subgradients of optimal value functions in dynamic programming the case of convex systems without optimal paths we study the first order behaviour of the optimal value function associated to a convex dynamic programming problem the optimization process takes place in a certain environment characterized by some perturbation parameters affecting the transition costs and or the evolution law of the dynamic system an important aspect of this work is that we do not assume the existence of optimal paths to the unperturbed problem

recursive combination tests we present a method that extends the flexibility of adaptive designs to the number of interim analyses and to the choice of decision boundaries at each stage of the trial the design of the next stage can be determined using all of the information gathered so far additionally one can specify the next stage as the final stage or can plan a further interim analysis the method is based on a recursive application of the two stage combination tests for p values the crucial point is the appropriate definition of a p value function that combines p values from two separate stages to a single p value formally we start with a two stage combination test however the p value of the second stage can be replaced by the p value of a further combination test this applies if the experimenter decides in the first interim analysis to perform another interim analysis thereby extending the design to at least three stages obviously this can be also done in a recursive way in the following interim analyses the test

applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

the contemplated average success probability for normally distributed models with an application to optimal sample sizes selection we analytically obtain the average success probability asp and the contemplated average success probability casp for normally distributed observed differences in the treatment group and the placebo group means of the early trial and the confirmatory trial assuming a uniform noninformative prior for the population treatment effect and a common known variance of the observations from both groups for the casp optimization problem with a fixed subtotal sample size of the early trial and the confirmatory trial of one arm larger than a threshold we obtain the optimal plan of the sample sizes in a theorem moreover in the theorem we obtain the analytical formula of the optimal casp as an increasing function of the subtotal sample size after that we calculate and compare the numerical values of the asp with those in table 1 of chuang stein 2006 finally we investigate the numerical features of the casp and find the optimal plan of the sample sizes for a given subtotal sample size

statistical inference for self designing clinical trials with a one sided hypothesis in the process of monitoring clinical trials it seems appealing to use the interim findings to determine whether the sample size originally planned will provide adequate power when the alternative hypothesis is true and to adjust the sample size if necessary in the present paper we propose a flexible sequential monitoring method following the work of fisher 1998 in which the maximum sample size does not have to be specified in advance the final test statistic is constructed based on a weighted average of the sequentially collected data where the weight function at each stage is determined by the observed data prior to that stage such a weight function is used to maintain the integrity of the variance of the final test statistic so that the overall type i error rate is preserved moreover the weight function plays an implicit role in termination of a trial when a treatment difference exists finally the design allows the trial to be stopped early when the efficacy result is sufficiently negative simulation studies confirm the performance of the method

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

parameter estimation following group sequential hypothesis testing summary parameter estimation techniques which fail to adjust for the interim analyses of group sequential test designs will introduce bias in much the same way that the repeated use of single sample hypothesis testing causes inflation of the type one statistical error rate methods based on the duality of hypothesis testing and interval estimation require definition of an ordering for the outcome space for the test statistic in this paper estimation following a group sequential hypothesis test for the mean of a normal distribution with known variance is investigated a proposed ordering of the sample space based on the maximum likelihood estimate of the mean is found to result in estimates which compare favourably with estimates computed from orderings investigated by tsiatis rosner mehta 1984 and chang o brien 1986 for a variety of group sequential designs the proposed ordering is then adapted for use when the sizes of groups accrued between analyses is random

effects of assumption violations on type i error rate in group sequential monitoring we compare the degree to which the type i error probability can be inflated by assumption violations under the different sequential monitoring schemes for methods that assume equally spaced time intervals between looks at the data we examine the effects of unequal spacings for methods that do not specify the look times in advance we examine the degree of type 1 error inflation that can result when future look times are chosen based on data trends we also evaluate a bayesian method of monitoring from this non bayesian perspective of type i error inflation

