data compatibility and conditional stability for an inverse source problem in the heat equation an inverse problem of determining a nonlinear source term in a heat equation via final observations is investigated by applying integral identity method data compatibilities are obtained with which the inverse source problem here is proved to be solvable furthermore with aid of an integral identity that connects unknown source terms with the known data a conditional stability is constructed theoretical examples are presented showing that theorems and conclusions given in this paper are reasonable and practicable

the method of central difference for the inverse time dependent heat source problem abstract this paper investigates the inverse problem of determining a heat source in a parabolic equation where data are given at some fixed location this problem is ill posed i e the solution if it exists does not depend continuously on the data a central difference regularization method is given and an order optimal stability estimate is obtained numerical results for several benchmark test problems indicate that the central difference regularization method is an accurate and flexible method to determine the unknown time dependent heat source

a mollification regularization method for the inverse spatial dependent heat source problem in this paper we consider the inverse problem of determining a heat source in a parabolic equation where data are given at a fixed time this problem is ill posed i e the solution if it exists does not depend continuously on the data the mollification method with gauss kernel is proposed to solve this problem an a priori error estimate between the exact solution and its regularized approximation is obtained moreover we also propose a new a posteriori parameter choice rule and get a good error estimate numerical results for several benchmark test problems indicate that the method is an accurate and flexible method to determine the unknown spatial dependent heat source

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

corrected confidence intervals after sequential testing with applications to survival analysis corrected confidence intervals are developed for an unknown parameter for data from a sequential experiment the corrections are obtained from asymptotic expansions for the distribution of the signed root transformation and they have a simple form the model does not require independent and identically distributed random variables or smooth stopping boundaries inference about the failure rate of an exponential distribution when data are subject to random censoring censoring by real time and optional stopping is used to illustrate the main results the accuracy of the method is confirmed through simulation studies

source term identification in 1 d ihcp we propose stable numerical solutions for the simultaneous identification of temperature temperature gradient and general source terms in the one dimensional inverse heat conduction problem ihcp the numerical solution consists of a regularization procedure based on the mollification method and a marching scheme for the solution of the stabilized problem the stability error analysis and implementation of the algorithm are presented together with a set of numerical results

the use of chebyshev cardinal functions for the solution of a partial differential equation with an unknown time dependent coefficient subject to an extra measurement a numerical technique is presented for the solution of a parabolic partial differential equation with a time dependent coefficient subject to an extra measurement the method is derived by expanding the required approximate solution as the elements of chebyshev cardinal functions using the operational matrix of derivative the problem can be reduced to a set of algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use

point and interval estimation following a sequential clinical trial summary the usual methodology employed in analysis after a sequential clinical trial is based on orderings of the possible samples resulting from the design however this approach lacks flexibility for use in wider applications in this paper two estimation techniques not based on orderings are considered and modified to obtain improved accuracy a bias adjusted maximum likelihood estimate together with a new and general method for setting confidence limits are discussed the realistic scenario of group sequential monitoring is assumed and methods for exact estimation are given accuracy of the methodology after a triangular test and an o brien fleming test are demonstrated through simulation

estimation of a secondary parameter in a group sequential clinical trial summary in this article we investigate estimation of a secondary parameter in group sequential tests we study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects the bias of the naive secondary parameter estimator is studied it is shown that the sampling proportions of the subgroup have a crucial effect on the bias as the sampling proportion of the subgroup at or just before the stopping time increases the bias of the naive subgroup parameter estimator increases as well an unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived using simulations we compare the mean squared error of the unbiased estimator to that of the naive estimator and we show that the differences are negligible as an example the methods of estimation are applied to an actual group sequential clinical trial the beta blocker heart attack trial

tests and confidence intervals for secondary endpoints in sequential clinical trials in a sequential clinical trial whose stopping rule depends on the primary endpoint inference on secondary endpoints is an important long standing problem ignoring the possibility of early stopping based on the primary endpoint may result in substantial bias to address this problem a commonly used approach is to develop bias correction by estimating the bias in the case of bivariate normal outcomes and appealing to joint asymptotic normality of the statistics associated with the primary and secondary endpoints we propose herein a new approach that uses resampling and a novel ordering scheme in the sample space of sequential statistics observed up to a stopping time this approach is shown to provide accurate inference in complex clinical trials including time sequential trials with survival endpoints and covariates

confidence interval calculation for a sequential clinical trial of binary responses consider a sequential test applied to two streams of binary responses in a comparative clinical trial after the completion of such a trial woodroofe s 1992 technique for accurate confidence interval estimation for a treatment difference can be applied in this paper the technique is extended to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial for which no current methodology exists accuracy is explored by simulation of coverage probabilities and individual confidence limit probabilities the simulations concern a triangular test and an o brien fleming test

estimation after sequential testing a simple approach for a truncated sequential probability ratio test an approximate pivot is constructed for the problem of estimating a normal mean following a truncated sequential probability ratio test and shown to provide a useful method for constructing confidence bounds and intervals letting t denote the sample size and s t the sum of observations the approximate pivot is constructed by standardizing s t t 1 2 s t t the mean and variance of which are no longer 0 and 1 due to the optional stopping

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

the method of simplified tikhonov regularization for dealing with the inverse time dependent heat source problem this paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location the problem is ill posed i e the solution if it exists does not depend continuously on the data a simplified tikhonov regularization method is given and an order optimal stability estimate is obtained a numerical example shows that the regularization method is effective and stable

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

nonhomogeneous heat equation identification and regularization for the inhomogeneous term we study the nonhomogeneous heat equation under the form ut uxx t f x where the unknown is the pair of functions u f under various assumptions about the function and the final value in t 1 i e g x we propose different regularizations on this ill posed problem based on the fourier transform associated with a lebesgue measure for 0 the solution is unique

an a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems the goal of this paper is to investigate a backward parabolic equation with time dependent coefficient by the truncation method tm the key point to our paper is that we give an a posteriori choice of regularization parameter for the tm with which the convergence estimates between the exact solution and the regularized approximation are obtained numerical implementation sheds light on the accuracy and efficiency of the proposed method

estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials pour des essais de confirmation destines a prendre une decision correcte il est important que des experiences adaptees fournissent une inference avec un niveau nominal correct des estimateurs sans biais et des intervalles de confiance pour comparer les traitements dans les essais en cours mais une estimation ponctuelle simple et son intervalle de confiance sont souvent biaises dans des experiences sequentielles adaptatives nous developpons une nouvelle procedure d estimation suivant un test a partir d une taille d echantillon d une experience de re estimation pour obtenir un intervalle de confiance exact et une estimation ponctuelle la methode et basee sur la propriete generale de la distribution d une fonction pivot d un essai clinique sequentiel de groupe de shen et fisher 1999 on propose un estimateur modifie tenant explicitement compte de la limite d arret avec biais reduits quand les tailles de blocs sont petites nous montrons que les estimateurs sont convergents le calcul des estimations est elementaire nous fournissons aussi une fonction de ponderation modifiee pour ameliorer la puissance du test de nombreuses etudes de simulation montrent que les intervalles de confiance exacts ont une probabilite nominale de recouvrement precise et les estimateurs ponctuels sont pratiquement sans biais pour les tailles d echantillon habituels

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

optimal error bound and fourier regularization for identifying an unknown source in the heat equation in this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation under an a priori condition we answer the question concerning the best possible accuracy for the problem the fourier regularization method is utilized for solving the problem and its convergent rate is analyzed numerical results are presented to illustrate the accuracy and efficiency of the method

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

nonstrictly hyperbolic nonlinear systems on etudie le caractere bien pose du probleme aux valeurs initiales localement en temps pour une classe de systemes hyperboliques non lineaires

a tikhonov type method for solving a multidimensional inverse heat source problem in an unbounded domain in this study we prove a stability estimate for an inverse heat source problem in the n dimensional case we present a revised generalized tikhonov regularization and obtain an error estimate numerical experiments for the one dimensional and two dimensional cases show that the revised generalized tikhonov regularization works well

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

a comparison of adaptive allocation rules for group sequential binary response clinical trials in clinical trials to compare two or more treatments with dichotomous responses group sequential designs may reduce the total number of patients involved in the trial and response adaptive designs may result in fewer patients being assigned to the inferior treatments in this paper we combine group sequential and response adaptive designs extending recent work on sample size re estimation in trials to compare two treatments with normally distributed responses to analogous binary response trials we consider the use of two parameters of interest in the group sequential design the log odds ratio and the simple difference between the probabilities of success in terms of the adaptive sampling rules we study two urn models the drop the loser rule and the randomized polya urn rule and compare their properties with those of two sequential maximum likelihood estimation rules which minimize the expected number of treatment failures we investigate two ways in which adaptive urn designs can be used in conjunction with group sequential designs the first method updates the urn at each interim analysis and the second method continually updates the urn after each patient response assuming immediate patient responses our simulation results show that the group sequential design which uses the drop the loser rule applied fully sequentially is the most effective method for reducing the expected number of treatment failures and the average sample number whilst still maintaining the nominal error rates over a range of success probabilities

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

identification of time dependent source terms and control parameters in parabolic equations from overspecified boundary data abstract this paper presents a semigroup approach for inverse source problems for the abstract heat equation when the measured output data is given in subject to the integral overspecification over the spatial domain the existence of a solution to the inverse source problem is shown in appropriate function spaces and a representation formula for the solution is proposed such representation permits the derivation of sufficient conditions for the uniqueness of the solution also an approximation method based on the optimal homotopy analysis method oham is designed and the error estimates are discussed using graphical analysis moreover we conjecture that our approach can be applied for the determination of a control parameter in an inverse problem with integral overspecialization data the proposed algorithm is examined through various numerical examples for the reconstruction of continuous sources and the determination of a control parameter in parabolic equations the accuracy and stability of the method are discussed and compared with several finite difference techniques computational results show efficiency and high accuracy of the proposed algorithm