estimation and inference for a spline enhanced population pharmacokinetic model summary this article is motivated by an application where subjects were dosed three times with the same drug and the drug concentration profiles appeared to be the lowest after the third dose one possible explanation is that the pharmacokinetic pk parameters vary over time therefore we consider population pk models with time varying pk parameters these time varying pk parameters are modeled by natural cubic spline functions in the ordinary differential equations mean parameters variance components and smoothing parameters are jointly estimated by maximizing the double penalized log likelihood mean functions and their derivatives are obtained by the numerical solution of ordinary differential equations the interpretation of pk parameters in the model and its flexibility are discussed the proposed methods are illustrated by application to the data that motivated this article the model s performance is evaluated through simulation

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

computer model calibration using high dimensional output this work focuses on combining observations from field experiments with detailed computer simulations of a physical process to carry out statistical inference of particular interest here is determining uncertainty in resulting predictions this typically involves calibration of parameters in the computer simulator as well as accounting for inadequate physics in the simulator the problem is complicated by the fact that simulation code is sufficiently demanding that only a limited number of simulations can be carried out we consider applications in characterizing material properties for which the field data and the simulator output are highly multivariate for example the experimental data and simulation output may be an image or may describe the shape of a physical object we make use of the basic framework of kennedy and o hagan however the size and multivariate nature of the data lead to computational challenges in implementing the framework to overcome these challenges we make use of basis repre

a weak galerkin finite element method with polynomial reduction the weak galerkin wg is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions in the implementation the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom the accuracy and the computational complexity of the corresponding wg scheme is significantly impacted by the selection of such polynomials this paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation for illustrative purpose the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction consequently a new weak galerkin finite element method is proposed and analyzed error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper presents some numerical results to demonstrate the power of the wg method in dealing with finite element partitions with arbitrary polygons in 2d or polyhedra in 3d the numerical examples include various finite element partitions such as triangular mesh quadrilateral mesh honeycomb mesh in 2d and mesh with deformed cubes in 3d

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

a robust wg finite element method for convection diffusion reaction equations this paper proposes and analyzes a weak galerkin wg finite element method for 2 and 3 dimensional convection diffusion reaction problems on conforming or nonconforming polygon polyhedral meshes the wg method uses piecewise polynomial approximations of degrees k k 0 for both the scalar function and its trace on the inter element boundaries we show that the method is robust in the sense that the derived a priori error estimates is uniform with respect to the coefficients for sufficient smooth true solutions numerical experiments confirm the theoretical results

a population pharmacokinetic model with time dependent covariates measured with errors we propose a population pharmacokinetic pk model with time dependent covariates measured with errors this model is used to model s oxybutynin s kinetics following an oral administration of ditropan and allows the distribution rate to depend on time dependent covariates blood pressure and heart rate which are measured with errors we propose two two step estimation methods the second order two step method with numerical solutions of differential equations 2ordernd and the second order two step method with closed form approximate solutions of differential equations 2orderad the proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear mixed model at the second step we apply the proposed methods to the analysis of the ditropan data and evaluate their performance using a simulation study our results show that the 2ordernd method performs well while the 2orderad method can yield pk parameter estimators that are subject to considerable biases

numerical methods for optimum experimental design in dae systems subject of this paper is the design of optimal experiments for chemical processes described by nonlinear dae models the optimization aims at maximizing the statistical quality of a parameter estimate from experimental data this leads to optimal control problems with an unusual and intricate objective function which depends implicitly on first derivatives of the solution of the underlying dae we treat these problems by the direct approach and solve them using a structured sqp method the required first and second derivatives of the solution of the dae are computed very efficiently by a special coupling of the techniques of internal numerical differentiation and automatic differentiation the performance of our approach is demonstrated for an application to chemical reaction kinetics

local extremes runs strings and multiresolution the paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes two methods the run method and the taut string multiresolution method are introduced and analyzed on standard test beds it is shown that the number and locations of local extreme values are consistently estimated rates of convergence are proved for both methods the run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers the rate of convergence of the taut string multiresolution method is almost optimal the method is extremely sensitive and can detect very low power peaks section 1 contains an introduction with special reference to the number of local extreme values the run method is described in section 2 and the taut string multiresolution method in section 3 low power peaks are considered in section 4 section 5 contains a comparison with other methods and section 6 a short conclusion the proofs are given in section 7 and the taut string algorithm is described in the appendix

gradient free descent shadowing and state estimation using limited derivative information shadowing trajectories can play an important role in assessing the reliability of forecasting models they can also play an important role in providing state estimates for ensemble forecasts gradient descent methods provide one approach for obtaining shadowing trajectories which have been shown to have many useful properties there remains the important question whether shadowing trajectories can be found in very high dimensional systems like weather and climate models the principle impediment is the need to compute the derivative or adjoint of the system dynamics in this paper we investigate gradient descent methods that use limited derivative information we demonstrate the methods with an application to a moderately high dimensional system using no derivative information at all

estimating varying coefficients for partial differential equation models partial differential equations pdes are used to model complex dynamical systems in multiple dimensions and their parameters often have important scientific interpretations in some applications pde parameters are not constant but can change depending on the values of covariates a feature that we call varying coefficients we propose a parameter cascading method to estimate varying coefficients in pde models from noisy data our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed the performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a pde model arising from lidar data

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

entire solutions in bistable reaction diffusion equations with nonlocal delayed nonlinearity this paper is concerned with entire solutions for bistable reaction diffusion equations with nonlocal delay in one dimensional spatial domain here the entire solutions are defined in the whole space and for all time t r assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x axis and annihilating at a finite time furthermore we show that such an entire solution is unique up to space time translations and is liapunov stable a key idea is to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions in order to illustrate our main results two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are considered

a weak galerkin finite element scheme with boundary continuity for second order elliptic problems abstract a new weak galerkin wg finite element method for solving the second order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed the wg method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces different combination gives rise to different weak galerkin finite element methods which makes wg methods highly flexible and efficient in practical computation this paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme yet without losing the accuracy of the numerical approximation error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper also presents some numerical experiments to demonstrate the power of the wg method the numerical results show that the wg method achieves the optimal convergence order with a relatively low computational cost which reveals a great promise of the flexibility and accuracy of the wg method

nonplanar traveling fronts in reaction diffusion equations with combustion and degenerate fisher kpp nonlinearities abstract this paper is concerned with nonplanar traveling fronts in reaction diffusion equations with combustion nonlinearity and degenerate fisher kpp nonlinearity our study contains two parts in the first part we establish the existence of traveling fronts of pyramidal shape in r 3 and in the second part we establish the existence and stability of v shaped traveling fronts in r 2

on the selection of ordinary differential equation models with application to predator prey dynamical models summary we consider model selection and estimation in a context where there are competing ordinary differential equation ode models and all the models are special cases of a full model we propose a computationally inexpensive approach that employs statistical estimation of the full model followed by a combination of a least squares approximation lsa and the adaptive lasso we show the resulting method here called the lsa method to be an asymptotically oracle model selection method the finite sample performance of the proposed lsa method is investigated with monte carlo simulations in which we examine the percentage of selecting true ode models the efficiency of the parameter estimation compared to simply using the full and true models and coverage probabilities of the estimated confidence intervals for ode parameters all of which have satisfactory performances our method is also demonstrated by selecting the best predator prey ode to model a lynx and hare population dynamical system among some well known and biologically interpretable ode models

periodic traveling curved fronts in reaction diffusion equation with bistable time periodic nonlinearity abstract this paper is concerned with the existence and stability of periodic traveling curved fronts for reaction diffusion equations with time periodic bistable nonlinearity in two dimensional space by constructing supersolution and subsolution we prove the existence of periodic traveling wave fronts furthermore we show that the front is globally stable

the uniqueness and asymptotic stability of pyramidal traveling fronts in the allen cahn equations abstract this paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three dimensional whole space for a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity for this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces moreover we characterize the pyramidal traveling front in another way that is we write it as a combination of two dimensional v form waves on the edges this characterization also uniquely determines a pyramidal traveling front

