analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

on the global convergence of the bfgs method for nonconvex unconstrained optimization problems this paper is concerned with the open problem of whether the bfgs method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems we propose a cautious bfgs update and prove that the method with either a wolfe type or an armijo type line search converges globally if the function to be minimized has lipschitz continuous gradients

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

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

auto backlund transformations for a differential delay equation discrete painleve equations have over recent years generated much interest one property of such equations that is considered to be particularly important is the existence of auto backlund transformations that is mappings between solutions of the equation in question usually involving changes in the values of parameters appearing as coefficients we have recently presented extensions of discrete painleve equations to equations involving derivatives as well as shifts in the independent variable here we show how auto backlund transformations can also be constructed for such differential delay equations we emphasise that this is the first time that an auto backlund transformation has been given for a differential delay equation

casorati determinant solutions for the discrete painleve ii equation we present a class of solutions to the discrete painleve ii equation for particular values of its parameters it is shown that these solutions can be expressed in terms of casorati determinants whose entries are discrete airy functions the analogy between the tau function for the discrete pii and that of the discrete toda molecule equation is pointed out

continuous and discrete painleve equations arising from the gap probability distribution of the finite n gaussian unitary ensembles in this paper we study the gap probability problem in the gaussian unitary ensembles of n by n matrices the probability that the interval j a a is free of eigenvalues in the works of tracy and widom adler and van moerbeke and forrester and witte on this subject it has been shown that two painleve type differential equations arise in this context the first is the jimbo miwa okomoto sigma form and the second is a particular painleve iv using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability denoted as sigma n a r n a and r n a we show that each one satisfies a second order painleve type differential equation as well as a discrete painleve type equation in particular in addition to providing an elementary derivation of the aforementioned sigma form and painleve iv we are able to show that the quantity r n a satisfies a particular case of chazy s second degree second order differential equation for the discrete equations we show that the quantity r n a satisfies a particular form of the modified discrete painleve ii equation obtained by grammaticos and ramani in the context of backlund transformations we also derive second order second degree difference equations for the quantities r n a and sigma n a

convergence properties of the bfgs algoritm the bfgs method is one of the most famous quasi newton algorithms for unconstrained optimization in 1984 powell presented an example of a function of two variables that shows that the polak ribiere polyak prp conjugate gradient method and the bfgs quasi newton method may cycle around eight nonstationary points if each line search picks a local minimum that provides a reduction in the objective function in this paper a new technique of choosing parameters is introduced and an example with only six cyclic points is provided it is also noted through the examples that the bfgs method with wolfe line searches need not converge for nonconvex objective functions

composite proximal bundle method we consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping this is a sufficiently rich class that includes max functions largest eigenvalue functions and norm 1 regularized functions the bundle method uses an oracle that is able to compute separately the function and subgradient information for the convex function and the function and derivatives for the smooth mapping with this information it is possible to solve approximately certain proximal linearized subproblems in which the smooth mapping is replaced by its taylor series linearization around the current serious step our numerical results show the good performance of the composite bundle method for a large class of problems

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

a new inversion free method for a rational matrix equation abstract motivated by the classical newton schulz method for finding the inverse of a nonsingular matrix we develop a new inversion free method for obtaining the minimal hermitian positive definite solution of the matrix rational equation x a x 1 a i where i is the identity matrix and a is a given nonsingular matrix we present convergence results and discuss stability properties when the method starts from the available matrix aa we also present numerical results to compare our proposal with some previously developed inversion free techniques for solving the same rational matrix equation

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

coalescence cascades and special function solutions for the continuous and discrete painleve equations we present a systematization of the special function type solutions of continuous and discrete painleve equations our method is to start from and construct its special solutions from the solutions of the hypergeometric equation and by coalescence obtain the lower painleve equations together with their special solutions in the discrete case we study the symmetric one component three point mapping forms of discrete painleve equations starting from d

miura transforms for discrete painleve equations a miura transformation is presented relating the discrete painleve ii equation to what plays the role of the modified d pii equation 34 in the painleve gambier classification the study of this transformation makes possible the derivation of the auto backlund transform of the discrete pii and allows one to obtain particular solutions to the latter moreover the use of this transformation makes it possible to construct a new mapping in the parameter space of pii the latter is a novel method for the construction of integrable discrete systems

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

discrete painleve equations from singularity patterns the asymmetric trihomographic case we derive the discrete painleve equations associated with the affine weyl group e8 1 that can be represented by an in the quispel roberts thompson sense asymmetric trihomographic system the method used in this paper is based on singularity confinement we start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns that cannot lead to confined singularities working with the remaining ones we implement the confinement conditions and derive the corresponding discrete painleve equations which involve two variables by eliminating either of these variables we obtain a symmetric equation examining all these equations of a single variable we find that they coincide exactly with those derived in previous works of ours thereby establishing the completeness of our results

the discrete painleve ii equations miura and auto backlund transformations we present miura transformations for all discrete painleve ii equations known to date we then use these miuras to derive special solutions in terms of discrete airy functions and to construct auto backlund transformations for the discrete painleve equations these transformations are then used to generate rational solutions some new forms of d pii and d p34 are obtained as well