locally adaptive greedy approximations for anisotropic parameter reduced basis spaces reduced order models in particular the reduced basis method rely on empirically built and problem dependent basis functions that are constructed during an off line stage in the on line stage the precomputed problem dependent solution space that is spanned by the basis functions can then be used in order to reduce the size of the computational problem for complex problems the number of basis functions required to guarantee a certain error tolerance can become too large in order to benefit computationally from the model reduction to overcome this the present work introduces a framework where local approximation spaces in parameter space are used to define the reduced order approximation in order to have explicit control over the on line cost this approach also adapts the local approximation spaces to local anisotropic behavior in the parameter space we present the algorithm and numerous numerical tests

an approach to the conditional error rate principle with nuisance parameters summary in the presence of nuisance parameters the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters to use the conditional error rate principle with nuisance parameters we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters we show that the optimization problem that defines such a test can be solved efficiently by existing algorithms

exponential rosenbrock integrators for option pricing in this paper we are concerned with the time integration of differential equations modeling option pricing in particular we consider the black scholes equation for american options as an alternative to existing methods we present exponential rosenbrock integrators these integrators require the evaluation of the exponential and related functions of the jacobian matrix the resulting methods have good stability properties they are fully explicit and do not require the numerical solution of linear systems in contrast to standard integrators we have implemented some numerical experiments in matlab showing the reliability of the new method

on sample size and inference for two stage adaptive designs proschan and hunsberger 1995 biometrics 51 1315 1324 proposed a two stage adaptive design that maintains the type i error rate for practical applications a two stage adaptive design is also required to achieve a desired statistical power while limiting the maximum overall sample size in our proposal a two stage adaptive design is comprised of a main stage and an extension stage where the main stage has sufficient power to reject the null under the anticipated effect size and the extension stage allows increasing the sample size in case the true effect size is smaller than anticipated for statistical inference methods for obtaining the overall adjusted p value point estimate and confidence intervals are developed an exact two stage test procedure is also outlined for robust inference

on tridiagonal linear complementarity problems for solving the tridiagonal linear complementarity problem numerically it is proposed to transform this into an optimization problem without constraints via construction of a weakly coupled program and its dualization if then e g newton s method is applied a locally superlinearly convergent algorithm for the complementarity problem originates

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

the generalized empirical interpolation method stability theory on hilbert spaces with an application to the stokes equation the generalized empirical interpolation method geim is an extension first presented by maday and mula in 1 in 2013 of the classical empirical interpolation method presented in 2004 by barrault maday nguyen and patera in 2 where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of banach spaces as outlined in 1 this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach in this paper we present a thorough analysis of the concept of stability condition of the generalized interpolant the lebesgue constant by relating it to an inf sup problem in the case of hilbert spaces in the second part of the paper it will be explained how geim can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number optimally placed measurements from the processes with their mathematical models parameter dependent pdes this idea is illustrated through a parameter dependent stokes problem in which it is shown that the pressure and velocity fields can e ciently be reconstructed with a relatively low dimensional interpolation space

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

a decomposition dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems in this paper we shall be concerned with the solution of constrained convex minimization problems the constrained convex minimization problems are proposed to be transformable into a convex additively decomposed and almost separable form e g by decomposition of the objective functional and the restrictions unconstrained dual problems are generated by using fenchel rockafellar duality this decomposition dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable moreover the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return formulas in quadratic programming the decomposition dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byh matrices numerical tests for solving obstacle problems in 1 discretized by using piecewise quadratic finite elements and in 2 by using the five point difference approximation are presented

the dual test safeguarding p value combination tests for adaptive designs many modern adaptive designs apply an analysis where p values from different stages are weighted together to an overall hypothesis test one merit of this combination approach is that the design can be made very flexible however combination tests violate the sufficiency and conditionality principles as a consequence combination tests may lead to absurd conclusions such as proving a positive effect while the average effect is negative we explore the possibility of modifying the test so that such illogical conclusions are no longer possible the dual test requires both the weighted combination test and a nave test ignoring the adaptations to be statistically significant the result is that the flexibility and type i error level control of the combination test are preserved while the nave test adds a safeguard against unconvincing results the dual test is by construction at least as conservative as the combination test however many design changes will not lead to any power loss a typical situation where the combination approach can be used is two stage sample size reestimation ssr for this case we give a complete specification of all sample size modifications for which the two tests are equally powerful we also study the overall power loss for some suggested ssr rules rules based on conditional power generally lead to ignorable power loss while a decision analytic approach exhibits clear discrepancies between the two tests

