numerical methods for systems of nonlinear parabolic equations with time delays abstract the purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays the system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations the mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system a theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained it is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero

optimal transport and skorokhod embedding the skorokhod embedding problem is to represent a given probability as the distribution of brownian motion at a chosen stopping time over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties these constructions employ a variety of techniques ranging from excursion theory to potential and pde theory and have been used in many different branches of pure and applied probability we develop a new approach to skorokhod embedding based on ideas and concepts from optimal mass transport in analogy to the celebrated article of gangbo and mccann on the geometry of optimal transport we establish a geometric characterization of skorokhod embeddings with desired optimality properties this leads to a systematic method to construct optimal embeddings it allows us for the first time to derive all known optimal skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings while previous constructions typically used particular properties of brownian motion our approach applies to all sufficiently regular markov processes

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

error estimates of the lumped mass finite element method for semilinear elliptic problems we are concerned with the semilinear elliptic problems we first investigate the l 2 error estimate for the lumped mass finite element method we then use the cascadic multigrid method to solve the corresponding discrete problem on the basis of the finite element error estimates we prove the optimality of the proposed multigrid method we also report some numerical results to support the theory

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

consistency of orthogonal series density estimators based on grouped observations in this correspondence we show that nonparametric orthogonal series estimators of probability densities retain the mean integrated square error mise consistency when observations are grouped to the points of a uniform grid prebinned this kind of grouping is typical for computer rounding errors and may also be useful in data compression before calculating estimates e g using the fft the main result shows that mise consistency holds for all square integrable densities under mild conditions on the grid step size

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

maximum entropy estimation of density and regression functions abstract this paper presents an information efficient technique to determine the functional forms of density and regression functions by maximizing entropy subject to given side conditions a flexible functional form approach is established as special cases of this approach exponential polynomial cobb douglas translog generalized cobb douglas generalized leontief fourier flexible form and minflex laurent series etc can be derived in a monte carlo experiment the maximum entropy regression method shows good performance in terms of goodness of fit boxplots and isoquants produced by the maximum entropy method are compared with those produced by other well known functions

a note on the continuity of free boundaries in finite horizon optimal stopping problems for one dimensional diffusions we provide sufficient conditions for the continuity of the free boundary in a general class of finite horizon optimal stopping problems arising for instance in finance and economics the underlying process is a strong solution of a one dimensional time homogeneous stochastic differential equation sde the proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory mild local regularity of the coefficients of the sde and smoothness of the gain function locally at the boundary are required

skorokhod embeddings minimality and non centred target distributions in this paper we consider the skorokhod embedding problem for target distributions with non zero mean in the zero mean case uniform integrability provides a natural restriction on the class of embeddings but this is no longer suitable when the target distribution is not centred instead we restrict our class of stopping times to those which are minimal and we find conditions on the stopping times which are equivalent to minimality

inexact block monotone methods for solving nonlinear elliptic problems abstract the paper deals with numerical solving of semilinear elliptic problems based on the method of upper and lower solutions inexact block monotone iterative methods are constructed where monotone linear systems are solved by the block jacobi or block gauss seidel methods only approximately the inexact block monotone methods combine the quadratic monotone iterative method at outer iterations and the block iterative methods at inner iterations and possess global monotone convergence results of numerical experiments implemented in the framework of an inexact newton method are presented where iteration counts and cpu times of the inexact block monotone methods are compared with the block monotone iterative methods whose convergence rate is linear

embedding laws in diffusions by functions of time we present a constructive probabilistic proof of the fact that if b bt t 0 is standard brownian motion started at 0 and is a given probability measure on r such that 0 0 then there exists a unique left continuous increasing function b 0 r and a unique left continuous decreasing function c 0 r such that b stopped at b c inf t 0 bt b t or bt c t has the law the method of proof relies upon weak convergence arguments arising from helly s selection theorem and makes use of the levy metric which appears to be novel in the context of embedding theorems we show that b c is minimal in the sense of monroe so that the stopped process b b c bt b c t 0 satisfies natural uniform integrability conditions expressed in terms of we also show that b c has the smallest truncated expectation among all stopping times that embed into b the main results extend from standard brownian motion to all recurrent diffusion processes on the real line

strong and weak discrete maximum principles for matrices associated with elliptic problems abstract we consider the strong and weak discrete maximum principles for matrix equations associated with the elliptic problems we also give some examples and an application to illustrate the usefulness of the discrete maximum principles

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

monotone finite difference schemes for nonlinear systems with mixed quasi monotonicity monotone finite difference schemes are proposed for nonlinear systems with mixed quasi monotonicity two monotone iteration processes for the corresponding discrete problems are presented which converge monotonically to the quasi solutions of the discrete problems the limits are the exact solutions under some conditions a monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed the numerical results coincide with theoretical analysis

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

orthogonal bases approach for comparing nonnormal continuous distributions summary we present an orthonormal bases approach for detecting general differences among continuous distributions an unknown density function is represented by a finite vector of its estimated fourier coefficients with respect to a suitable orthonormal basis for a wide class of orthonormal bases we establish asymptotic normality of the vector of estimated fourier coefficients and propose an unbiased and consistent estimator of its asymptotic covariance matrix fourier coefficients are modelled as functions of fixed and possibly random effects this approach allows simultaneous detection of distributional differences attributable to various factors in clustered and correlated data with sufficiently large numbers of observations per each cluster with the same fixed and random effects realisations this work was motivated by multi level clustered non gaussian datasets from genetic studies

numerical solutions for some coupled systems of nonlinear boundary value problems in the well known volterra lotka model concerning two competing species with diffusion the densities of the species are governed by a coupled system of reaction diffusion equations the aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems this iterative scheme is based on the method of upper lower solutions which leads to two monotone sequences from some uncoupled linear systems it is shown that each of the two sequences converges to a nontrivial solution of the discrete equations the model under consideration may have one two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration numerical results are given for these solutions under both the dirichlet boundary condition and the mixed type boundary condition

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

estimating the density of unemployment duration based on contaminated samples or small samples in estimating a density function for the duration of unemployment we consider two departures from what would be ideal conditions if the so called digit preference effect produces local distortion in observed samples we can apply a maximum entropy density estimation method to establish the functional form of the density we maximize entropy subject to moment restrictions the global shape of the density is determined by the lower ordered sample moments which are not affected much by the digit preference effect as a by product of this method we can establish the local transition structure of the digit preference effect as a second case of departure from an ideal condition we consider coarse sample observations where unemployment duration was observed only for 4 10 14 26 and 52 weeks once the unemployment duration density is derived quintile behavior over time the lorenz curve and the gini coefficient for the distribution of unemployment duration can be obtained finally we discuss the ramifications of only focusing on the headcount ratio of unemployment when other information is available

the selection of terms in an orthogonal series density estimator abstract we show that kronmal and tarter s well known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases these difficulties arise when the underlying density has a nonmonotone sequence of fourier coefficients as is likely to be the case with sharply peaked or multimodal distributions we suggest a way of overcoming these shortcomings

the two sided stefan problem with a spatially dependent latent heat we prove existence and uniqueness of solutions to a problem which generalizes the two sided stefan problem the initial temperature distribution and variable latent heat may be given by positive measures rather than point functions and the free boundaries which result are essentially arbitrary increasing functions which need not exhibit any degree of smoothness in general nevertheless the solutions are classical in the sense that all derivatives and boundary values have the classical interpretation we also study connections with the skorohod embedding problem of probability theory and with a general class of optimal stoppmg problems

block monotone iterations for numerical solutions of fourth order nonlinear elliptic boundary value problems this paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth order elliptic boundary value problems in a two dimensional domain the boundary value problem is discretized by the finite difference method and two iterative processes called block jacobi and block gauss seidel monotone iterations are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration it is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi monotone nondecreasing a sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system similar results are obtained for quasi monotone nonincreasing functions an analytical comparison relation between the block jacobi and block gauss seidel monotone iterations is obtained it is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically