optimal approximation of sparse hessians and its equivalence to a graph coloring problem we consider the problem of approximating the hessian matrix of a smooth non linear function using a minimum number of gradient evaluations particularly in the case that the hessian has a known fixed sparsity pattern we study the class of direct methods for this problem and propose two new ways of classifying direct methods examples are given that show the relationships among optimal methods from each class the problem of finding a non overlapping direct cover is shown to be equivalent to a generalized graph coloring problem the distance 2 graph coloring problem a theorem is proved showing that the general distance k graph coloring problem is np complete for all fixedk 2 and hence that the optimal non overlapping direct cover problem is also np complete some worst case bounds on the performance of a simple coloring heuristic are given an appendix proves a well known folklore result which gives lower bounds on the number of gradient evaluations needed in any possible approximation method

population models with diffusion strong allee effect and nonlinear boundary conditions abstract we consider a population model with diffusion a strong allee effect per capita growth function and constant yield harvesting in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain nonlinear boundary condition we establish our existence results by the method of sub supersolutions

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

a correction for a paper by j smoller and a wasserman abstract in j differential equations 39 1981 269 290 j smoller and a wasserman presented some rich ideas and results on the bifurcation of steady state solutions of a reaction diffusion equation in one space variable however the proof of their theorem 2 1 is incorrect the case a 0 and to correct it we find one additional hypothesis is necessary

bifurcation studies in reaction diffusion ii and estimated the critical lengths of the domain at which bifurcation occurs in the cases b 0 a and 1 also we were able to characterise some of the non constant steady state solutions which appear beyond the points of instability by a steady state solution of the form u x

a scalable parallel algorithm for incomplete factor preconditioning we describe a parallel algorithm for computing incomplete factor ilu preconditioners the algorithm attains a high degree of parallelism through graph partitioning and a two level ordering strategy both the subdomains and the nodes within each subdomain are ordered to preserve concurrency we show through an algorithmic analysis and through computational results that this algorithm is scalable experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors the resulting preconditioned krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors

particle filters for partially observed diffusions in this paper we introduce novel particle filters for a class of partially observed continuous time dynamic models where the signal is given by a multivariate diffusion process we consider a variety of observation schemes including diffusion observed with error observation of a subset of the components of the multivariate diffusion and arrival times of a poisson process whose intensity is a known function of the diffusion cox process unlike currently available methods our particle filters do not require approximations of the transition and or the observation density using time discretisations instead they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in beskos et al 2006 in particular we introduce the generalised poisson estimator which generalises the poisson estimator of beskos et al 2006 thus our filters avoid the systematic biases caused by time discretisations and they have significant computational advantages over alternative continuous time filters these advantages are supported theoretically by a central limit theorem

rigorous analysis and estimates of s shaped bifurcation curves in a combustion problem with general arrhenius reaction rate laws we study the bifurcation of solutions of the combustion problem with general arrhenius reaction rate laws d 2 u d x 2 1 e u m e u 1 e u 0 1 x 1 u 1 u 1 0 for this two point boundary value problem we give a bifurcation analysis in the physically important ranges 0 m 1 and m 0 it is proved that the bifurcation curve is s shaped in certain parameter ranges and a lower bound is given for the frank kamenetskii transition value

quenching for solutions of a plasma type equation here 0 l the so called slow diffusion case the analysis will be carried out in the manner of 5 in which levine gave a complete qualitative study of problem i with m 1 the plan of our paper is as follows in the next section we present the comparison theorem and local existence of solutions for i in the third section we characterize the set of stationary solutions of i finally in the last section we establish stability and quenching results for problem i

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

series expansion approximations of brownian motion for non linear kalman filtering of diffusion processes in this paper we describe a novel application of sigma point methods to continuous discrete filtering the nonlinear continuous discrete filtering problem is often computationally intractable to solve assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractable probability distributions filters such as these are usually decompose the problem into two sub problems the first of these is a prediction step in which one uses the known dynamics of the signal to predict its state at time tk 1 given observations up to time tk in the second step one updates the prediction upon arrival of the observation at time tk 1 the aim of this paper is to describe a novel method that improves the prediction step we decompose the brownian motion driving the signal in a generalised fourier series which is truncated after a number of terms this approximation to brownian motion can be described using a relatively small number of fourier coefficients and allows us to compute statistics of the filtering distribution with a single application of a sigma point method assumed density filters that exist in the literature usually rely on discretisation of the signal dynamics followed by iterated application of a sigma point transform or a limiting case thereof iterating the transform in this manner can lead to loss of information about the filtering distribution in highly non linear settings we demonstrate that our method is better equipped to cope with such problems