monte carlo smoothing for nonlinear time series we develop methods for performing smoothing computations in general state space models the methods rely on a particle representation of the filtering distributions and their evolution through time using sequential importance sampling and resampling ideas in particular novel techniques are presented for generation of sample realizations of historical state sequences this is carried out in a forward filtering backward smoothing procedure that can be viewed as the nonlinear non gaussian counterpart of standard kalman filter based simulation smoothers in the linear gaussian case convergence in the mean squared error sense of the smoothed trajectories is proved showing the validity of our proposed method the methods are tested in a substantial application for the processing of speech signals represented by a time varying autoregression and parameterized in terms of time varying partial correlation coefficients comparing the results of our algorithm with those from a simple smoother based on the filte

physiological pharmacokinetic analysis using population modeling and informative prior distributions abstract we describe a general approach using bayesian analysis for the estimation of parameters in physiological pharmacokinetic models the chief statistical difficulty in estimation with these models is that any physiological model that is even approximately realistic will have a large number of parameters often comparable to the number of observations in a typical pharmacokinetic experiment e g 28 measurements and 15 parameters for each subject in addition the parameters are generally poorly identified akin to the well known ill conditioned problem of estimating a mixture of declining exponentials our modeling includes a hierarchical population modeling which allows partial pooling of information among different experimental subjects b a pharmacokinetic model including compartments for well perfused tissues poorly perfused tissues fat and the liver and c informative prior distributions for population parameters which is possible because the parameters represent real physiological

computer model validation with functional output a key question in evaluation of computer models is does the computer model adequately represent reality a six step process for computer model validation is set out in bayarri et al technometrics 49 2007 138 154 and briefly summarized below based on comparison of computer model runs with field data of the process being modeled the methodology is particularly suited to treating the major issues associated with the validation process quantifying multiple sources of error and uncertainty in computer models combining multiple sources of information and being able to adapt to different but related scenarios two complications that frequently arise in practice are the need to deal with highly irregular functional data and the need to acknowledge and incorporate uncertainty in the inputs we develop methodology to deal with both complications a key part of the approach utilizes a wavelet representation of the functional data applies a hierarchical version of the scalar validation methodology to the wavelet coefficients and transforms back to ultimately compare computer model output with field output the generality of the methodology is only limited by the capability of a combination of computational tools and the appropriateness of decompositions of the sort wavelets employed here the methods and analyses we present are illustrated with a test bed dynamic stress analysis for a particular engineering system

adaptive m estimation in nonparametric regression a method for robust nonparametric regression is discussed a method for robust nonparametric regression is discussed we consider kernel m estimates of the regression function using huber s function and extend results of hardle and gasser to the case of random designs a practical adaptive procedure is proposed consisting of simultaneously minimising a cross validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the function this method is shown to possess a theoretical asymptotic optimality property while some simulated examples confirm that the approach is practicable

a weak galerkin finite element scheme for solving the stationary stokes equations a weak galerkin wg finite element method for solving the stationary stokes equations in two or three dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed the variational form we considered is based on two gradient operators which is different from the usual gradient divergence operators the wg method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity optimal order error estimates are established for the corresponding wg finite element solutions in various norms numerical results are presented to illustrate the theoretical analysis of the new wg finite element scheme for stokes problems

convex compact sets in rn 1 give traveling fronts of cooperation diffusion systems in rn abstract this paper studies traveling fronts to cooperation diffusion systems in r n for n 3 we consider n 2 dimensional smooth surfaces as boundaries of strictly convex compact sets in r n 1 and define an equivalence relation between them we prove that there exists a traveling front associated with a given surface and show its stability the associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation

a symmetric weak galerkin method for solving non divergence form elliptic equations abstract we reformulate the second order non divergence form elliptic equations in a symmetric variation form i e a least squares form we design a weak galerkin finite element method for this high regularity formulation the optimal order of convergence is proved numerical results verify the theory in addition numerical results compared to the existing weak galerkin method for the non symmetric form and the new methods show advantage on the accuracy and the degree of freedoms

