stability of time stepping methods for abstract time dependent parabolic problems we consider an abstract nonautonomous parabolic problem u t a t u t quad u t 0 u 0 where a t d t subset x to x t 0 le t le t 1 is a family of sectorial operators in a banach space x this problem is discretized in time by means of either an a theta stable runge kutta or an a theta stable linear multistep method we prove that the resulting discretization is stable under some natural assumptions on the relative total variation of a t with respect to t for strongly a theta stable runge kutta methods stability holds even for variable time step sizes our results are applicable to the analysis of time dependent parabolic problems in the lp norms

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

modeling and analyzing respondent driven sampling as a counting process summary respondent driven sampling rds is an approach to sampling design and analysis which utilizes the networks of social relationships that connect members of the target population using chain referral rds sampling will typically oversample participants with many acquaintances naive estimators such as the sample average will thus be biased towards the state of the most highly connected individuals current methodology cannot estimate population size from rds and promotes inverse probability weighted estimators for population parameters such as hiv prevalence we propose to use the timing of recruitment typically collected and discarded in order to estimate the population size via a counting process model once population size and degree frequencies are made available prevalence can be debiased in a post stratified framework we adapt methods developed for inference in epidemiology and software reliability to estimate the population size degree counts and frequencies a fundamental advantage of our approach is that it makes the assumptions of the sampling design explicit this enables verification of the assumptions maximum likelihood estimation extension with covariates and model selection we develop large sample theory proving consistency and asymptotic normality we further compare our estimators to other estimators in the rds literature through simulation and real world data in both cases we find our estimators to outperform current methods the likelihood problem in the model we present is separable and thus efficiently solvable we implement these estimators in an accompanying r package chords available on cran

a splitting technique of higher order for the navier stokes equations this article presents a splitting technique for solving the time dependent incompressible navier stokes equations using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time the integration of adaptive methods in space and time in the splitting are discussed in this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm results on the flow around a cylinder s and the driven cavity s problem are presented

on the stability of interpolatory variable stepsize adams methods in nordsieck form this paper considers the stability of adams methods that vary the stepsize using the interpolation technique of nordsieck sufficient conditions sometimes necessary on the stepsize ratios h n 1 h n that imply the stability of adams methods with different strategies of stepsize variation are obtained this study extends previous results on this subject by gear and tu siamj numer anal 11 1974 pp 1025 1043 and skeel and jackson siam j numer anal 20 1983 pp 840 853

inference for exponential order statistic models based on an integrated likelihood function abstract methods of statistical inference are developed for the exponential order statistic eos model where only a subset of order statistics from a collection of n iid exponential random detection times is observable when the rate parameter for detections is unknown the maximum likelihood estimator mle of the unknown integer parameter n can be infinite with substantial probability inference for n is developed using a pseudolikelihood function obtained by integrating out the rate parameter the estimator that maximizes this function called the integrated likelihood estimator ile is shown to be finite and to have better sampling properties than the mle parameter based asymptotics are developed for the case where n is large application of the methodology is illustrated using two datasets

estimation in multitype epidemics a multitype epidemic model is analysed assuming proportionate mixing between types estimation procedures for the susceptibilities and infectivities are derived for three sets of data complete data meaning that the whole epidemic process is observed continuously the removal processes are observed continuously only the final state is observed under the assumption of a major outbreak in a population of size n it is shown that for all three data sets the susceptibility estimators are always efficient i e consistent with a n rate of convergence the infectivity estimators are in most cases respectively efficient efficient and unidentifiable however if some susceptibilities are equal then the corresponding infectivity estimators are respectively barely consistent log n rate of convergence not consistent and unidentifiable the estimators are applied to simulated data

improved inference for respondent driven sampling data with application to hiv prevalence estimation respondent driven sampling is a form of link tracing network sampling which is widely used to study hard to reach populations often to estimate population proportions previous treatments of this process have used a with replacement approximation which we show induces bias in estimates for large sample fractions and differential network connectedness by characteristic of interest we present a treatment of respondent driven sampling as a successive sampling process unlike existing representations our approach respects the essential without replacement feature of the process while converging to an existing with replacement representation for small sample fractions and to the sample mean for a full population sample we present a successive sampling based estimator for population means based on respondent driven sampling data and demonstrate its superior performance when the size of the hidden population is known we present sensitivity analyses for unknown population sizes in addition we note tha

two high order multistep methods with offstep points applied with a variable stepsize abstract in this paper we derive new generalized multistep methods with variable stepsize for initial value problems in ordinary differential equations some numerical examples are given

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

asymptotic properties of some estimators for the infection rate in the general stochastic epidemic model summary some estimators for the infection rate in the general stochastic epidemic model are presented the first estimator follows the approach of maximum likelihood however this approach requires us to observe most if not all of the epidemic process as an alternative an estimator which uses less detailed data is derived from a suitable martingale both estimators are shown to be consistent only when a major outbreak of disease is observed minor outbreaks do not provide enough information on the infection rate the asymptotic distributions of these estimators are also obtained asymptotic normality holds only for a major outbreak finally the estimators are applied to data on a smallpox outbreak

stability of multistep methods on variable grids this paper is concerned with the stability of multistep methods for ordinary initial value problems on grids with variable mesh sizes a necessary and sufficient condition for stability is given from which generalizations of recent results by gear et al and by zlatev can be obtained as special cases as an application the stability of the variable bdf formulas is treated

convergence of the lambert mcleod trajectory solver and of the celf method a trajectory problem is an initial value problemd y dt f y y 0 where the interest lies in obtaining the curve traced by the solution the trajectory rather than in finding the actual correspondanc between values of the parametert and points on that curve we prove the convergence of the lambert mcleod scheme for the numerical integration of trajectory problems we also study the celf method an explicit procedure for the integration in time of semidiscretizations of pdes which has some useful conservation properties the proofs rely on the concept of restricted stability introduced by stetter in order to show the convergence of the methods an idea of strang is also employed whereby the numerical solution is compared with a suitable perturbation of the theoretical solution rather than with the theoretical solution itself

a family of multistep methods to integrate orbits on spheres a trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution rather than in finding the actual correspondence between the values of the parameter and the points on that curve this paper introduces a family of multi stage multi step numerical methods to integrate trajectory problems whose solution is on a spherical surface it has been shown that this kind of algorithms has good numerical properties consistency stability convergence and others that are not standard the latest ones make them a better choice for certain problems

a hierarchy of polynomial time lattice basis reduction algorithms we present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from lenstra lenstra lovasz reduction to korkine zolotareff reduction let l be the length of a shortest nonzero element of a lattice l we present an algorithm which for k n finds a nonzero lattice vector b so that b 2 6k2 nk l 2 this algorithm uses o n2 kk o k n2 log b arithmetic operations on o n log b bit integers this holds provided that the given basis vectors b1 bn zn are integral and have the length bound b this algorithm successively applies korkine zolotareff reduction to blocks of length k of the lattice basis we also improve kannan s algorithm for korkine zolotareff reduction

low dimensional lattices v integral coordinates for integral lattices we say that an n dimensional classically integral lattice lambda is s integrable for an integer s if it can be described by vectors s frac 1 2 x 1 x k with all x i in mathbb z in a euclidean space of dimension k geqslant n equivalently lambda is s integrable if and only if any quadratic form f x corresponding to lambda can be written as s 1 times a sum of k squares of linear forms with integral coefficients or again if and only if the dual lattice lambda contains a eutactic star of scale s this paper gives many techniques for s integrating low dimensional lattices such as e 8 and the leech lattice a particular result is that any one dimensional lattice can be 1 integrated with k 4 this is lagrange s four squares theorem let phi s be the smallest dimension n in which there is an integral lattice that is not s integrable in 1937 ko and mordell showed that phi 1 6 we prove that phi 2 12 phi 3 14 21 leqslant phi 4 leqslant 25 16 leqslant phi 5 leqslant 22 phi s 4 leqslant 4s 2 s odd phi s leqslant 2 pi es 1 o 1 s even and phi s geqslant 2 ln ln s ln ln ln s 1 o 1

on the zero stability of the variable order variable stepsize bdf formulas this paper contains bounds for the mesh size ratiosh j 1 h j of nonequidistant grids such that any combination of the widely used backward differentiation formulas with ordersl 1 l m m 3 4 5 performs zerostable

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

multiderivative variable stepsize variable formula methods abstract in this paper we present a study of consistency stability and convergence properties of linear multiderivative multistep variable stepsize variable formula methods

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

representations of integral quadratic forms by sums of squares let g n be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares in this article we first discuss representations of integers by unimodular lattices we then estimate the order of magnitude of the diameter of the 2 graph of unimodular lattices of rank n combining these results we prove g n o 3 n 2n log n we also provide a lower bound for g n finally we discuss s integrable lattices as an application of our method