exact structure of positive solutions for some p laplacian equations this paper establishes the exact multiplicities and properties of positive solutions for some second order differential equations involving p laplacian operator

particle smoothing for hidden diffusion processes adaptive path integral smoother smoothing methods are used for inference of stochastic processes given noisy observations the estimation of the marginal posterior distribution given all observations is typically a computationally intensive task we propose a novel algorithm based on path integral control theory to efficiently estimate the smoothing distribution of continuous time diffusion processes from partial observations in particular we use an adaptive importance sampling method to improve the effective sampling size of the posterior and the reliability of the estimation of the marginals this is achieved by estimating a feedback controller together with an adaptive initialization and an annealing scheme to sample efficiently from the joint smoothing distribution we compare the results with estimations obtained from the standard forward filter backward simulator ffbsi for two diffusion processes of different complexity we show that the proposed method gives more accurate estimates than the standard ffbsi

a sequential smoothing algorithm with linear computational cost in this paper we propose a new particle smoother that has a computational complexity of o n where n is the number of particles this compares favourably with the o n2 computational cost of most smoothers the new method also overcomes some degeneracy problems in existing algorithms through simulation studies we show that substantial gains in efficiency are obtained for practical amounts of computational cost it is shown both through these simulation studies and by the analysis of an athletics dataset that our new method also substantially outperforms the simple filter smoother the only other smoother with computational cost that is o n

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

particle smoothing in continuous time a fast approach via density estimation we consider the particle smoothing problem for state space models where the transition density is not available in closed form in particular for continuous time nonlinear models expressed via stochastic differential equations sdes conventional forward backward and two filter smoothers for the particle filter require a closed form transition density with the linear gaussian euler maruyama discretization usually applied to the sdes to achieve this we develop a pair of variants using kernel density approximations to relieve the dependence and in doing so enable use of faster and more accurate discretization schemes such as runge kutta in addition the new methods admit arbitrary proposal distributions providing an avenue to deal with degeneracy issues experimental results on a functional magnetic resonance imaging fmri deconvolution task demonstrate comparable accuracy and significantly improved runtime over conventional techniques

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance the abstract boundary value problem lu gu f u dom l h is considered here h is used to denote a real separable hilbert space l a closed symmetric linear operator and g a nonlinear operator assumed to be lipschitz continuous and strongly monotone in addition l is assumed to have a complete set of eigenfunctions in h and is allowed to have an infinite dimensional null space the existence of unique solutions depending continuously on f is established by a constructive approach galerkin approximations are considered and error estimates are given as an application of the main result the existence of time periodic weak solutions of the n dimensional wave equation is shown

a finite element method for computing the bifurcation function for semilinear elliptic bvps abstract the method of alternative problems can be used to show that a semilinear elliptic boundary value problem lu g x u 0 with gu x u bounded below is equivalent to a finite dimensional problem f c 0 r d c r d in the sense that their solution sets which are not necessarily singletons are in a one to one correspondence this correspondence is based on a map from low frequency to high frequency fourier components of solutions a numerical method is presented for approximating and hence also solutions of the bvp the method uses finite element approximations and avoids the use of eigenfunction expansions existence uniqueness and error estimates for the approximations of and solutions u are derived

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

estimation of sparse jacobian matrices and graph coloring blems given a mapping with a sparse jacobian matrix the problem of minimizing the number of function evaluations needed to estimate the jacobian matrix by differences is investigated this problem can be attacked as a graph coloring problem and this approach leads to very efficient algorithms the behavior of these algorithms is studied and in particular it is proved that two of the algorithms are optimal for band graphs numerical evidence is presented which indicates that these two algorithms are nearly optimal on practical problems

diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary abstract the structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined in particular a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary our existence results are established via the well known sub super solution method

comparison theorems for radial solutions of semilinear elliptic equations on obtient des resultats sur les solutions radiales des equations elliptiques de la forme u f u 0

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

a quaternion widely linear model for nonlinear gaussian estimation this paper deals with the nonlinear minimum mean squared error estimation problem by using a quaternion widely linear model on the basis of the information supplied by a gaussian signal and its square a quaternion observation process is defined and then by applying a widely linear processing an optimal estimator for the continuous time setting is provided the special structure of the estimator proves its superiority over the complex valued widely linear solution the continuous discrete version of the problem is also studied where the solution takes the form of a suboptimal estimator useful in practical applications in addition the particular case of signal plus noise is considered in which the suboptimal solution and its associated error can be implemented through an iterative algorithm two numerical simulation examples are presented showing the advantages of the proposed approach

