numerical discretization based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering physics and biomedical sciences parameter estimation of differential equation models is a challenging problem because of high computational cost and high dimensional parameter space in this article we propose a novel class of methods for estimating parameters in ordinary differential equation ode models which is motivated by hiv dynamics modeling the new methods exploit the form of numerical discretization algorithms for an ode solver to formulate estimating equations first a penalized spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ode solver to obtain the ode parameter estimates via a regression approach we consider three different order of discretization methods euler s method trapezoidal rule and runge kutta method a higher order numerical algorithm reduces numerical error in the approximation of the derivative which produces a more accurate estimate but its computational cost is higher to balance the computational cost and estimation accuracy we demonstrate via simulation studies that the trapezoidal discretization based estimate is the best and is recommended for practical use the asymptotic properties for the proposed numerical discretization based estimators are established comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade off between computational cost and estimation accuracy we apply the proposed methods t an hiv study to further illustrate the usefulness of the proposed approaches

generalized ordinary differential equation models existing estimation methods for ordinary differential equation ode models are not applicable to discrete data the generalized ode gode model is therefore proposed and investigated for the first time we develop the likelihood based parameter estimation and inference methods for gode models we propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving odes the simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ode model estimation approach and the extended smoothing based esb method supplementary materials for this article are available online

existence of non spurious solutions to discrete dirichlet problems with lower and upper solutions abstract this paper investigates the solvability of discrete dirichlet boundary value problems by the lower and upper solution method here the second order difference equation with a nonlinear right hand side f is studied and f t u v can have a superlinear growth both in u and in v moreover the growth conditions on f are one sided we compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution it is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations

topological methods for some boundary value problems we establish existence of solutions for a finite difference approximation to y f x y y on 0 1 subject to nonlinear two point sturm liouville boundary conditions of the form g i y i y i 0 i 0 1 assuming s satisfies one sided growth bounds with respect to y c 2001 elsevier science ltd all rights reserved

modeling price dynamics in ebay auctions using differential equations empirical research of online auctions has grown dramatically in recent years studies using publicly available bid data from such websites as ebay com have found many divergences of bidding behavior and auction outcomes compared with ordinary offline auctions and auction theory among the main differences between online and offline auctions are the former s longer duration anonymity of bidders and sellers and low barriers of entry all of these factors lead to dynamics in the bid arrival and price process that change throughout the auction in this work we examine the price process in a large and diverse set of ebay auctions for both low and high valued items in terms of item auction bidder and seller characteristics we propose a family of differential equation models that captures online auction dynamics in particular we show that a second order linear differential equation well approximates the dynamics that occur in our diverse set of auctions we also introduce a multiple comparisons test fo

infinitely many solutions for a class of discrete non linear boundary value problems the existence of infinitely many solutions for a discrete non linear dirichlet problem involving the p laplacian under appropriate oscillating behaviours of the non linear term is established the approach is based on the critical point theory

systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations we establish existence results for solutions to boundary value problems for systems of second order difference equations associated with systems of second order ordinary differential equations subject to nonlinear boundary conditions

bifurcation from infinity and multiple solutions for some discrete sturm liouville problems in this paper we study nonlinear discrete boundary value problems of the form d p t 1 dy t 1 q t y t l email protected y t f t y t h t email protected t a 1 1y a a 1 email protected a 0 a 2 1y b 1 a 2 email protected b 1 0 where l k is an eigenvalue of the associated linear problem l is a parameter and f satisfies the sublinear growth condition f t s a s a b email protected a 1 b 1 email protected r for some email protected email protected 1 and a email protected 0 we study the existence and multiplicity of solutions for the above mentioned problems by establishing some a priori bounds together with leray schauder continuation and bifurcation arguments