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

sampling the posterior an approach to non gaussian data assimilation the viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a bayesian framework in the absence of model error the correct solution to the data assimilation problem is to find the posterior distribution implied by this bayesian setting methods for dealing with data assimilation should then be judged by their ability to probe this distribution in this paper we propose a range of techniques for probing the posterior distribution based around the langevin equation and we compare these new techniques with existing methods when the underlying dynamics is deterministic the posterior distribution is on the space of initial conditions leading to a sampling problem over this space when the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths by writing down a density and conditioning on observations it is possible to define a range of markov chain monte carlo mcmc methods which sample from the desired posterior distribution and thereby solve the data assimilation problem the basic building blocks for the mcmc methods that we concentrate on in this paper are langevin equations which are ergodic and whose invariant measures give the desired distribution in the case of path space sampling these are stochastic partial differential equations spdes two examples are given to show how data assimilation can be formulated in a bayesian fashion the first is weather prediction and the second is lagrangian data assimilation for oceanic velocity fields furthermore the relationship between the bayesian approach outlined here and the commonly used kalman filter based techniques prevalent in practice is discussed two simple pedagogical examples are studied to illustrate the application of bayesian sampling to data assimilation concretely finally a range of open mathematical and computational issues arising from the bayesian approach are outlined

efficient gradient computation for dynamical models data assimilation is a fundamental issue that arises across many scales in neuroscience ranging from the study of single neurons using single electrode recordings to the interaction of thousands of neurons using fmri data assimilation involves inverting a generative model that can not only explain observed data but also generate predictions typically the model is inverted or fitted using conventional tools of convex optimization that invariably extremise some functional norms minimum descriptive length variational free energy etc generally optimisation rests on evaluating the local gradients of the functional to be optimized in this paper we compare three different gradient estimation techniques that could be used for extremising any functional in time i finite differences ii forward sensitivities and a method based on iii the adjoint of the dynamical system we demonstrate that the first order gradients of a dynamical system linear or non linear can be computed most efficiently using the adjoint method this is particularly true for systems where the number of parameters is greater than the number of states for such systems integrating several sensitivity equations as required with forward sensitivities proves to be most expensive while finite difference approximations have an intermediate efficiency in the context of neuroimaging adjoint based inversion of dynamical causal models dcms can in principle enable the study of models with large numbers of nodes and parameters

asymptotic behavior and traveling wave solutions for parabolic functional differential equations on presente une generalisation de la theorie des equations kpp et de diffusion bistables non lineaires on montre que des solutions onde de propagation existent pour des equations fonctionnelles paraboliques non lineaires

estimating a predator prey dynamical model with the parameter cascades method ordinary differential equations odes are widely used in ecology to describe the dynamical behavior of systems of interacting populations however systems of odes rarely provide quantitative solutions that are close to real field observations or experimental data because natural systems are subject to environmental and demographic noise and ecologists are often uncertain about the correct parameterization in this article we introduce parameter cascades as an improved method to estimate ode parameters such that the corresponding ode solutions fit the real data well this method is based on the modified penalized smoothing with the penalty defined by odes and a generalization of profiled estimation which leads to fast estimation and good precision for ode parameters from noisy data this method is applied to a set of odes originally developed to describe an experimental predator prey system that undergoes oscillatory dynamics the new parameterization considerably improves the fit of the ode model to the experimental data sets at the same time our method reveals that important structural assumptions that underlie the original ode model are essentially correct the mathematical formulations of the two nonlinear interaction terms functional responses that link the odes in the predator prey model are validated by estimating the functional responses nonparametrically from the real data we suggest two major applications of parameter cascades to ecological modeling it can be used to estimate parameters when original data are noisy missing or when no reliable priori estimates are available it can help to validate the structural soundness of the mathematical modeling approach

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

existence and global stability of traveling curved fronts in the allen cahn equations abstract this paper is concerned with existence and stability of traveling curved fronts for the allen cahn equation in the two dimensional space by using the supersolution and the subsolution we construct a traveling curved front and show that it is the unique traveling wave solution between them our supersolution can be taken arbitrarily large which implies some global asymptotic stability for the traveling curved front

spatial spline regression models summary we describe a model for the analysis of data distributed over irregularly shaped spatial domains with complex boundaries strong concavities and interior holes adopting an approach that is typical of functional data analysis we propose a spatial spline regression model that is computationally efficient allows for spatially distributed covariate information and can impose various conditions over the boundaries of the domain accurate surface estimation is achieved by the use of piecewise linear and quadratic finite elements