increasing the sample size when the unblinded interim result is promising increasing the sample size based on unblinded interim result may inflate the type i error rate and appropriate statistical adjustments may be needed to control the type i error rate at the nominal level we briefly review the existing approaches which allow early stopping due to futility or change the test statistic by using different weights or adjust the critical value for final test or enforce rules for sample size recalculation the implication of early stopping due to futility and a simple modification to the weighted z statistic approach are discussed in this paper we show that increasing the sample size when the unblinded interim result is promising will not inflate the type i error rate and therefore no statistical adjustment is necessary the unblinded interim result is considered promising if the conditional power is greater than 50 per cent or equivalently the sample size increment needed to achieve a desired power does not exceed an upper bound the actual sample size increment may be determined by important factors such as budget size of the eligible patient population and competition in the market the 50 per cent conditional power approach is extended to a group sequential trial with one interim analysis where a decision may be made at the interim analysis to stop the trial early due to a convincing treatment benefit or to increase the sample size if the interim result is not as good as expected the type i error rate will not be inflated if the sample size may be increased only when the conditional power is greater than 50 per cent if there are two or more interim analyses in a group sequential trial our simulation study shows that the type i error rate is also well controlled

parameter multi domain hp empirical interpolation summary in this paper we introduce two parameter multi domain hp techniques for the empirical interpolation method eim in both approaches we construct a partition of the original parameter domain into parameter subdomains h refinement we apply the standard eim independently within each subdomain to yield local in parameter approximation spaces p refinement further for a particularly simple case we introduce a priori convergence theory for the partition procedure we show through two numerical examples that our approaches provide significant reduction in the eim approximation space dimension and thus significantly reduce the computational cost associated with eim approximations copyright 2012 john wiley sons ltd

a dual algorithm for convex concave data smoothing by cubicc2 splines in this paper the problem of smoothing a given data set by cubicc 2 splines is discussed the spline may required to be convex in some parts of the domain and concave in other parts application of splines has the advantage that the smoothing problem is easily discretized moreover the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained therefore the latter program is treated numerically much more easier than the original program further the validity of a return formula is of importance by which a minimizer of the orginal program is obtained from a maximizer of the dual program the theoretical background of this general approach is discussed and above all the details for applying the strategy to the present smoothing problem are elaborated also some numerical tests are presented

confidence intervals following group sequential tests in clinical trials tsiatis rosner and mehta 1984 biometrics 40 797 803 proposed a procedure for constructing confidence intervals following group sequential tests of a normal mean this method is first extended for group sequential tests for which the sample sizes between interim analyses are not identical or the times are not equally spaced then properties of this confidence interval estimation procedure are studied by simulation the extension accommodates the flexible procedure by lan and demets 1983 biometrika 70 659 663 for constructing discrete group sequential boundaries to form a structure for monitoring and estimation following a class of group sequential tests finally it is demonstrated how to combine the procedures by lan and demets and by tsiatis rosner and mehta using a fortran program

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

an hp certified reduced basis method for parametrized elliptic partial differential equations we present a new hp parameter multidomain certified reduced basis rb method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations we propose and provide theoretical justification for a new procedure for adaptive partition h refinement of the parameter domain into smaller parameter subdomains we pursue a hierarchical splitting of the parameter sub domains based on proximity to judiciously chosen parameter anchor points within each subdomain subsequently we construct individual standard rb approximation spaces p refinement over each subdomain greedy parameter sampling procedures and a posteriori error estimation play important roles in both the h type and p type stages of the new algorithm we present illustrative numerical results for a convection diffusion problem the new hp approach is considerably faster respectively more costly than the standard p type reduced basis method in the online respectively offline stage

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

epsilon maximum principle of pontryagin type and perturbation analysis of convex optimal control problems in this paper we study the first order behaviour of the optimal value function associated with a convex parametric problem of optimal control a formula for the subdifferential of the optimal value function is derived without assuming the existence of optimal solutions to the unperturbed problem the so called epsilon maximum principle is a key ingredient in the writing of our main sensitivity result