residual methods for the large scale matrix pth root and some related problems abstract the problem of finding the p th root of a matrix has received special attention in the last few years standard approaches for this problem include and combine some variations of newton s method which in turn involve matrix factorizations that in general are not suitable for large scale problems motivated by some recently developed low cost iterative schemes for nonlinear problems we consider and analyze specialized residual methods that only require a few matrix matrix products per iteration and hence are suitable for the large scale case as a by product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate preliminary and encouraging numerical results are presented for computing p th roots of large scale symmetric and positive definite matrices for different values of p

convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization we study the gradient sampling algorithm of burke lewis and overton for minimizing a locally lipschitz function f on mathbb r n that is continuously differentiable on an open dense subset we strengthen the existing convergence results for this algorithm and introduce a slightly revised version for which stronger results are established without requiring compactness of the level sets of f in particular we show that with probability 1 the revised algorithm either drives the f values to infty or each of its cluster points is clarke stationary for f we also consider a simplified variant in which the differentiability check is skipped and the user can control the number of f evaluations per iteration

convergence properties of some block krylov subspace methods for multiple linear systems in the present paper we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems using the schur complement formulae and a new matrix product we give expressions of the approximate solutions and the corresponding residuals we also derive some useful relations between the norm of the residuals

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

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

parallel matrix function evaluation via initial value ode modeling the purpose of this article is to propose ode based approaches for the numerical evaluation of matrix functions f a a question of major interest in the numerical linear algebra for that we model f a as the solution at a finite time t of a time dependent equation we use parallel algorithms such as the parareal method on the time interval 0 t in order to solve the obtained evolution equation when f a is reached as a stable steady state it can be computed by combining parareal algorithms and optimal control techniques numerical illustrations are given

delay differential equations and the painleve transcendents abstract we apply the recently proposed integrability criterion for differential difference systems that blends the classical painleve analysis with singularity confinement for discrete systems to a class of first order differential delay equations our analysis singles out the family of bi riccati equations as integrability candidates among these equations that pass the test some are integrable in a straightforward way usually by reduction to a standard riccati equation for some transformed variable while the remaining ones define new hysterodifferential forms of the painleve transcendental equations

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

investigating relations between discrete painleve equations the multistep approach we show how starting from a mapping where the independent variable advances one step at a time one can obtain versions of the mapping corresponding to a multi step evolution the same procedure is applied to discrete painleve equations and we proceed to establish miura relations between the single step and the multi step versions in the present study multi referring to double triple and quintuple these miura relations are discrete painleve equations on their own right we show that while in some cases it is impossible to obtain a multi step equation for a single variable deriving a miura system is still possible we perform our analysis for equations associated with the affine weyl groups e8 1 e7 1 e6 1 and a4 1

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

nonsmooth optimization via quasi newton methods we investigate the behavior of quasi newton algorithms applied to minimize a nonsmooth function f not necessarily convex we introduce an inex act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the armijo and wolfe conditions if f is absolutely continuous along the line furthermore the line search is guaranteed to terminate if f is semi algebraic it seems quite difficult to establish a convergence theorem for quasi newton methods applied to such general classes of functions so we give a care ful analysis of a special but illuminating case the euclidean norm in one variable using the inexact line search and in two variables assuming that the line search is exact in practice we find that when f is locally lipschitz and semi algebraic with bounded sublevel sets the bfgs broyden fletcher goldfarb shanno method with the inexact line search almost always generates sequences whose cluster points are clarke stationary and with function values converging r linearly to a clarke station ary value we give references documenting the successful use of bfgs in a variety of nonsmooth applications particularly the design of low order controllers for linear dynamical systems we conclude with a challenging open question

the bfgs method with exact line searches fails for non convex objective functions this work shows that the bfgs method and other methods in the broyden class with exact line searches may fail for non convex objective functions

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

a family of variable metric proximal methods we consider conceptual optimization methods combining two ideas the moreau yosida regularization in convex analysis and quasi newton approximations of smooth functions we outline several approaches based on this combination and establish their global convergence then we study theoretically the local convergence properties of one of these approaches which uses quasi newton updates of the objective function itself also we obtain a globally and superlinearly convergent bfgs proximal method at each step of our study we single out the assumptions that are useful to derive the result concerned

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

quasi newton bundle type methods for nondifferentiable convex optimization in this paper we provide implementable methods for solving nondifferentiable convex optimization problems a typical method minimizes an approximate moreau yosida regularization using a quasi newton technique with inexact function and gradient values which are generated by a finite inner bundle algorithm for a bfgs bundle type method global and superlinear convergence results for the outer iteration sequence are obtained

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

backlund transformations for a discrete second painleve hierarchy we present a new general formulation of our approach to obtaining auto backlund transformations this formulation is much more straightforward and can be applied directly to discrete equations as an example we give a derivation of auto backlund transformations for a discrete second painleve hierarchy to the best of our knowledge this is the first time such results have been given for a discrete hierarchy

bilinear discrete painleve ii and its particular solutions by analogy to the continuous painleve ii equation we present particular solutions of the discrete painleve ii d pii equation these solutions are of a rational and special function airy type our analysis is based on the bilinear formalism that allows us to obtain the tau function for d pii two different forms of bilinear d pii are obtained and we show that they can be related by a simple gauge transformation