travelling wave fronts in reaction diffusion systems with spatio temporal delays this paper deals with the existence of travelling wave fronts in reaction diffusion systems with spatio temporal delays our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system new iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions following this the existence of travelling wave fronts for reaction diffusion systems with spatio temporal delays is established finally we apply the main results to a single species diffusive model with spatio temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels

traveling wavefronts for delayed reaction diffusion systems via a fixed point theorem abstract by using schauder s fixed point theorem we prove some existence results for traveling wavefronts of reaction diffusion systems with quasimonotonicity reactions more precisely we reduce the existence of traveling wavefronts to the existence of an admissible pair of supersolution and subsolution which are easy to construct in practice finally to illustrate our main results we study the existence of traveling wavefronts for a delayed predator prey model with diffusion as well as the reaction diffusion system with the well known belousov zhabotinskii reaction and the obtained results improve the existing ones

linear mixed effects modeling by parameter cascading a linear mixed effects model lme is a familiar example of a multilevel parameter structure involving nuisance and structural parameters as well as parameters that essentially control the model s complexity marginalization over nuisance parameters such as the restricted maximization likelihood method has been the usual estimation strategy but it can involve onerous and complex algorithms to achieve the integrations involved parameter cascading is described as a multicriterion optimization algorithm that is relatively simple to program and leads to fast and stable computation the method is applied to lme where well developed marginalization methods are already available our results suggest that parameter cascading is at least as good as if not better than the available methods we also extend the lme model to multicurve data smoothing by introducing a basis partitioning scheme and defining roughness penalty terms for both functional fixed effect and random effects the results are substantially

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

recursive monte carlo filters algorithms and theoretical analysis recursive monte carlo filters also called particle filters are a powerful tool to perform the computations in general state space models we discuss and compare the accept reject version with the more common sampling importance resampling version of the algorithm in particular we show how auxiliary variable methods and stratification can be used in the accept reject version and we compare dierent resampling techniques in a second part we show laws of large numbers and a central limit theorem for these monte carlo filters by simple induction arguments that need only weak conditions we also show that under stronger conditions the required sample size is independent of the length of the observed series ams 2000 subject classifications primary 62m09 secondary 60g35 60j22 65c05

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

failure of maximum likelihood methods for chaotic dynamical systems the maximum likelihood method is a basic statistical technique for estimating parameters and variables and is the starting point for many more sophisticated methods like bayesian methods this paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system because there are trajectories that appear to be far more infinitely more likely than truth this failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems the reason for the failure is rather simple in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed and hence their likelihood dominates truth the residuals of these truth dominating trajectories are not consistent with the noise distribution they would typically have too small standard deviation and many outliers and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

efficient local estimation for time varying coefficients in deterministic dynamic models with applications to hiv 1 dynamics recently deterministic dynamic models have become very popular in biomedical research and other scientific areas examples include modeling human immunodeficiency virus hiv dynamics pharmacokinetic pharmacodynamic analysis tumor cell kinetics and genetic network modeling in this article we propose estimation methods for the time varying coefficients in deterministic dynamic systems that are usually described by a set of differential equations three two stage local polynomial estimators are proposed and their asymptotic normality is established an alternative approach a discretization method that is widely used in stochastic diffusion models is also investigated we show that the discretization method that uses the simple euler discretization approach for the deterministic dynamic model does not achieve the optimal convergence rate compared with the proposed two stage estimators we use monte carlo simulations to study the finite sample performance and use a real data application to hiv dynamic

the weak galerkin method for solving the incompressible brinkman flow the brinkman equations are used to describe the dynamics of fluid flows in complex porous media with the high variability in the permeability coefficients which may take extremely large or small values this paper is devoted to the numerical analysis of a family of weak galerkin wg finite element methods for solving the time dependent brinkman problems this wg method is equipped with stable finite elements consisting of usual polynomials of degree k 1 for the velocity and polynomials of degree k 1 for the pressure the velocity element is enhanced by polynomials of degree k on the interface of the finite element partition all the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly defined spaces we further establish a priori error estimates in l 2 norm and h 1 norm and we provide a few numerical experiments to illustrate the behavior of the proposed scheme and confirm our theoretical findings regarding optimal convergence of the approximate solutions

consistent nonlinear dynamics identifying model inadequacy empirical modelling often aims for the simplest model consistent with the data a new technique is presented which quantifies the consistency of the model dynamics as a function of location in state space as is well known traditional statistics of nonlinear models like root mean square rms forecast error can prove misleading testing consistency is shown to overcome some of the deficiencies of rms error both within the perfect model scenario and when applied to data from several physical systems using previously published models in particular testing for consistent nonlinear dynamics provides insight towards i identifying when a delay reconstruction fails to be an embedding ii allowing state dependent model selection and iii optimising local neighbourhood size it also provides a more relevant state dependent threshold for identifying false nearest neighbours

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

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

pyramidal traveling fronts in a nonlocal delayed diffusion equation abstract recently there have been great progresses on the study of nonplanar traveling wave solutions of reaction diffusion equations in this paper we study the pyramidal traveling fronts of nonlocal delayed diffusion equation in r n with n 2 by using the comparison principle and establishing super and subsolution since the effect of nonlocal delay we show that nonlocal delayed diffusion equation in n dimensional space has a pyramidal traveling front u t x v x x n s t toward x n axis for each s c 0 in particular two dimensional traveling curved front and three dimensional pyramidal fronts for nonlocal delayed diffusion equation in r 2 and r 3 are also established respectively moreover we also extend our results to generally nonlocal delayed reaction diffusion equation

asymptotic efficiency and finite sample properties of the generalized profiling estimation of parameters in ordinary differential equations ordinary differential equations odes are commonly used to model dynamic behavior of a system because many parameters are unknown and have to be estimated from the observed data there is growing interest in statistics to develop efficient estimation procedures for these parameters among the proposed methods in the literature the generalized profiling estimation method developed by ramsay and colleagues is particularly promising for its computational efficiency and good performance in this approach the ode solution is approximated with a linear combination of basis functions the coefficients of the basis functions are estimated by a penalized smoothing procedure with an ode defined penalty however the statistical properties of this procedure are not known in this paper we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations then we use this bound to prove the consistency and asymptotic normality of this estimation procedure we show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation therefore this procedure is asymptotically efficient for a fixed sample and fixed basis functions we study the limiting behavior of the approximation when the smoothing parameter tends to infinity we propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the odes

existence and stability of time periodic traveling waves for a periodic bistable lotka volterra competition system abstract this paper is concerned with the time periodic lotka volterra competition diffusion system u t d 1 t u x x u r 1 t a 1 t u b 1 t v v t d 2 t v x x v r 2 t a 2 t u b 2 t v x r t 0 where d i t r i t a i t b i t c 2 r are t periodic functions 0 1 i 1 2 under certain conditions the system admits two stable semi trivial periodic solutions p t 0 and 0 q t and a unique coexistence periodic solution u 0 t v 0 t which is unstable and satisfies 0 u 0 t p t and 0 v 0 t q t for t 0 t in this paper we prove that the system admits a time periodic traveling wave solution u x t v x t u x c t t v x c t t connecting two periodic solutions p t 0 and 0 q t as x where c is the wave speed by using a dynamical method we show that the time periodic traveling wave solution u x c t t v x c t t is asymptotically stable and unique modulo translation for front like initial values

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

time delay embedding increases estimation precision of models of intraindividual variability this paper investigates the precision of parameters estimated from local samples of time dependent functions we find that time delay embedding i e structuring data prior to analysis by constructing a data matrix of overlapping samples increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data we show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function hence we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions we support these conclusions both by mathematical analysis and two simulations

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

indistinguishable states ii the imperfect model scenario given a perfect model of a chaotic system and a set of noisy observations of arbitrary duration it is not possible to determine the state of this system precisely rather one must consider a set of states which are indistinguishable from one another given the observations k judd l a smith indistinguishable states i physica d 151 2001 yet the perfect model scenario is a fiction in practice all models are imperfect how do the results from the perfect model scenario change under imperfect models it is shown to be essential to take even small model imperfections into account failure to do so can systematically degrade state estimation or prediction of nonlinear systems with an imperfect model the system state space and model state space are rarely if ever equivalent and so one must consider a projection of the system state into the model state space furthermore it is almost certain that no trajectory of the model is consistent with an infinite series of observations thus there is no consistent way to estimate the projection of system state using trajectories there are pseudo orbits however that are consistent with observations and these can be used to estimate the projection of the system state using pseudo orbits one finds that as in the perfect model scenario there is a set of states that are indistinguishable from the projection of the system state estimation of this set of indistinguishable states and the probability density on these states is discussed the main conclusions are i that there is no state of the model that can be identified with the state of the system and ii that great care must be taken when using an imperfect model to forecast the system because the initialization of the model state from observations can provide a poor analogue for the system the forecast may not shadow the future behaviour of the system for very long even if one were able to obtain a noise free projection of the system state the ultimate aims of probability forecasts should be re examined in light of these results

network reconstruction from high dimensional ordinary differential equations abstractwe consider the task of learning a dynamical system from high dimensional time course data for instance we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points we model the dynamical system nonparametrically as a system of additive ordinary differential equations most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations this is known to be challenging and inefficient we propose a novel approach that does not involve derivative estimation we show that the proposed method can consistently recover the true network structure even in high dimensions and we demonstrate empirical improvement over competing approaches supplementary materials for this article are available online

monte carlo state space likelihoods by weighted posterior kernel density estimation maximum likelihood estimation and likelihood ratio tests for nonlinear non gaussian state space models require numerical integration for likelihood calculations several methods including monte carlo mc expectation maximization mc likelihood ratios direct mc integration and particle filter likelihoods are inefficient for the motivating problem of stage structured population dynamics models in experimental settings an mc kernel likelihood mckl method is presented that estimates classical likelihoods up to a constant by weighted kernel density estimates of bayesian posteriors mckl is derived by using bayesian posteriors as importance sampling densities for unnormalized kernel smoothing integrals mc error and mode bias due to kernel smoothing are discussed and two methods for reducing mode bias are proposed zooming in on the maximum likelihood parameters using a focused prior based on an initial estimate and using a posterior cumulant based approximation of mode bias a simulated example show

existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay this paper is concerned with the existence uniqueness and globally asymptotic stability of traveling wave fronts in the quasi monotone reaction advection diffusion equations with nonlocal delay under bistable assumption we construct various pairs of super and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front up to translation which is monotonically increasing and globally asymptotically stable with phase shift the influence of advection on the propagation speed is also considered comparing with the previous results our results recovers and or improves a number of existing ones in particular these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay some new results are obtained

forcing function diagnostics for nonlinear dynamics this article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations odes i propose modeling lack of fit as a time varying correction to the right hand side of a proposed differential equation this correction can be described as being a set of additive forcing functions estimated from data representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements i derive lack of fit tests based on estimated forcing functions model building in partially observed systems of odes is particularly difficult and i consider the problem of identification of forcing functions in these systems the methods are illustrated with examples from computational neuroscience

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

bistable wavefronts in a diffusive and competitive lotka volterra type system with nonlocal delays abstract a diffusive lotka volterra type model with nonlocal delays for two competitive species is considered the existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four dimensional system without delay furthermore in order to prove the asymptotic stability up to translation of bistable wavefronts of the system the existence regularity and comparison theorem of solutions of the corresponding cauchy problem are first established for the systems on r by appealing to the theory of abstract functional differential equations the asymptotic stability up to translation of bistable wavefronts are then proved by spectral methods in particular we also prove that the spreading speed is unique by upper and lower solutions technique from the point of view of ecology our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation

robust estimation for ordinary differential equation models summary applied scientists often like to use ordinary differential equations odes to model complex dynamic processes that arise in biology engineering medicine and many other areas it is interesting but challenging to estimate ode parameters from noisy data especially when the data have some outliers we propose a robust method to address this problem the dynamic process is represented with a nonparametric function which is a linear combination of basis functions the nonparametric function is estimated by a robust penalized smoothing method the penalty term is defined with the parametric ode model which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ode model the basis coefficients and ode parameters are estimated in two nested levels of optimization the coefficient estimates are treated as an implicit function of ode parameters which enables one to derive the analytic gradients for optimization using the implicit function theorem simulation studies show that the robust method gives satisfactory estimates for the ode parameters from noisy data with outliers the robust method is demonstrated by estimating a predator prey ode model from real ecological data