invertibility of irreducible matrices abstract we obtain new sufficient conditions for invertibility of an irreducible complex matrix remarks are also given on eigenvalues and the associated eigenvectors that lie on the boundary of various spectrum inclusion regions of an irreducible matrix our results extend strengthen or clarify the recent work of brualdi brualdi and mellendorff farid solov ev and zhang and gu

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

design and analysis of phase i clinical trials the phase i clinical trial is a study intended to estimate the so called maximum tolerable dose mtd of a new drug although there exists more or less a standard type of design for such trials its development has been largely ad hoc as usually implemented the trial design has no intrinsic property that provides a generally satisfactory basis for estimation of the mtd in this paper the standard design and several simple alternatives are compared with regard to the conservativeness of the design and with regard to point and interval estimation of an mtd 33rd percentile with small sample sizes using a markov chain representation we found several designs to be nearly as conservative as the standard design in terms of the proportion of patients entered at higher dose levels in monte carlo simulations two two stage designs are found to provide reduced bias in maximum likelihood estimation of the mtd in less than ideal dose response settings of the three methods considered for determining confidence intervals the delta method a method based on fieller s theorem and a likelihood ratio method none was able to provide both usefully narrow intervals and coverage probabilities close to nominal

the use of a canonical form in the construction of locally optimal designs for non linear problems optimal experimental designs for non linear problems depend on the values of the underlying undknown parameters in the model for various reasons there is interest in providing explicit formulae for the optimal designs as a function of the unknown parameters this paper shows that for a certain class of generalized linear models the problem can be reduced to a canonical form this simplifies the underlying problem and designs are constructed for several contexts with a single variable using geometric and other arguments

discontinuous galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales an analysis of a multiscale symmetric interior penalty discontinuous galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed this new method first described in a abdulle c r acad sci paris ser i 346 2008 is based on numerical homogenization it allows to significantly reduce the computational cost of a fine scale discontinuous galerkin method by probing the fine scale data on sampling domains within a macroscopic partition of the computational domain macroscopic numerical fluxes an essential ingredient of discontinuous galerkin finite elements can be recovered from the computation on the sampling domains with negligible computation overhead fully discrete a priori error bounds are derived in the l 2 and h 1 norms

spectral inequalities for matrix exponentials abstract this note generalizes an inequality of bernstein as follows if c is an n n complex matrix and c k is the k th compound of c 1 k n n n k and if the eigenvalues of c k are labeled in order of decreasing magnitude 1 c k 2 c k n c k define the partial trace tr k i c by tr k i c h 1 i h c k i 1 n then for any complex n n matrix a tr k i e a e a tr k i e a a i 1 n with equality if a is normal or k n a spectral inequality of k fan is also generalized through the use of compound matrices

the high road to an exponential formula a conjectured exponential formula is proved using a recent result of klyachko linear algebra appl 319 2000 37 then the exponential formula is applied to confirm the equality of two types of spectral triples used to describe all homogeneous eigenvalue inequalities for sum of hermitian matrices and product of positive definite matrices

a new certification framework for the port reduced static condensation reduced basis element method abstract in this paper we introduce a new certification framework for the port reduced static condensation reduced basis element pr scrbe method which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides in an offline computational stage we construct a library of interoperable parametrized reference components in the subsequent online stage we instantiate and connect the components at the interfaces ports to form a system of components to compute a truth finite element approximation of the say coercive elliptic partial differential equation on the component based system we use a domain decomposition approach for an efficient simulation we employ two different types of model reduction a reduced basis rb approximation within the interior of the component huynh et al 2013 and empirical port reduction eftang and patera 2013 on the ports where the components connect we demonstrate the well posedness of the pr scrbe approximation and introduce a new certification framework to assess the quality of the port reduction we use conservative fluxes we adapt the standard estimators from rb methods to the scrbe setting to derive an a posteriori error estimator for the rb error contribution in order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for pr scrbe we adapt techniques from multi scale methods and component mode synthesis finally we prove that the effectivity of the derived estimator can be bounded we provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational online costs

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

discontinuous galerkin finite element heterogeneous multiscale method for advection diffusion problems with multiple scales a discontinuous galerkin finite element heterogeneous multiscale method is proposed for advection diffusion problems with highly oscillatory coefficients the method is based on a coupling of a discontinuous galerkin discretization for an effective advection diffusion problem on a macroscopic mesh whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element the computational work involved is independent of the high oscillations in the problem at the smallest scale the stability of our method depending on both macro and micro mesh sizes is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data fully discrete a priori error bounds are derived for locally periodic data numerical experiments confirm the theoretical error estimates