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

adaptive finite element methods for hydrodynamic lubrication with cavitation we present an adaptive finite element method for a cavitation model based on reynolds equation a posteriori error estimates and adaptive algorithms are discussed and numerical examples illustrating the theory are supplied

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

convergence of a penalty finite element approximation for an obstacle problem this study establishes an error estimate for a penalty finite element approximation of the variational inequality obtained by a class of obstacle problems by special identification of the penalty term we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality the rate of convergence of the penalization is where is the penalty parameter to obtain the error of finite element approximation we apply the results obtained by brezzi hager and raviart for the mixed finite element method to the variational inequality

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

finite termination of a dual newton method for convex best c1 interpolation and smoothing given the data x i y i 2 i 0 1 ldots n which are in convex position the problem is to choose the convex best c 1 interpolant with the smallest mean square second derivative among all admissible cubic c 1 splines on the grid this problem can be efficiently solved by its dual program developed by schmdit and his collaborators in a series of papers the newton method remains the core of their suggested numerical scheme it is observed through numerical experiments that the method terminates in a small number of steps and its total computational complexity is only of o n the purpose of this paper is to establish theoretical justification for the newton method in fact we are able to prove its finite termination under a mild condition and on the other hand we illustrate that the newton method may fail if the condition is violated consistent with what is numerically observed for the newton method corresponding results are also obtained for convex smoothing

issues in designing flexible trials we outline the general framework of adaptive combination tests and discuss their relationship to flexible group sequential designs an important field of applications is sample size reassessment we discuss reassessment rules based on conditional power arguments using either the observed or the prefixed effect size these rules tend to lead to large expected sample sizes for small actual effects however the application of a maximal bound for the second stage sample size leads to more favourable properties additionally we consider an optimized reassessment rule in terms of expected sample sizes since the adaptive design does not use the classical test statistics for some types of sample size reassessments the adaptive test may reject the null hypothesis while the classical one sample test does not we characterize sample size reassessment rules where such inconsistencies are avoided finally the extension of flexibility to the number of stages is explored in the first interim analysis a second interim analysis is only planned if the chance to achieve a decision there is high this leads to savings in the average number of interim analysis performed without paying a noticeable price in terms of expected sample size

modification of sample size in group sequential clinical trials in group sequential clinical trials sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type i error in most practical situations a new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two sample mean test the new test has the type i error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size generalization of the new procedure is discussed

nonlinear model reduction via a locally weighted pod method summary in this article we propose a new approach for model reduction of parameterized partial differential equations pdes by a locally weighted proper orthogonal decomposition lwpod method the presented approach is particularly suited for large scale nonlinear systems characterized by parameter variations instead of using a global basis to construct a global reduced model lwpod approximates the original system by multiple local reduced bases each local reduced basis is generated by the singular value decomposition of a weighted snapshot matrix compared with global model reduction methods such as the classical proper orthogonal decomposition lwpod can yield more accurate solutions with a fixed subspace dimension as another contribution we combine lwpod with the chord iteration to solve elliptic pdes in a computationally efficient fashion the potential of the method for achieving large speedups while maintaining good accuracy is demonstrated for both elliptic and parabolic pdes in a few numerical examples copyright 2016 john wiley sons ltd

sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets let g y 1 y 2 be the optimal value of the abstract program which consists in minimizing a convex function f x to mathbb r cup infty over a feasible set of the form x in x a 1 x y 1 a 2 x leq k y 2 without assuming the existence of optimal solutions to this minimization problem we derive formulas for the subdifferential and the approximate subdifferential of g several applications are discussed

adaptive sample size modification in clinical trials start small then ask for more we consider sample size re estimation in a clinical trial in particular when there is a significant delay before the measurement of patient response mehta and pocock have proposed methods in which sample size is increased when interim results fall in a promising zone where it is deemed worthwhile to increase conditional power by adding more subjects our analysis reveals potential pitfalls in applying this approach mehta and pocock use results of chen demets and lan to identify when increasing sample size but applying a conventional level significance test at the end of the trial does not inflate the type i error rate we have found the greatest gains in power per additional observation are liable to lie outside the region defined by this method mehta and pocock increase sample size to achieve a particular conditional power calculated under the current estimate of treatment effect this leads to high increases in sample size for a small range of interim outcomes whereas we have found it more efficient to make moderate increases in sample size over a wider range of cases if the aforementioned pitfalls are avoided we believe the broad framework proposed by mehta and pocock is valuable for clinical trial design working in this framework we propose sample size rules that apply explicitly the principle of adding observations when they are most beneficial the resulting trial designs are closely related to efficient group sequential tests for a delayed response proposed by hampson and jennison copyright 2015 john wiley sons ltd

subgradients of value functions in parametric dynamic programming we study in this paper the first order behavior of value functions in parametric dynamic programming with linear constraints and nonconvex cost functions by establishing an abstract result on the frechet subdifferential of value functions of parametric mathematical programming problems some new formulas on the frechet subdifferential of value functions in parametric dynamic programming are obtained

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

smoothness of the policy function in discrete time economic models the theory applying to dynamic programming has furnished a useful set of techniques for the analysis of many types of sequential models this theory however has not yielded heretofore much information about the differentiability properties of optimal solutions this aspect is of particular interest as regards the qualitative analysis of optimal paths where differentiable methods are often called into play this paper shows roughly that if the objective is twice continuously differentiable and strongly concave then any interior optimal path is continuously differentiable with respect to the initial state copyright 1991 by the econometric society

on two level optimization decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

unconstrained duals to partially separable constrained programs in this note partially separable convex programs are dualized in such a way that under certain assumptions unconstrained concave duals arise a return formula is given by which the solution of the primal is directly computed if a solution of the dual is known further the solvability of both the primal and the dual is shown to depend essentially on the behaviour of the lower dimensional programs for determining the fenchel conjugates

localized discrete empirical interpolation method this paper presents a new approach to construct more efficient reduced order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method deim whereas deim projects the nonlinear term onto one global subspace our localized discrete empirical interpolation method ldeim computes several local subspaces each tailored to a particular region of characteristic system behavior then depending on the current state of the system ldeim selects an appropriate local subspace for the approximation of the nonlinear term in this way the dimensions of the local deim subspaces and thus the computational costs remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes ldeim uses machine learning methods in the offline computational phase to discover these regions via clustering local deim approximations are then computed for each cluster in the online computational phase machine

type i error in sample size re estimations based on observed treatment difference sample size re estimation based on an observed difference can ensure an adequate power and potentially save a large amount of time and resources in clinical trials one of the concerns for such an approach is that it may inflate the type i error however such a possible inflation has not been mathematically quantified in this paper the mathematical mechanism of this inflation is explored for two sample normal tests a conditional type i error function based on normal data is derived this function not only provides the quantification but also gives mathematical mechanisms of possible inflation in the type i error due to the sample size re estimation theoretically based on their decision rules certain upper and lower bounds people can calculate this function and exactly visualize the changes in type i error computer simulations are performed to ensure the results if there are no bounds for the adjustment the inflation is evident if proper adjusting rules are used the inflation can be well controlled in some cases the type i error can even be reduced the trade off is to give up some unrealistic power we investigated several scenarios in which the mechanisms to change the type i error are different our simulations show that similar results may apply to other distributions copyright 2001 john wiley sons ltd

exact confidence intervals following a group sequential test a numerical method is used to compute confidence intervals which have exact coverage probabilities for the mean of a normal distribution following a group sequential test this method which uses an ordering of the sample space similar to that employed by siegmund 1978 biometrika 65 341 349 is contrasted with the usual confidence interval for the mean

certified dual corrected radiation patterns of phased antenna arrays by offline online order reduction of finite element models this paper presents a fast numerical method for computing certified far field patterns of phased antenna arrays over broad frequency bands as well as wide ranges of steering and look angles the proposed scheme combines finite element analysis dual corrected model order reduction and empirical interpolation to assure the reliability of the results improved a posteriori error bounds for the radiated power and directive gain are derived both the reduced order model and the error bounds algorithm feature offline online decomposition a real world example is provided to demonstrate the efficiency and accuracy of the suggested approach

e optimal solutions in nondifferentiable convex programming and some related questions in this paper we present e optimality conditions of the kuhn tucker type for points which are within e of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints such e optimality conditions are of interest for theoretical consideration as well as from the computational point of view some illustrative applications are made thus we derive an expression for the e subdifferential of a general convex max function we also show how the e optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints

on the existence of convex decompositions of partially separable functions the concept of a partially separable functionf developed in 4 is generalized to include all functionsf that can be expressed as a finite sum of element functionsf i whose hessians have nontrivial nullspacesn i such functions can be efficiently minimized by the partitioned variable metric methods described in 5 provided that each element functionf i is convex if this condition is not satisfied we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf i such that the resulting modified element functions are at least locally convex to avoid tests on the numerical value of the hessian we study the totally convex case where all locally convexf with the separability structuren i 1 have a convex decomposition it is shown that total convexity only depends on the associated linear conditions on the hessian matrix in the sparse case when eachn i is spanned by cartesian basis vectors it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a permuted ldlt factorization without fill in

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

optimal flexible sample size design with robust power it is well recognized that sample size determination is challenging because of the uncertainty on the treatment effect size several remedies are available in the literature group sequential designs start with a sample size based on a conservative smaller effect size and allow early stop at interim looks sample size re estimation designs start with a sample size based on an optimistic larger effect size and allow sample size increase if the observed effect size is smaller than planned different opinions favoring one type over the other exist we propose an optimal approach using an appropriate optimality criterion to select the best design among all the candidate designs our results show that 1 for the same type of designs for example group sequential designs there is room for significant improvement through our optimization approach 2 optimal promising zone designs appear to have no advantages over optimal group sequential designs and 3 optimal designs with sample size re estimation deliver the best adaptive performance we conclude that to deal with the challenge of sample size determination due to effect size uncertainty an optimal approach can help to select the best design that provides most robust power across the effect size range of interest copyright 2016 john wiley sons ltd

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

sample size recalculation using conditional power the sample size required to achieve a given power at a prespecified absolute difference in mean response may depend on one or more nuisance parameters which are usually unknown proposed methods for using an internal pilot to recalculate the sample size using estimates of these parameters have been well studied most of these methods ignore the fact that data on the parameter of interest from within this internal pilot will contribute towards the value of the final test statistic we propose a method which involves recalculating the target sample size by computing the number of further observations required to maintain the probability of rejecting the null hypothesis at the end of the study under the prespecified absolute difference in mean response conditional on the data observed so far we do this within the framework of a two group error spending sequential test modified so as to prevent inflation of the type i error rate

regularized model of post touchdown configurations in electrostatic mems equilibrium analysis abstract in canonical models of micro electro mechanical systems mems an event called touchdown whereby the electrical components of the device come into contact is characterized by a blow up in the governing equations and a non physical divergence of the electric field in the present work we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria we employ techniques from variational calculus dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations

adaptive finite elements for elastic bodies in contact to avoid interpenetration of matter under the small strain assumption the linear contact condition is frequently applied where the distance of bodies is controlled only along a certain direction the standard direction is the normal on the surface where interpenetration might occur in this paper we allow other directions as well we address questions such as the correct mathematical model existence of solutions the penalty method for regularization of the variational inequality finite element discretization and a priori and a posteriori error estimates but exclude the error of penalization the computable upper error bound leads to a criterion for automatic mesh refinements within a finite element method numerical simulations of the hertzian contact problem and a supported cantilever beam are included