nonparametric spline regression with prior information summary by using prior information about the regression curve we propose new nonparametric regression estimates we incorporate two types of information first we suppose that the regression curve is similar in shape to a family of parametric curves characterized as the solution to a linear differential equation the regression curve is estimated by penalized least squares with the differential operator defining the smoothness penalty we discuss in particular growth and decay curves and take a time transformation to obtain a tractable solution the second type of prior information is linear equality constraints we estimate unknown parameters by generalized cross validation or maximum likelihood and obtain efficient 0 n algorithms to compute the estimate of the regression curve and the cross validation and maximum likelihood criterion functions

unbounded perturbations of nonlinear discrete periodic problem at resonance abstract in this paper we study the existence of solutions of nonlinear discrete boundary value problems 2 u t 1 k u t g t u t h t u 0 u t u 1 u t 1 where t 1 t h t r k is the k th eigenvalue of the linear problem 2 u t 1 u t 0 u 0 u t u 1 u t 1 g n r r satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u r

sieve estimation of constant and time varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error this article considers estimation of constant and time varying coefficients in nonlinear ordinary differential equation ode models where analytic closed form solutions are not available the numerical solution based nonlinear least squares nls estimator is investigated in this study a numerical algorithm such as the runge kutta method is used to approximate the ode solution the asymptotic properties are established for the proposed estimators considering both numerical error and measurement error the b spline is used to approximate the time varying coefficients and the corresponding asymptotic theories in this case are investigated under the framework of the sieve approach our results show that if the maximum step size of the p order numerical algorithm goes to zero at a rate faster than n 1 p 4 the numerical error is negligible compared to the measurement error this result provides a theoretical guidance in selection of the step size for numerical evaluations of odes moreover we have shown that the numerical solution based nls estimator and the sieve nls estimator are strongly consistent the sieve estimator of constant parameters is asymptotically normal with the same asymptotic co variance as that of the case where the true ode solution is exactly known while the estimator of the time varying parameter has the optimal convergence rate under some regularity conditions the theoretical results are also developed for the case when the step size of the ode numerical solver does not go to zero fast enough or the numerical error is comparable to the measurement error we illustrate our approach with both simulation studies and clinical data on hiv viral dynamics

existence of multiple solutions for second order discrete boundary value problems we give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two point boundary value problem yk 1 2yk yk 1 f k yk vk 0 for k 1 n 1 y0 0 yn where f is continuous and vk yk yk 1 for k 1 n in the special case f k t p f t greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and peterson and are in the spirit of our results for the continuous analogue c 2002 elsevier science ltd all rights reserved

sparse additive ordinary differential equations for dynamic gene regulatory network modeling the gene regulation network grn is a high dimensional complex system which can be represented by various mathematical or statistical models the ordinary differential equation ode model is one of the popular dynamic grn models high dimensional linear ode models have been proposed to identify grns but with a limitation of the linear regulation effect assumption in this article we propose a sparse additive ode sa ode model coupled with ode estimation methods and adaptive group least absolute shrinkage and selection operator lasso techniques to model dynamic grns that could flexibly deal with nonlinear regulation effects the asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach an application example for identifying the nonlinear dynamic grn of t cell activation is used to illustrate the usefulness of the proposed method

nonlinear difference equations investigated via critical point methods abstract the multiplicity of solutions for nonlinear difference equations involving the p laplacian is investigated the approach is based on critical point theorems in the setting of finite dimensional banach spaces

parameter estimation and variable selection for big systems of linear ordinary differential equations a matrix based approach abstractordinary differential equations odes are widely used to model the dynamic behavior of a complex system parameter estimation and variable selection for a big system with linear odes are very challenging due to the need of nonlinear optimization in an ultra high dimensional parameter space in this article we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares sls simulation studies demonstrate that the proposed matrix based sls method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ode system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector based two stage method that are currently available we applied this new method to two real datasets a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the standard poor 1500 index stock price dat

classifying non uniformly sampled vector valued curves we propose a novel means of classifying vector valued curves using functional principal components this uses cross validation to select curve components the degree of smoothing and scores associated with the best classification performance our approach is well suited to data generated by sensors of different modalities and time varying sampling frequencies experimental comparisons show this approach has significant advantages over conventional techniques using non uniformly sampled data our approach also generates novel forms of derivative analysis a widely used technique for classifying spectral functions from contaminated data

