constrained variational inequalities

periods for transversal maps via lefschetz numbers for periodic points let f m m be a c1 map on a c differentiable manifold the map f is called transversal if for all m e n the graph of fm intersects transversally the diagonal of m x m at each point x x such that x is a fixed point of fm we study the set of periods of f by using the lefschetz numbers for periodic points we focus our study on transversal maps defined on compact manifolds such that their rational homology is ho z q hi q e q and hk t 0 for k 0 1

a geometric newton method for oja s vector field newton s method for solving the matrix equation runs up against the fact that its zeros are not isolated this is due to a symmetry of f by the action of the orthogonal group we show how differential geometric techniques can be exploited to remove this symmetry and obtain a geometric newton algorithm that finds the zeros of f the geometric newton method does not suffer from the degeneracy issue that stands in the way of the original newton method

trust region algorithms for solving nonsmooth equations two globally convergent trust region algorithms are presented for solving nonsmooth equations where the functions are only locally lipschitzian the first algorithm is an extension of the classic levenberg marquardt method by approximating the locally lipschitzian function with a smooth function and using the derivative of the smooth function in the algorithm wherever a derivative is needed global convergence for this algorithm is established under a regular condition in the second algorithm successive smooth approximation functions and their derivatives are used global convergence for the second algorithm is established under mild assumptions both objective functions of subproblems of these two algorithms are quadratic functions

newton and quasi newton methods for a class of nonsmooth equations and related problems the paper presents concrete realizations of quasi newton methods for solving several standard problems including complementarity problems special variational inequality problems and the karush kuhn tucker kkt system of nonlinear programming a new approximation idea is introduced in this paper the q superlinear convergence of the newton method and the quasi newton method are established under suitable assumptions in which the existence of f x is not assumed the new algorithms only need to solve a linear equation in each step for complementarity problems the qr factorization on the quasi newton method is discussed

robust solution of richards equation for nonuniform porous media capillary pressure saturation relative permeability relations described using the van genuchten 1980 and mualem 1976 models for nonuniform porous media lead to numerical convergence difficulties when used with richards equation for certain auxiliary conditions these difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure convergence difficulties are illustrated using standard numerical approaches to simulate such problems we investigate constitutive relations interblock permeability nonlinear algebraic system approximation methods and two time integration approaches an integral permeability approach approximated by hermite polynomials is recommended and shown to be robust and economical for a set of test problems which correspond to sand loam and clay loam media

finite element approximation of a model reaction diffusion problem with a non lipschitz nonlinearity given 0 andp 0 1 we consider the following problem findu such that begin gathered delta u lambda u p 0in omega hfill u 1on partial omega hfill end gathered where 2 is a smooth convex domain we prove optimalh 1 andl error bounds for the standard continuous piecewise linear galerkin finite element approximation in addition we analyse a more practical approximation using numerical integration on the nonlinear term finally we consider a modified nonlinear sor algorithm which is shown to be globally convergent for solving the algebraic system derived from the more practical approximation

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

global solution of the critical burgers equation in n dimensions abstract in this paper we first prove the local well posedness of the fractional burgers equations in n dimensions combining the local well posedness and the method of modulus of continuity we show the global well posedness of the n d critical burgers equation in critical besov spaces b p 1 n p r n with p n

a new filled function method for nonlinear equations in this paper a new global optimization approach based on the filled function method is proposed for solving box constrained systems of nonlinear equations we first convert the nonlinear system into an equivalent global optimization problem and then propose a new filled function method to solve the converted global optimization problem several numerical examples are presented and solved by using different local minimization methods which illustrate the efficiency of the present approach

indices of the iterates of r 3 homeomorphisms at lyapunov stable fixed points abstract given any positive sequence c n n n we construct orientation preserving homeomorphisms f r 3 r 3 such that fix f per f 0 0 is lyapunov stable and lim sup i f m 0 c m we will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms

a system of nonsmooth equations solver based upon subgradient method in this paper a subgradient method is developed to solve the system of nonsmooth equations first the system of nonsmooth equations is transformed into a nonsmooth optimization problem with zero minimal objective function value then a subgradient method is applied to solve the nonsmooth optimization problem during the processes the pre known optimal objective function value is adopted to update step sizes the corresponding convergence results are established as well several numerical experiments and applications show that the proposed method is efficient and robust

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

a globally convergent successive approximation method for severely nonsmooth equations this paper presents a globally convergent successive approximation method for solving f x 0 where f is a continuous function at each step of the method f is approximated by a smooth function f k with pa f k f pa rightarrow 0 as k rightarrow infty the direction f k x k 1 f x k is then used in a line search on a sum of squares objective the approximate function f k can be constructed for nonsmooth equations arising from variational inequalities maximal monotone operator problems nonlinear complementarity problems and nonsmooth partial differential equations numerical examples are given to illustrate the method

blowup for the nonlinear heat equation with small initial data in scale invariant besov norms abstract we consider the cauchy problem of the nonlinear heat equation u t u u b u 0 x u 0 with b 2 and b n we prove that initial data u 0 s r n the schwartz class arbitrarily small in the scale invariant besov norm b n b 1 b 2 q 2 b r n can produce solutions that blow up in finite time the case b 3 answers a question raised by yves meyer our result also proves that the smallness assumption put in an earlier work by c miao b yuan and b zhang for the global in time solvability is essentially optimal

secant methods for semismooth equations some generalizations of the secant method to semismooth equations are presented in the one dimensional case the superlinear convergence of the classical secant method for general semismooth equations is proved moreover a new quadratically convergent method is proposed that requires two function values per iteration for the n dimensional cases we discuss secant methods for two classes of composite semismooth equations most often studied semismooth equations are of such form

a trust region method for parabolic boundary control problems in this paper we develop a trust region algorithm for constrained parabolic boundary control problems for the computation of a trust region step we propose an iterative scheme which is a projected form of the steihaug trust region conjugate gradient method to ensure the good local convergence properties in the terminal phase a smoothing step at each iteration is added this step and the projection require the modification of the standard trust region algorithm and its convergence proof the algorithm has sup norm convergence in the terminal phase and l2 convergence in the global phase the results are illustrated for a parabolic boundary control problem

affine scaling interior levenberg marquardt method for bound constrained semismooth equations under local error bound conditions we develop and analyze a new affine scaling levenberg marquardt method with nonmonotonic interior backtracking line search technique for solving bound constrained semismooth equations under local error bound conditions the affine scaling levenberg marquardt equation is based on a minimization of the squared euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable the global convergence results are developed in a very general setting of computing trial directions by a semismooth levenberg marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set we establish that close to the solution set the affine scaling interior levenberg marquardt algorithm is shown to converge locally q superlinearly depending on the quality of the semismooth and levenberg marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition that is bd regular condition under nonsmooth case a nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature

global mild solutions of navier stokes equations we establish a global well posedness of mild solutions to the three dimensional incompressible navier stokes equations if the initial data are in the space x 1 defined by x 1 f e d r 3 3 1 f d and if the norms of the initial data in x 1 are bounded exactly by the viscosity coefficient

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

on the convergence of modified newton methods for solving equations containing a non differentiable term we provide a semilocal convergence analysis for certain modified newton methods for solving equations containing a non differentiable term the sufficient convergence conditions of the corresponding newton methods are often taken as the sufficient conditions for the modified newton methods that is why the latter methods are not usually treated separately from the former however here we show that weaker conditions as well as a finer error analysis than before can be obtained for the convergence of modified newton methods numerical examples are also provided

realization of all dold s congruences with stability the main goal of this paper is to prove that for each n 2 every sequence of integers satisfying dold s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving r n homeomorphism at an isolated stable fixed point

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

hedging with a correlated asset solution of a nonlinear pricing pde hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk even if the residual risk is considered diversifiable the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument if the residual risk is not considered diversifiable then this risk can be priced using an actuarial standard deviation principle in infinitesimal time in both cases these models result in the same nonlinear partial differential equation pde a fully implicit monotone discretization method is developed for solution of this pricing pde this method is shown to converge to the viscosity solution certain grid conditions are required to guarantee monotonicity an algorithm is derived which given an initial grid inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied at each timestep the nonlinear discretized algebraic equations are solved using an iterative algorithm which is shown to be globally convergent monte carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option

a quasisecant method for solving a system of nonsmooth equations in this paper the solution of nonsmooth equations is studied we first transform the problem into an equivalent nonsmooth optimization problem and then the quasisecant method is introduced to solve it some nonsmooth equations that have arisen from bilevel programming problems are solved by our proposed method the numerical results show the effectiveness and efficiency of our proposed method

a regularization semismooth newton method based on the generalized fischer burmeister function for p0 ncps we consider a regularization method for nonlinear complementarity problems with f being a p 0 function which replaces the original problem with a sequence of the regularized complementarity problems in this paper this sequence of regularized complementarity problems are solved approximately by applying the generalized newton method for an equivalent augmented system of equations constructed by the generalized fischer burmeister fb ncp functions f p with p 1 we test the performance of the regularization semismooth newton method based on the family of ncp functions through solving all test problems from mcplib numerical experiments indicate that the method associated with a smaller p for example email protected 1 1 2 usually has better numerical performance and the generalized fb functions f p with email protected 1 1 2 can be used as the substitutions for the fb function f 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

the existence of viable trajectories in state dependent impulsive systems abstract suitable topological tools for studying the existence of viable and periodic viable trajectories in state dependent impulsive differential equations and inclusions are provided some results based on the wazewski retract method are applied the techniques presented in the paper are illustrated with several examples

quasi newton methods for solving underdetermined nonlinear simultaneous equations abstract we analyze iterative processes of type xk 1 xk xk ek f xk for solving f x 0 f r n r m m n parameters ek are updated at each iteraction using least change secant update procedures we prove local linear and superlinear convergence results we introduce two new superlinearly convergent methods of this type and one linearly convergent quasi newton generalization of cimmino s parallel algorithm for solving linear systems some numerical experiments are presented

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

new control allocation algorithms in fixed point framework for overactuated systems with actuator saturation abstractin this paper two control allocation algorithms have been proposed for overactuated systems the algorithms are developed by formulating constrained control allocation problem into an equivalent fixed point framework the first algorithm follows sequential solving method while the other one involves a zero finding technique by the newton method in order to consider limiting constraints the saturation function has been taken into account that leads to a nonsmooth zero finding problem and the proposed method guarantees superlinear convergence the second algorithm is easy to implement and faster than the algorithm developed based on the sequential solving technique to demonstrate the effectiveness of the proposed algorithms a detail simulation study has been carried out in which one example considers both the actuator rate and amplitude constraints

the burnside ring of the infinite cyclic group and its relations to the necklace algebra λ rings and the universal ring of witt vectors abstract it is shown that well known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the burnside ring of the infinite cyclic group on the one hand and grothendieck s ring of formal power series with constant term 1 as well as the universal ring of witt vectors on the other hand

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm