approximate fundamental solutions and wave fronts for general anisotropic materials abstract the time dependent differential equations of elastodynamics for homogeneous solids with a general structure of anisotropy are considered in the paper a new method of computation of the fundamental solution for these equations is proposed this method consists of the following applying the fourier transformation with respect to space variables to these equations we obtain a system of second order ordinary differential equations whose coefficients depend on fourier parameters using the matrix transformations and properties of the coefficients the fourier image of the fundamental solution is computed finally the fundamental solution is calculated by the inverse fourier transformation to the obtained fourier transform the implementation and justification of the suggested method have been made by computational experiments in matlab these experiments confirm the robustness of the suggested method the visualization of the displacement components in general homogeneous anisotropic solids by modern computer tools allows us to see and evaluate the dependence between the structure of solids and the behavior of the displacement field our method allows users to observe the elastic wave propagation arising from pulse point forces of the form em x t in monoclinic triclinic and other anisotropic solids the visualization of displacement components gives knowledge about the form of fronts of elastic wave propagation in sodium thiosulfate with monoclinic and copper sulphate pentahydrate with triclinic structures of anisotropy

a practical relative error criterion for augmented lagrangians this paper develops a new error criterion for the approximate minimization of augmented lagrangian subproblems this criterion is practical since it is readily testable given only a gradient or subgradient of the augmented lagrangian it is also relative in the sense of relative error criteria for proximal point algorithms in particular it uses a single relative tolerance parameter rather than a summable parameter sequence our analysis first describes an abstract version of the criterion within rockafellar s general parametric convex duality framework and proves a global convergence result for the resulting algorithm specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented lagrangian method with a novel inexact solution condition for the subproblems finally we present computational results drawn from the cute test set including many nonconvex problems indicating that the approach works well in practice

finite element calculation of elastodynamic stress field around a notch tip via contour integrals abstract direct computation of the mixed mode dynamic asymptotic stress field around a notch tip is difficult because the mode i and mode ii stresses are in general governed by different orders of singularity in this paper we propose a pair of elastodynamic contour integrals j kr t the integrals are shown to be path independent in a modified sense and so they can be accurately evaluated with finite element solutions also by defining a pair of generalized stress intensity factors sifs k i t and k ii t the relationship between j kr t and the sif s is derived and expressed as functions of the notch angle once the j kr t integrals are accurately computed the generalized sif s and consequently the asymptotic mixed mode stress field can then be properly determined no particular singular elements are required in the calculation the proposed numerical scheme can be used to investigate the dynamic amplifying effect in the near tip stress field

multiple test procedures for identifying the maximum safe dose we consider dose response studies for safety assessment of crop protection compounds and drugs and we offer a hypothesis testing approach for identifying the maximum dose level that is guaranteed to be safe with preassigned confidence the focus is on step down sd multiple test procedures for identifying the maximum safe dose we propose two classes of contrasts among the dose means as test statistics for these procedures pairwise contrasts pc and helmert contrasts hc the first procedure sd2pc consists of a sequence of ordinary t tests and is thus easy to apply but it can have very low power for certain step response functions the second procedure sd1hc does not suffer from this drawback but it requires a weak monotonicity assumption for its mathematical validity the powers of sd2pc and sd1hc are studied via monte carlo simulation we recommend sd2pc for linear response functions and sd1hc for step response functions the procedures are illustrated by applying them to data from an aquatic

modified sequentially rejective multiple test procedures abstract suppose that n hypotheses h 1 h 2 h n with associated test statistics t 1 t 2 t n are to be tested by a procedure with experimentwise significance level the probability of rejecting one or more true hypotheses smaller than or equal to some specified value a commonly used procedure satisfying this condition is the bonferroni b procedure which consists of rejecting h i for any i iff the associated test statistic t i is significant at the level n holm 1979 introduced a modified bonferroni procedure with greater power than the b procedure under holm s sequentially rejective bonferroni srb procedure if any hypothesis is rejected at the level n the denominator of for the next test is n 1 and the criterion continues to be modified in a stagewise manner with the denominator of reduced by 1 each time a hypothesis is rejected so that tests can be conducted at successively higher significance levels holm proved that the experimentwise significance level

multiple test procedures for identifying the minimum effective and maximum safe doses of a drug we address the problem of determining the therapeutic window of a drug by finding its minimum effective and maximum safe doses mined and maxsd the mined is the lowest dose that exceeds the mean efficacy of the zero dose by a specified threshold and the maxsd is the highest dose that does not exceed the mean toxicity of the zero dose by a specified threshold step down multiple test procedures are proposed to identify the mined and maxsd assuming a bivariate normal model these procedures control the type i familywise error probability of declaring any ineffective dose as effective or any unsafe dose as safe at a prespecified level a new multivariate t distribution is introduced whose critical points are required to implement the exact normal theory procedures because these critical points depend on the unknown correlation coefficient between the efficacy and safety variables the bonferroni method is proposed as an alternative which amounts to separately testing for efficacy and safety each at t

on closed test procedures for dose response analysis this paper concerns the testing of a dose response effect in medical studies we consider two situations the first is when the drug effect on a parameter of interest is likely to increase or decrease with increasing doses we propose a procedure based on the closure principle of peritz combined with an application of a closed testing procedure of marcus et al to a test proposed by turkey et al the second situation is the analysis of a drug effect where one might observe a reversal at higher doses for this situation we propose simultaneous examination of contrasts among the doses at each of several stages of the testing scheme we calculate critical values for this procedure incorporating the correlation structure among the contrasts both procedures strongly control the family wise error rate at the pre determined alpha level they also provide information about the shape of the dose response lacking in other commonly used procedures for testing against an ordered alternative hypothesis we illustrate the procedures on two datasets

projective splitting with forward steps this work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators for the projective splitting framework recently proposed by combettes and eckstein we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps the resulting algorithms have the same kind of coordination procedure and can be implemented in the same block iterative and highly flexible manner but may perform backward steps on some operators and forward steps on others prior algorithms in the projective splitting family have used only backward steps forward steps can be used for any lipschitz continuous operators provided the stepsize is bounded by the inverse of the lipschitz constant if the lipschitz constant is unknown a simple backtracking linesearch procedure may be used for affine operators the stepsize can be chosen adaptively without knowledge of the lipschitz constant and without any additional forward steps we close the paper by empirically studying the performance of several kinds of splitting algorithms on a large scale rare feature selection problem

convergence rates with inexact non expansive operators in this paper we present a convergence rate analysis for the inexact krasnosel ski mann iteration built from non expansive operators the presented results include two main parts we first establish the global pointwise and ergodic iteration complexity bounds then under a metric sub regularity assumption we establish a local linear convergence for the distance of the iterates to the set of fixed points the obtained results can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature including the forward backward splitting the generalized forward backward the douglas rachford splitting alternating direction method of multipliers and primal dual splitting methods for these methods we also develop easily verifiable termination criteria for finding an approximate solution which can be seen as a generalization of the termination criterion for the classical gradient descent method we finally develop a parallel analysis for the non stationary krasnosel ski mann iteration

a one sided studentized range test for testing against a simple ordered alternative abstract consider the usual balanced one way fixed effects analysis of variance anova model x ij i e ij 1 j n 1 i k where the e ij are independent n 0 sgrave 2 random variables and i is the mean of the ith treatment 1 i k the i and sgrave 2 are unknown parameters let x i be the sample mean of the ith treatment based on n observations 1 i k and let s 2 be an unbiased estimate of sgrave 2 which is distributed independently of the x i as a 2 2 v v random variable usually the anova mean squared error with v k n 1 df is used as the estimate s 2 a problem that has received considerable attention is that of testing the null hypothesis h 0 1 k against the simple ordered alternative hypothesis h a 1 k with at least one strict inequality see e g barlow bartholomew bremner and brunk 1972 and robertson wright and dykstra 1988 various test procedures have been proposed for this problem such as the likelihood ratio test

a note on multiple testing procedures in dose finding it is shown that in some recently proposed multiple test procedures for dose finding tamhane hochberg and dunnett 1996 biometrics 52 21 37 strong control of the familywise error is only guaranteed if order relations are assumed to hold among the means procedures that can be applied safely without assuming any restrictions on the means can be based on the classical many one comparisons of doses with the zero dose

multiple test procedures in clinical dose finding studies abstract test procedures in clinical dose finding studies are considered when it can be assumed that increasing dose levels lead to a larger or at least equal efficacy if one is interested in various comparisons within the set of dose levels one arrives at a problem of multiple testing against ordered alternatives some multiple test procedures controlling the multiple level of significance are compared in a simulation study it follows that a simple procedure that only uses comparisons between neighboring dose levels can be recommended the degree of approximation of this bonferroni type procedure is investigated under normal assumptions

closure procedures for monotone bi factorial dose response designs les essais en doses multiples ont deux buts i demontrer une amelioration de l efficacite pour une combinaison donnee de chacun des composants ii identifier un intervalle de doses efficaces et sures les auteurs s interessent aux deux aspects quoique avec un interet plus marque pour le second avec une procedure de cloture qui garantit un controle fort du risque global d erreur on etudie deux familles differentes d hypotheses nulles pour des dispositifs bi factoriels monotones par rapport a un ordre partiel l une est adaptee a la recherche d une ou plusieurs doses efficaces minimums l autre est suffisamment large pour permettre d identifier les doses efficaces les plus elevees on applique des tests du rapport de vraisemblance et des contrastes multiples appropries a un exemple d essai clinique tire de hung 2000 statistics in medicine 19 2079 2087 le code ecrit en langage r est disponible dans sa totalite sur internet

stepwise confidence intervals without multiplicity adjustment for dose response and toxicity studies abstract not all simultaneous inferences need multiplicity adjustment if the sequence of individual inferences is predefined and failure to achieve the desired inference at any step renders subsequent inferences unnecessary then multiplicity adjustment is not needed this can be justified using the closed testing principle to test appropriate hypotheses that are nested in sequence starting with the most restrictive one but what hypotheses are appropriate may not be obvious in some problems we give a fundamentally different confidence set based justification by partitioning the parameter space naturally and using the principle that exactly one member of the partition contains the true parameter in dose response studies designed to show superiority of treatments over a placebo negative control or a drug known to be efficacious active control the confidence set approach generates methods with meaningful guarantee against incorrect decision whereas previous applications of the closed testing app

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

sequential implementation of stepwise procedures for identifying the maximum tolerated dose this article considers the problem of finding the maximum tolerated dose mtd of a drug in human trials the mtd is defined as the maximum test dose with toxicity probability less than or equal to a target toxicity rate we adopt the multiple test framework with step down tests used in an escalation stage and step up tests used in a deescalation stage to allow sequential dose assignments for ethical purposes by formulating the estimation problem as a testing problem the proposed procedures formally control the error probability of selecting an unsafe dose in addition we can control the probability of correctly selecting the mtd under a parameter subspace where no toxicity probability lies in an interval bracketed by the target toxicity rate and an unacceptably high toxicity rate the so called indifference zone this frequentist property which is currently lacking in the conduct of dose finding trials in humans is appealing from a regulatory standpoint we give the general expressions of the se

relative error approximate versions of douglas rachford splitting and special cases of the admm we derive a new approximate version of the alternating direction method of multipliers admm which uses a relative error criterion the new version is somewhat restrictive and allows only one of the two subproblems to be minimized approximately but nevertheless covers commonly encountered special cases the derivation exploits the long established relationship between the admm and both the proximal point algorithm ppa and douglas rachford dr splitting for maximal monotone operators along with a relative error of the ppa due to solodov and svaiter in the course of analysis we also derive a version of dr splitting in which one operator may be evaluated approximately using a relative error criterion we computationally evaluate our method on several classes of test problems and find that it significantly outperforms several alternatives on one problem class

a step up multiple test procedure abstract we consider the problem of simultaneously testing k 2 hypotheses on parameters 1 k in a typical application the s may be a set of contrasts for instance a set of orthogonal contrasts among population means or a set of differences between k treatment means and a standard treatment mean it is assumed that least squares estimators 1 k are available that are jointly normally distributed with a common variance known up to a scalar namely the error variance 2 and a common known correlation an independent 2 distributed unbiased estimator of 2 is also assumed to be available we propose a step up multiple test procedure for this problem which tests the t statistics for the k hypotheses in order starting with the least significant one and continues as long as an acceptance occurs by contrast the step down approach which is usually used starts with the most significant and continues as long as a rejection occurs critical constants required by this step up procedure to co

local linear convergence of the alternating direction method of multipliers on quadratic or linear programs we introduce a novel matrix recurrence yielding a new spectral analysis of the local transient convergence behavior of the alternating direction method of multipliers admm for the particular case of a quadratic program or a linear program we identify a particular combination of vector iterates whose convergence can be analyzed via a spectral analysis the theory predicts that admm should go through up to four convergence regimes such as constant step convergence or linear convergence ending with the latter when close enough to the optimal solution if the optimal solution is unique and satisfies strict complementarity

step down multiple tests for comparing treatments with a control in unbalanced one way layouts we show how a well known multiple step down significance testing procedure for comparing treatments with a control in balanced one way layouts can be applied in unbalanced layouts unequal sample sizes for the treatments the method we describe has the advantage that it provides p values for each treatment versus control comparison that take account of the multiple step down testing nature of the procedure these joint p values can be used with any value of alpha the fixed type i family wise error rate bound that may be specified by the investigator to determine the p values it is necessary to compute a multivariate student t integral for which a computer program is available this procedure is more powerful than the step down bonferroni procedure of holm and the single step procedure of dunnett an example from the pharmaceutical literature is used to illustrate the procedure

a two stage stepwise estimation procedure this article proposes a two stage simultaneous confidence procedure for the comparisons of k pairs of population means without using multiplicity adjustment of more than two populations the proposed procedure can be broadly applied to parametric or nonparametric models it is robust and versatile because its derivation only utilizes a partitioning approach in conjunction with a bivariate adjustment without any assumption on the underlying distribution to elucidate the application the proposed procedure is intertwined with the estimation of the therapeutic window of a drug it provides confidence limits for the efficacy and the toxicity of the effective doses highest ineffective dose safe doses and lowest unsafe dose simultaneously such estimation information facilitates follow up studies in clinical trials as an illustrative example the new procedure is applied to analyze a data set on molecular cancer therapeutics regarding the apoptotic killing effects of different chemical compounds on two leukemia cell lines

multiple test procedures for dose finding the problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered a general framework for stepwise testing procedures that use contrasts among the dose level means is proposed using this framework several new procedures are derived these and some existing procedures including that of williams 1971 biometrics 27 103 117 1972 biometrics 28 519 531 are compared analytically and by an extensive simulation study for the normal theory balanced one way layout case it is pointed out that the procedures based on the so called step and basin contrasts proposed by ruberg 1989 journal of american statistical association 84 816 822 have excessively high type i familywise error rates fwes and hence they should not be used some findings of the simulation study are as follows for monotone dose mean configurations williams procedure and two step down test procedures based on helmert and linear contrasts offer the best performance for nonmonotone dose mean configurations the performance of williams procedure does degrade somewhat but the other two procedures are still the best for more complex designs a simple step down test procedure that uses any alpha level tests not necessarily t tests to compare each dose level with the zero dose level controls the fwe and is the only alternative available but its power is rather low especially under nonmonotone configurations step up procedures are generally dominated by step down procedures when the same contrasts are used although the differences are not great

step up multiple testing of parameters with unequally correlated estimates we consider the problem of simultaneously testing k or to 2 hypotheses on parameters theta 1 theta k using test statistics t 1 t k such that a specified familywise error rate alpha is achieved dunnett and tamhane 1992a proposed a step up multiple test procedure in which testing starts with the hypothesis corresponding to the least significant test statistic and proceeds towards the most significant stopping the first time a significant test result is obtained and rejecting the hypotheses corresponding to that and any remaining test statistics the parameter estimates used in the t statistics were assumed to be normally distributed with a common variance which was a known multiple of an unknown sigma 2 and known correlations which were equal in the present article we show how the procedure can be extended to include unequally correlated parameter estimates unequal correlations occur for example in experiments involving comparisons among treatment groups with unequal sample sizes we also compare the step up and step down multiple testing approaches and discuss applications to some biopharmaceutical testing problems

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

path independent integrals for computing stress intensity factors at sharp notches in elastic plates a set of path independent integrals is constructed for the calculation of the generalized stress intensity factors occurring in elastic plates having sharp re entrant corners or notches with stress free faces and subjected to mode i ii or iii type loading the mode i integral is then demonstrated to enjoy a reasonable degree of numerical path independence in a finite element analysis of a test problem having an exact solution finally this integral is used on the same problem in conjunction with a regularizing finite element procedure or superposition method the results indicate that sufficiently accurate estimates of these stress intensity factors for engineering purposes can be achieved with little computational effort

optimal designs for dose finding studies understanding and properly characterizing the dose response relationship is a fundamental step in the investigation of a new compound be it a herbicide or fertilizer a molecular entity an environmental toxin or an industrial chemical in this article we investigate the problem of deriving efficient designs for the estimation of target doses in the context of clinical dose finding we propose methods to determine the appropriate number and actual levels of the doses to be administered to patients as well as their relative sample size allocations more specifically we derive local optimal designs that minimize the asymptotic variance of the minimum effective dose estimate under a particular dose response model we investigate the small sample properties of these designs together with their sensitivity to a misspecification of the true parameter values and of the underlying dose response model through simulation finally we demonstrate that the designs derived for a fixed model are rather sensitive

identifying effective and or safe doses by stepwise confidence intervals for ratios typical randomized clinical dose finding studies consist of the comparison of several doses of a drug versus a placebo interest lies in estimating relevant doses among those under investigation for efficacy and safety variables such as the minimum effective dose or the maximum safe dose or estimating both doses simultaneously step down procedures have been proposed for comparing the standardized differences of the dose groups against placebo in this paper we consider the ratio of population means and propose stepwise confidence intervals for these ratios these confidence intervals do not require multiplicity adjustments and yield the same decisions as the associated test procedures in addition several power concepts are investigated within the present framework the results allow sample size determination in the design phase of a study for the probability of estimating correctly the dose of interest auxiliary results of a numerical study show the range of application of these methods

threshold estimation based on a p value framework in dose response and regression settings we use p values to identify the threshold level at which a regression function leaves its baseline value a problem motivated by applications in toxicological and pharmacological dose response studies and environmental statistics we study the problem in two sampling settings one where multiple responses can be obtained at a number of different covariate levels and the other the standard regression setting involving limited number of response values at each covariate our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value and then computing the potentially approximate p value of the test an estimate of the threshold is obtained by fitting a piecewise constant function with a single jump discontinuity known as a stump to these observed p values as they behave in markedly different ways on the two sides of the threshold the estimate is shown to be consistent and its finite sample properties are studied through simulations our approach is computationally simple and extends to the estimation of the baseline value of the regression function heteroscedastic errors and to time series it is illustrated on some real data applications