a numerical approach to nonlinear two point boundary value problems for odes in this paper we propose a numerical approach to solve some problems connected with the implementation of the newton type methods for the resolution of the nonlinear system of equations related to the discretization of a nonlinear two point bvps for odes with mixed linear boundary conditions by using the finite difference method

statistical regularization of inverse problems in experimental sciences we often need to solve inverse problems that is we want to obtain information about the internal structure of a physical system from indirect noisy observations often the problem is not whether a solution exists on the contrary there are too many solutions that fit the data to a chosen tolerance level the goal is to use prior information to determine a physically meaningful solution here we present some of the basic questions that arise we describe methods that can be used to find inversion estimates as well as ways to assess their performance

nontrivial solutions of boundary value problems of second order difference equations we employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second order difference equations

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

functional components of variation in handwriting abstract functional data analysis techniques are used to analyze a sample of handwriting in chinese the goals are a to identify a differential equation that satisfactorily models the data s dynamics and b to use the model to classify handwriting samples taken from differential individuals after preliminary smoothing and registration steps a second order linear differential equation for which the forcing function is small is found to provide a good reconstruction of the original script records the equation is also able to capture a substantial amount of the variation in the scripts across replication the cross validated classification process is 100 effective for the samples analyzed

parameter estimation for differential equation models using a framework of measurement error in regression models differential equation de models are widely used in many scientific fields including engineering physics and biomedical sciences the so called forward problem the problem of simulations and predictions of state variables for given parameter values in the de models has been extensively studied by mathematicians physicists engineers and other scientists however the inverse problem the problem of parameter estimation based on the measurements of output variables has not been well explored using modern statistical methods although some least squares based approaches have been proposed and studied in this article we propose parameter estimation methods for ordinary differential equation ode models based on the local smoothing approach and a pseudo least squares psls principle under a framework of measurement error in regression models the asymptotic properties of the proposed psls estimator are established we also compare the psls method to the corresponding simulation extrapolation

on sturm liouville boundary value problems for second order nonlinear functional finite difference equations motivated by the interesting paper i karaca discrete third order three point boundary value problem j comput appl math 205 2007 458 468 this paper is concerned with a class of boundary value problems for second order functional difference equations sufficient conditions for the existence of at least one solution of a sturm liouville boundary value problem for second order nonlinear functional difference equations are established we allow f to be at most linear superlinear or sublinear in obtained results

existence and multiple solutions for a second order difference boundary value problem via critical point theory in this paper the critical point theory is employed to establish existence and multiple solutions for a second order difference boundary value problem

minimum state unsteady aerodynamic shape sensitivities using sensitivity of optimal solutions to problem parameters parameters involved including the aerodynamic lag roots its solution is unique and convergence of the solution process based on standard mathematical programming techniques allows accurate determination of all unknown variables an improvement of the math programming minimum state fitting is presented here and it is developed further to include sensitivity analysis for the resulting minimum state series free of the nonuniqueness and convergence issues that plagued previous attempts to obtain sensitivities of minimum state fits a variable planform double swept forward wing case is used to demonstrate performance of the fitting technique itself shape sensitivity calculations and taylor series based approximations that can be used to replace detailed full order analysis in the course of multidisciplinary design optimization of flight vehicles nomenclature a0 a1 a2 minimum state approximation matrices corresponding to aerodynamic stiffness damping and inertia respectively a 0 a 1 a 2 elements of the matrices a0 a1 a2 b reference semichord length of wing c structural viscous damping matrix c reference chord length of wing d e minimum state approximation matrices

functional data analysis of the dynamics of the monthly index of nondurable goods production functional data analysis techniques pioneered by ramsay and bernard silverman are used to analyze the dynamics of a monthly nonseasonally adjusted index of production after determining the order three of the ordinary differential equation that describes the data and estimating its coefficients the time variation of each vector of coefficients is examined in detail to determine the evolutionary dynamics of each series over a 70 year span the dynamical properties of the solution paths are also analyzed for their qualitative dynamics the resulting models are used to examine the interaction between seasonal dynamics dynamics at business cycle frequencies and long term growth

existence of multiple solutions for a second order difference equation with a parameter applying the critical point theorem we establish existence and multiple solutions for a second order difference boundary value problem and show the explicit intervals of l such that the equation has at least 2n distinct solutions

parameter estimation of partial differential equation models partial differential equation pde models are commonly used to model complex dynamic systems in applied sciences such as biology and finance the forms of these pde models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system parameters in pde models often have interesting scientific interpretations but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors most pdes used in practice have no analytic solutions and can only be solved with numerical methods currently methods for estimating pde parameters require repeatedly solving pdes numerically under thousands of candidate parameter values and thus the computational load is high in this article we propose two methods to estimate parameters in pde models a parameter cascading method and a bayesian approach in both methods the underlying dynamic process modeled with the pde model is represented via basis function

nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups abstract in this paper we study the existence of nontrivial solutions for a class of second order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and morse theory

aeroservoelastic aspects of wing control surface planform shape optimization equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters rational function approximations for unsteady aerodynamic forces their shape sensitivities and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined the goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy effects of structural and unsteady aerodynamic modeling errors are studied examination of approximation accuracy using alternative approximation techniques and the resulting move limits provide insight and experience on the way to realistic wing control surface shape optimization with active controls and aeroservoelastic constraints

minimum state unsteady aerodynamics for aeroservoelastic configuration shape optimization of flight vehicles the minimum state technique for creating linear time invariant state space models of unsteady aerodynamic loads leads to major savings in terms of the dimension of the resulting aeroservoelastic systems compared to alternative techniques the upfront effort of the generation of minimum state approximants however is large and involves an iterative double least squares function fitting process when the overall shape of a configuration varies during the repetitive analysis required for configuration design optimization the cost of creating new minimum state approximations becomes prohibitive this paper presents an efficient method for obtaining shape sensitivity analysis of minimum state approximants the resulting sensitivities are then used based on a single detailed analysis and sensitivity analysis at some reference design to generate approximate aeroelastic poles divergence speed and flutter speed results for configurations that are subject to considerable planform shape changes

existence of sign changing periodic solutions of second order difference equations abstract we are concerned with determining values of r for which there exist sign changing solutions of nonlinear periodic boundary value problems 2 u t 1 rf u t 0 t t u 0 u t u 1 u t 1 the proof of our main result is based upon global bifurcation techniques

sensitivity calculations for iteratively solved problems the calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated a modified finite difference procedure is presented which improves the accuracy of the calculated derivatives the procedure is demonstrated for a simple algebraic example as well as an element by element preconditioned conjugate gradient iterative solution technique applied to truss examples

principal differential analysis data reduction by differential operators functional data are observations that are either themselves functions or are naturally representable as functions when these functions can be considered smooth it is natural to use their derivatives in exploring their variation principal differential analysis pda identifies a linear differential operator l w 0 i w 1 d w m 1 d m 1 d m that comes as close as possible to annihilating a sample of functions convenient procedures for estimating the m weighting functions w j are developed the estimated differential operator l is analogous to the projection operator used as the data annihilator in principal components analysis and thus can be viewed as a type of data reduction or exploration tool the corresponding linear differential equation may also have a useful substantive interpretation modelling and regularization features can also be incorporated into pda

existence of multiple solutions for second order boundary value problems we give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y f x y y 0 x epsilon 0 1 y 0 0 y 1 in the special case f x y y f y greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and of lakshmikantham et al

solvability of discrete neumann boundary value problems in this article we gain solvability to a nonlinear second order difference equation with discrete neumann boundary conditions our methods involve new inequalities on the right hand side of the difference equation and schaefer s theorem in the finite dimensional space setting