operating characteristics of sample size re estimation with futility stopping based on conditional power various methods have been described for re estimating the final sample size in a clinical trial based on an interim assessment of the treatment effect many re weight the observations after re sizing so as to control the pursuant inflation in the type i error probability alpha lan and trost estimation of parameters and sample size re estimation proceedings of the american statistical association biopharmaceutical section 1997 48 51 proposed a simple procedure based on conditional power calculated under the current trend in the data cpt the study is terminated for futility if cpt or cu or re sized by a factor m to yield cpt cu if cl cpt cu where cl and cu are pre specified probability levels the overall level alpha can be preserved since the reduction due to stopping for futility can balance the inflation due to sample size re estimation thus permitting any form of final analysis with no re weighting herein the statistical properties of this approach are described including an evaluation of the probabilities of stopping for futility or re sizing the distribution of the re sizing factor m and the unconditional type i and ii error probabilities alpha and beta since futility stopping does not allow a type i error but commits a type ii error then as the probability of stopping for futility increases alpha decreases and beta increases an iterative procedure is described for choice of the critical test value and the futility stopping boundary so as to ensure that specified alpha and beta are obtained however inflation in beta is controlled by reducing the probability of futility stopping that in turn dramatically increases the possible re sizing factor m the procedure is also generalized to limit the maximum sample size inflation factor such as at m max 4 however doing so then allows for a non trivial fraction of studies to be re sized at this level that still have low conditional power these properties also apply to other methods for sample size re estimation with a provision for stopping for futility sample size re estimation procedures should be used with caution and the impact on the overall type ii error probability should be assessed

the reassessment of trial perspectives from interim data a critical view if an interim analysis is performed during a trial it is tempting to determine the conditional power to reach a rejection in the trial given the observed results in the interim analysis since the true effect size is unknown the conditional power may be calculated by using the effect size which the study has been powered for in the planning phase or by using an interim estimate of the true size or a combination of both in either case the conditional power is a random variable and its density is investigated depending on the analysis time and the true effect size under the null hypothesis in early interim analyses after a small proportion of sample units the conditional power typically will be close to the overall power when the effect size from the planning stage is used for calculation in this case the majority of observations must still be made and the small first stage sample in general will be dominated by the hypothetical second stage chance based on the wrong parameter value it is shown that the conditional power in moderately underpowered studies can have a distribution symmetric around 0 5 when using the interim estimate for calculating the conditional power the density in general will be u shaped the impact of using conditional power to reassess the sample size using flexible two stage combination tests is shown for a specific example in terms of overall power and average sample size as compared to the corresponding group sequential design for small true effect sizes mid trial sample size recalculation based on an interim estimate may lead to an overly large price to be paid in average sample size in relation to the gain in overall power finally the problem is discussed in terms of estimating the true conditional power

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected

parametric domain decomposition for accurate reduced order models applications of mp lrom methodology abstract the multivariate predictions of local reduced order models mp lrom methodology recently proposed by the authors moosavi et al 0000 uses machine learning based regression methods to predict the errors of reduced order models this study considers two applications of mp lrom first the error model is used in conjunction with a greedy sampling algorithm to generate decompositions of one dimensional parametric domains with overlapping regions such that the associated local reduced order models meet the prescribed accuracy requirements once a parametric domain decomposition is constructed any parametric configuration belongs to at least one of the partitions the local reduced order model associated with that partition approximates the full order model at the given parameters within an accuracy level that is estimated a priori the parameter domain decomposition creates a database of the available local bases local reduced order and high fidelity models and identifies the most accurate solutions for an arbitrary parametric configuration next this database is used to enhance the accuracy of the reduced order models using 1 lagrange interpolation of reduced bases in the matrix space 2 lagrange interpolation of reduced bases in the tangent space of the grassmann manifold 3 concatenation of reduced bases followed by a gram schmidt orthogonalization process and 4 lagrange interpolation of high fidelity model solutions numerical results with a viscous burgers model illustrate the potential of the mp lrom methodology to improve the design of parametric reduced order models

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

internal pilot studies i type i error rate of the naive t test when sample size is recalculated using unblinded interim data use of the usual t test at the end of a study may lead to an elevated type i error rate this paper describes a numerical quadrature investigation to calculate the true probability of rejection as a function of the time of the recalculation the magnitude of the detectable treatment effect and the ratio of the guessed to the true variance we consider both restricted designs those that require final sample size at least as large as the originally calculated size and unrestricted designs those that permit smaller final sample sizes than originally calculated our results indicate that the bias in the type i error rate is often negligible especially in restricted designs some sets of parameters however induce non trivial bias in the unrestricted design