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

nonlinear gaussian smoothers with colored measurement noise this paper is concerned with the gaussian approximation ga smoothing estimation for the nonlinear system with the colored measurement noise modeled as an autoregressive process firstly based on measurement differencing scheme designing the ga smoothers with the colored measurement noise is transformed into deriving the ga ones with delayed state secondly the novel fixed interval fixed point and fixed lag ga smoothers are proposed via the recursive operation of analytical computation and nonlinear integrals as the general and unifying frameworks they are applicable for both linear and nonlinear systems by setting the noise correlation parameter as zero they can automatically reduce to the standard ga smoothers with uncorrelated white noises many implementations of the ga frameworks can be developed through utilizing different numerical technologies for computing such nonlinear integrals e g the cubature rule based cubature kalman smoothers ckss with the colored measurement noise finally the superior performance in estimation accuracy and computation efficiency of the proposed smoothing methods is demonstrated with a multi sensor target tracking example

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

a numerical study of spontaneous bifurcation abstract in the study of nonlinear boundary value problems it has been observed that bifurcations may take place which are not local to the origin in function space in such problems the analysis of the bifurcation and the computation of solutions is complicated by the nonlocal nature of the bifurcation and the multiplicity of solutions in this work a model problem exhibiting this phenomena which has been of recent interest is numerically studied the method of alternative problems is employed to provide a complete numerical description of the bifurcation occurring in the model problem

representation and asymptotic formulas for 1 dimensional linearized eigenvalue problems with dirichlet boundary condition abstract in this article nontrivial solutions to a nonlinear dirichlet boundary value problem with a small parameter and the corresponding linearized eigenvalue problems are investigated in a typical case of nonlinearity asymptotic formulas of all eigenvalues and asymptotic profiles of all eigenfunctions are proved proofs are based on the method of representation equation and asymptotic results in the corresponding neumann problem obtained in the recent papers by the author and s yotsutani

bifurcation and stability of stationary solutions of the fitz hugh nagumo equations the fitz hugh nagumo equations have been of some interest to both mathematicians and theoretical biologists for several years the reason for this stems from the fact that they can be considered as a simpler model for the celebrated hodgkin huxley equations in that they exhibit many of the features of this latter system indeed mathematicians have studied them because their structure is different from the usually encountered equations in physics and they therefore admit solutions with less familiar properties homoclinic travelling waves threshold effects etc the equations can be written as

a singularity theory approach to a semilinear boundary value problem where a b 3 are real parameters with 0 a b 1 this problem arises in population genetics with the unknown function u corresponding to the relative frequency of a gene the parameter a is given by a combination of the death rates of the various genotypes involved b gives the frequency of the gene on the boundary of the region and 1 is the reciprocal of the diffusion rate of the gene see aronson and weinberger 1 for further details solutions of 1 1 correspond to possible steady states of the gene populations we are interested mainly in how the bifurcation diagram changes as a and b vary and we investigate this question by exploiting singularity theory for bifurcation problems as developed by golubitsky and schaeffer c51 equation 1 1 is a simple example of a semilinear elliptic boundary value problem and a standard method of investigating the bifurcation of

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

semi widely simulation and estimation of continuous time bbc eta proper quaternion random signals the repercussions of quaternion bbc eta properness on the continuous time simulation and estimation problems are studied as a first step a semi widely linear quaternion karhunen loeve expansion is derived then this series representation is used in both problems to give the solutions the simulation technique proposed is useful for transformations of nonstationary gaussian quaternion signals the estimation problem is also analyzed in a continuous discrete setting and by considering two different formulations the cambanis formulation and the signal plus noise model initially the estimators are developed on the basis of a semi widely linear processing afterwards the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi widely nonlinear processing in this last scenario we assume gaussian observations and unknown probability distributions of the signal of interest

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

new acyclic and star coloring algorithms with application to computing hessians acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited we present an algorithmic paradigm for finding heuristic solutions for these two np hard problems the underlying common technique is the exploitation of the structure of two colored induced subgraphs for a graph g on n vertices and m edges the time complexity of our star coloring algorithm is o n overline d 2 where overline d k a generalization of vertex degree denotes the average number of distinct paths of length at most k edges starting at a vertex in g the time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of ackermann s function the space complexity of both algorithms is o m to the best of our knowledge our work is the first practical algorithm for the acyclic coloring problem for the star coloring problem our algorithm uses fewer colors and is considerably faster than a previously known o n overline d 3 time algorithm computational results from experiments on various large size test graphs demonstrate that the algorithms are fast and produce highly effective solutions the use of these algorithms in hessian computation is expected to reduce overall runtime drastically

distributed memory parallel algorithms for distance 2 coloring and related problems in derivative computation the distance 2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than 2 from each other its applications include derivative computation in numerical optimization and channel assignment in radio networks we present efficient distributed memory parallel heuristic algorithms for this np hard problem as well as for two related problems used in the computation of jacobians and hessians parallel speedup is achieved through graph partitioning speculative iterative coloring and a bulk synchronous parallel like organization of parallel computation results from experiments conducted on a pc cluster employing up to 96 processors and using large size real world as well as synthetically generated test graphs show that the algorithms are scalable in terms of quality of solution the algorithms perform remarkably well the numbers of colors used by the parallel algorithms are observed to be very close to the numbers used by their sequential counterparts which in turn are quite often near optimal moreover the experimental results show that the parallel distance 2 coloring algorithm compares favorably with the alternative approach of solving the distance 2 coloring problem on a graph mathcal g by first constructing the square graph mathcal g 2 and then applying a parallel distance 1 coloring algorithm on mathcal g 2 implementations of the algorithms are made available via the zoltan toolkit

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

estimation of sparse hessian matrices and graph coloring problems large scale optimization problems often require an approximation to the hessian matrix if the hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem the problem of estimating hessian matrices by diferences can be phrased as follows given the sparsity structure of a symmetric matrix a obtain vectors d 1 d 2 ldots d p such that ad 1 ad 2 ldots ad p determine a uniquely with p as small as possible we approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation the complexity of the problem is analyzed and efficient practical heuristic procedures are developed numerical results illustrate the differences between the various approaches

blind deconvolution via sequential imputations abstract the sequential imputation procedure is applied to adaptively and sequentially reconstruct discrete input signals that are blurred by an unknown linear moving average channel and contaminated by additive gaussian noises a problem known as blind deconvolution in digital communication a rejuvenation procedure for improving the efficiency of sequential imputation is introduced and theoretically justified the proposed method does not require the channel to be nonminimum phase and can be used in real time signal restoration two simulated systems are studied to illustrate the proposed method our result shows that the ideas of multiple imputations and flexible simulation techniques are as powerful in engineering as in survey sampling

low dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem abstract a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered it is shown that for some values of the problem parameters there is only the zero solution while for other values of these parameters two nontrivial solutions also appear these nontrivial solutions do not lie in small neighborhoods of the zero function so that the bifurcation analysis is not local to the origin the method of reduction to alternative finite dimensional problems is applied to give a numerical description of the bifurcation and in particular to describe a surface of bifurcating solutions in addition the solution set of the 1 dimensional galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem

a self organizing state space model a self organizing filter and smoother for the general nonlinear non gaussian state space model is proposed an expanded state space model is defined by augmenting the state vector with the unknown parameters of the original state space model the state of the augmented state space model and hence the state and the parameters of the original state space model are estimated simultaneously by either a non gaussian filter smoother or a monte carlo filter smoother in contrast to maximum likelihood estimation of model parameters in ordinary state space modeling for which the recursive filter computation has to be done many times model parameter estimation in the proposed self organizing filter smoother is achieved with only two passes of the recursive filter and smoother operations examples such as automatic tuning of dispersion and the shape parameters adaptation to changes of the amplitude of a signal in seismic data state estimation for a nonlinear state space model with unknown parameters and seasonal adjustment with a nonlinear model with changing variance parameters are shown to exemplify the usefulness of the proposed method

verification of bifurcation diagrams for polynomial like equations the results of our recent paper p korman y li t ouyang computing the location and the direction of bifurcation math res lett 12 2005 933 944 appear to be sufficient to justify computer generated bifurcation diagram for any autonomous two point dirichlet problem here we apply our results to polynomial like nonlinearities

bifurcation diagrams of population models with nonlinear diffusion we develop analytical and numerical tools for the equilibrium solutions of a class of reaction diffusion models with nonlinear diffusion rates such equations arise from population biology and material sciences we obtain global bifurcation diagrams for various nonlinear diffusion functions and several growth rate functions

what color is your jacobian graph coloring for computing derivatives graph coloring has been employed since the 1980s to efficiently compute sparse jacobian and hessian matrices using either finite differences or automatic differentiation several coloring problems occur in this context depending on whether the matrix is a jacobian or a hessian and on the specifics of the computational techniques employed we consider eight variant vertex coloring problems here this article begins with a gentle introduction to the problem of computing a sparse jacobian followed by an overview of the historical development of the research area then we present a unifying framework for the graph models of the variant matrix estimation problems the framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance 2 coloring an appropriate graph representation the unified framework helps integrate earlier work and leads to fresh insights enables the design of more efficient algorithms for many problems leads to new algorithms for others and eases the task of building graph models for new problems we report computational results on two of the coloring problems to support our claims most of the methods for these problems treat a column or a row of a matrix as an atomic entity and partition the columns or rows or both a brief review of methods that do not fit these criteria is provided we also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1