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

a reduced basis localized orthogonal decomposition in this work we combine the framework of the reduced basis method rb with the framework of the localized orthogonal decomposition lod in order to solve parametrized elliptic multiscale problems the idea of the lod is to split a high dimensional finite element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information the low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems however for parameter dependent multiscale problems the local basis has to be computed repeatedly for each choice of the parameter to overcome this issue we propose an rb approach to compute in an offline stage lod for suitable representative parameters the online solution of the multiscale problems can then be obtained in a coarse space thanks to the lod decomposition and for an arbitrary value of the parameters thanks to a suitable interpolation of the selected rb the online rb lod has a basis with local support and leads to sparse systems applications of the strategy to both linear and nonlinear problems are given

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

the spectrum of a hermitian matrix sum abstract an announcement by lidskii b v lidskii functional anal appl 10 1982 76 77 russian 139 140 english claimed to establish the explicit description of the spectrum of a hermitian matrix sum in the form conjectured by horn a horn pacific j math 16 1962 225 241 but no supporting proof has been published this paper begins an analysis of the claim and is the first step towards bridging the distance between the elementary methods of horn s 1962 paper and the partial solution of the same problem using noncommutative harmonic analysis by dooley et al i dooley r repka n wildberger lin mult alg 36 1993 79 102 the methods of horn lidskii lead to combinatorial issues of independent interest

an extension of the continual reassessment method using decision theory the primary goal of a phase i trial is to find the maximally tolerated dose mtd of a treatment the mtd is usually defined in terms of a tolerable probability q of toxicity our objective is to find the highest dose with toxicity risk that does not exceed q a criterion that is often desired in designing phase i trials this criterion differs from that of finding the dose with toxicity risk closest to q that is used in methods such as the continual reassessment method we use the theory of decision processes to find optimal sequential designs that maximize the expected number of patients within the trial allocated to the highest dose with toxicity not exceeding q among the doses under consideration the proposed method is very general in the sense that criteria other than the one considered here can be optimized and that optimal dose assignment can be defined in terms of patients within or outside the trial it includes as an important special case the continual reassessment method numerical study indicates the strategy compares favourably with other phase i designs

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

a short and versatile finite element multiscale code for homogenization problems we describe a multiscale finite element fe solver for elliptic or parabolic problems with highly oscillating coefficients based on recent developments of the so called heterogeneous multiscale method hmm the algorithm relies on coupled macro and microsolvers the framework of the hmm allows to design a code whose structure follows the classical finite elements implementation at the macro level to account for the fine scales of the problem elementwise numerical integration is replaced by micro fe methods on sampling domains we discuss a short and flexible fe implementation of the multiscale algorithm which can accommodate simplicial or quadrilateral fe and various coupling conditions for the constrained micro simulations extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

adaptive reduced basis finite element heterogeneous multiscale method an adaptive reduced basis finite element heterogeneous multiscale method rb fe hmm is proposed for elliptic problems with multiple scales the multiscale method is based on the rb fe hmm introduced in a abdulle y bai reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems j comput phys 231 21 2012 7014 7036 it couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains unlike classical numerical homogenization methods the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions the reduced basis obtained by a greedy algorithm in an offline stage in this paper we present a residual based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest for both type of adaptive strategies rigorous a posteriori error estimates are derived and corresponding error estimators are proposed in contrast to the adaptive finite element heterogeneous multiscale method fe hmm there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement up to an offline preliminary stage the rb fe hmm has the same computational complexity as a standard adaptive fem for the effective problem two and three dimensional numerical experiments confirm the efficiency of the rb fe hmm and illustrate the improvements compared to the adaptive fe hmm c 2013 elsevier b v all rights reserved

experimental design for binary data abstract models for binary data are usually such that the information matrix depends on the unknown parameters thus the standard criteria for optimality in regression experiments cannot be applied without modification methods of going around this difficulty include the use of initial point estimates sequential methods and bayesian analysis this article is mainly concerned with the robustness and the number of design points for methods involving initial estimates and for sequential methods in a small number of stages a final section discusses the criterion of constant information for models involving one or two parameters and summarizes recent results in this field

optimal design via curve fitting of monte carlo experiments abstract this article explores numerical methods for stochastic optimization with special attention to bayesian design problems a common and challenging situation occurs when the objective function in bayesian applications the expected utility is very expensive to evaluate perhaps because it requires integration over a space of very large dimensionality our goal is to explore a class of optimization algorithms designed to gain efficiency in such situations by exploiting smoothness of the expected utility surface and borrowing information from neighboring design points the central idea is that of implementing stochastic optimization by curve fitting of monte carlo samples this is done by simulating draws from the joint parameter sample space and evaluating the observed utilities fitting a smooth surface through these simulated points serves as estimate for the expected utility surface the optimal design can then be found deterministically in this article we introduce a general algorithm for cu

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

the localized reduced basis multiscale method for two phase flows in porous media summary in this work we propose a novel model order reduction approach for two phase flow in porous media by introducing a global pressure formulation in which the total mobility which realizes the coupling between saturation and pressure is regarded as a parameter to the pressure equation in our approach the mobility itself is an optimal fit of mobility profiles that are precomputed using the time of flight for the initial saturation using this formulation and the localized reduced basis multiscale method we obtain a low dimensional surrogate of the high dimensional pressure equation by applying ideas from model order reduction for parametrized partial differential equations we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two phase flow simulation which is thereby largely accelerated usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online steps to reach an ideal runtime at acceptable error increase for the two phase flow simulation this is achieved by constructing reduced dimensional local spaces that lead to a non conforming global approximation space as one example for a coupling on the global space we introduce a discontinuous galerkin scheme copyright 2014 john wiley sons ltd

bayesian design for dose response curves with penalized risk this paper considers some bayesian design problems in quantal response analysis an experimenter must choose a set of dose levels and number of independent observations to take at these levels subject to a total sample size in order to estimate some characteristic phi e g ed50 of a tolerance distribution f theta where theta is the vector of unknown parameters it is shown for the logistic model that the sampling variability of the posterior variance of phi is such that the predicted posterior variance alone is an undesirable criterion for design selection a family of penalty functions is introduced that penalizes any excess in the posterior variance over the expected or predicted variance and protects against unexpected outcomes the goal is to find a design that avoids experimental results with little information at the expense of a small sacrifice in the bayes risk numerical results indicate that the chance of an extreme posterior variance can be reduced by sacrificing a small amount of posterior risk

adaptive finite element heterogeneous multiscale method for homogenization problems in this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method unlike standard finite element methods our discretization scheme relies on macro and microfinite elements the desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data as the macroscopic data such as the macroscopic diffusion tensor are not available beforehand appropriate error indicators have to be defined for designing adaptive methods we show that such indicators based only on the available macro and microsolutions used to compute the actual macrosolution can be defined allowing for a macroscopic mesh refinement strategy which is both reliable and efficient the corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm in the case of a uniformly oscillating tensor we recover the standard residual based a posteriori error estimate for the finite element method applied to the homogenized problem numerical experiments confirm the efficiency and reliability of the adaptive multiscale method

the numerical range of derivations abstract let p q n be integers satisfying 1 p q n the p q numerical range of an n n complex matrix a is defined by w p q a e p uau q u unitary where for an n n complex matrix x x q denotes its q q leading principal submatrix and e p x q denotes the p th elementary symmetric function of the eigenvalues of x q when 1 p q the set reduces to the classical numerical range of a which is well known to be convex many authors have used the concept of classical numerical range to study different classes of matrices in this note we extend the results to the generalized cases besides obtaining new results we collect existing ones and give alternative proofs for some of them we also study the p q numerical radius of a defined by r p q a max w p q a

completion of partial matrices to contractions an n by m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction necessary and sufficient condition are given for the pattern of specified entries such that any n by m partial contraction with this pattern may be completed to a full n by m contraction

bayesian model averaging continual reassessment method in phase i clinical trials the continual reassessment method crm is a popular dose finding design for phase i clinical trials this method requires that practitioners prespecify the toxicity probability at each dose such prespecification can be arbitrary and different specifications of toxicity probabilities may lead to very different design properties to overcome the arbitrariness and further enhance the robustness of the design we propose using multiple parallel crm models each with a different set of prespecified toxicity probabilities in the bayesian paradigm we assign a discrete probability mass to each crm model as the prior model probability the posterior probabilities of toxicity can be estimated by the bayesian model averaging bma approach dose escalation or deescalation is determined by comparing the target toxicity rate and the bma estimates of the dose toxicity probabilities we examine the properties of the bma crm approach through extensive simulation studies and also compare this new method and its vari

reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems a new finite element method for the efficient discretization of elliptic homogenization problems is proposed these problems characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed recently the finite element heterogeneous multiscale method fe hmm has been proposed for such problems based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh departing from the approach used in the fe hmm we show that interpolation techniques based on the reduced basis methodology an offline online strategy allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions this new method called the reduced basis finite element heterogeneous multiscale method rb fe hmm is significantly more efficient than the fe hmm for high order macroscopic discretizations and for three dimensional problems when the repeated computation of micro problems over the whole computational domain is expensive a priori error estimates of the rb fe hmm are derived numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

bayesian dose finding in phase i ii clinical trials using toxicity and efficacy odds ratios a bayesian adaptive design is proposed for dose finding in phase i ii clinical trials to incorporate the bivariate outcomes toxicity and efficacy of a new treatment without specifying any parametric functional form for the drug dose response curve we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy after observing all the responses of each cohort of patients the dosage for the next cohort is escalated deescalated or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities a novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes we conduct simulation studies to evaluate the operating characteristics of the proposed method under various scenarios the new bayesian design based on the toxicity efficacy odds ratio trade offs exhibits good properties and treats most patients at the desirable dose levels the method is illustrated with a real trial design for a breast medical oncology study

fast computation of multiphase flow in porous media by implicit discontinuous galerkin schemes with optimal ordering of elements we present a family of implicit discontinuous galerkin schemes for purely advective multiphase flow in porous media in the absence of gravity and capillary forces to advance the solution one time step one must solve a discrete system of nonlinear equations by reordering the grid cells the nonlinear system can be shown to have a lower triangular block structure where each block corresponds to the degrees of freedom in a single or a small number of cells to reorder the system we view the grid cells and the fluxes over cell interfaces as vertices and edges in a directed graph and use a standard topological sorting algorithm then the global system can be computed by processing the blocks sequentially using a standard newton raphson algorithm for the degrees of freedom in each block decoupling the system offers greater control over the nonlinear solution procedure and reduces the computational costs memory requirements and complexity of the scheme significantly in particular the first order version of the method may be at least as efficient as modern streamline methods when accuracy requirements or the dynamics of the flow allow for large implicit time steps

numerical ranges of principal submatrices abstract it is shown that the ratio of the area of the convex hull of the fields of values of the n 1 by n 1 principal submatrices of an n by n matrix a to the area of the field of values of a is bounded below by a function of n which approaches 1 as n approaches since this convex hull is necessarily contained in the field of values of a an interpretation is that asymptotically in the dimension the field of any given matrix is filled up by the fields of the submatrices collectively some new inequalities for the eigenvalues of principal submatrices of hermitian matrices which are not implied by interlacing are employed

bayesian inference on protective antibody levels using case control data in the study of immune responses to infectious pathogens the minimum protective antibody concentration mpac is a quantity of great interest we use case control data to estimate the posterior distribution of the conditional risk of disease given a lower bound on antibody concentration in an at risk subject the concentration bound beyond which there is high credibility that infection risk is zero or nearly so is a candidate for the mpac a very simple gibbs sampling procedure that permits inference on the risk of disease given antibody level is presented in problems involving small numbers of patients the procedure is shown to have favorable accuracy and robustness to choice misspecification of priors frequentist evaluation indicates good coverage probabilities of credibility intervals for antibody dependent risk and rules for estimation of the mpac are illustrated with epidemiological data

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

eigenvalue inequalities for products of matrix exponentials motivated by models from stochastic population biology and statistical mechanics we prove new inequalities of the form cp eae 2 cp eatr where a and b are n x n complex matrices 1 n 00 and cp is a real valued continuous function of the eigenvalues of its matrix argument for example if a is essentially nonnegative b is diagonal real and cp is the spectral radius then holds if in addition a is irreducible and b has at least two different diagonal elements then the inequality is strict the proof uses kingman s theorem on the logconvexity of the spectral radius lie s product formula and perturbation theory we conclude with conjectures

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

an atlas for tridiagonal isospectral manifolds abstract let t be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix with simple spectrum we introduce bidiagonal coordinates charts defined on open dense domains forming an explicit atlas for t in contrast to the standard inverse variables consisting of eigenvalues and norming constants every matrix in t now lies in the interior of some chart domain we provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics both for continuous and discrete time

minimax problems on grassmann manifolds sums of eigenvalues courbures principales et energies de hartree fock grassmanniens des domaines de forme quadratique cycles de schubert le plongement de plucker fonctions de morse quadratiques caracterisation minimax de sommes de valeurs propres

quadratic forms and the graph isomorphism problem abstract we relate the graph isomorphism problem to the classical problem of equivalence of integer quadratic forms we give some necessary number theoretic conditions that the isomorphic graphs have to satisfy

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

consistency of continual reassessment method under model misspecification summary this paper studies large sample behaviour of the continual reassessment method used to determine the target dose levels in phase i clinical trials a sequential maximum likelihood procedure is investigated for local estimation of a given targeted toxicity level consistency and asymptotic normality for the maximum likelihood estimates are established under model misspecification and it is shown that in general the recommended dose level will converge to the target level there are however situations in which the recommended dose may converge to a level close to the target level but not necessarily the closest this contrasts with the conclusions of o quigley pepe fisher 1990 and we describe this situation as well as providing the conditions under which convergence is assured

a multiscale two point flux approximation method a large number of multiscale finite volume methods have been developed over the past decade to compute conservative approximations to multiphase flow problems in heterogeneous porous media in particular several iterative and algebraic multiscale frameworks that seek to reduce the fine scale residual towards machine precision have been presented common for all such methods is that they rely on a compatible primal dual coarse partition which makes it challenging to extend them to stratigraphic and unstructured grids herein we propose a general idea for how one can formulate multiscale finite volume methods using only a primal coarse partition to this end we use two key ingredients that are computed numerically i elementary functions that correspond to flow solutions used in transmissibility upscaling and ii partition of unity functions used to combine elementary functions into basis functions we exemplify the idea by deriving a multiscale two point flux approximation mstpfa method which is robust with regards to strong heterogeneities in the permeability field and can easily handle general grids with unstructured fine and coarse scale connections the method can easily be adapted to arbitrary levels of coarsening and can be used both as a standalone solver and as a preconditioner several numerical experiments are presented to demonstrate that the mstpfa method can be used to solve elliptic pressure problems on a wide variety of geological models in a robust and efficient manner

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

optimal observation times in experimental epidemic processes this article describes a method for choosing observation times for stochastic processes to maximise the expected information about their parameters two commonly used models for epidemiological processes are considered a simple death process and a susceptible infected si epidemic process with dual sources for infection spreading within and from outwith the population the search for the optimal design uses bayesian computational methods to explore the joint parameter data design space combined with a method known as moment closure to approximate the likelihood to make the acceptance step efficient for the processes considered a small number of optimally chosen observations are shown to yield almost as much information as much more intensively observed schemes that are commonly used in epidemiological experiments analysis of the simple death process allows a comparison between the full bayesian approach and locally optimal designs around a point estimate from the prior based on asymptotic results the robustness of the approach to misspecified priors is demonstrated for the si epidemic process for which the computational intractability of the likelihood precludes locally optimal designs we show that optimal designs derived by the bayesian approach are similar for observational studies of a single epidemic and for studies involving replicated epidemics in independent subpopulations different optima result however when the objective is to maximise the gain in information based on informative and non informative priors this has implications when an experiment is designed to convince a naive or sceptical observer rather than consolidate the belief of an informed observer some extensions to the methods including the selection of information criteria and extension to other epidemic processes with transition probabilities are briefly addressed

a counterexample connected with gersgorin s theorem abstract a conjectured generalization of gersgorin s eigenvalue location theorem is shown to be false

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

two remarks on matrix exponentials it is shown that for two square matrices a and b with algebraic elements eaeb ebea if and only if ab ba moreover the in some sense best possible perturbation inequality ea eb 2 ea 2 e a b 2 1 for a normal matrix a is proven

designs for phase i clinical trials with multiple courses of subjects at different doses the goal of this article is to provide a new design framework and its corresponding estimation for phase i trials existing phase i designs assign each subject to one dose level based on responses from previous subjects yet it is possible that subjects with neither toxicity nor efficacy responses can be treated at higher dose levels and their subsequent responses to higher doses will provide more information in addition for some trials it might be possible to obtain multiple responses repeated measures from a subject at different dose levels in this article a nonparametric estimation method is developed for such studies we also explore how the designs of multiple doses per subject can be implemented to improve design efficiency the gain of efficiency from single dose per subject to multiple doses per subject is evaluated for several scenarios our numerical study shows that using multiple doses per subject and the proposed estimation method together increases the efficiency substantially

discontinuous galerkin approximation of two phase flows in heterogeneous porous media with discontinuous capillary pressures we design and investigate a sequential discontinuous galerkin method to approximate two phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures the nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation an accurate reconstruction of the total velocity is considered in the raviart thomas nedelec finite element spaces together with diffusivity dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities the proposed method is assessed on one dimensional test cases exhibiting rough solutions degeneracies and capillary barriers stable and accurate solutions are obtained without limiters

analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems a fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed in contrast to previous results obtained for such problems in dimension d leq2 for the h 1 norm and for a semi discrete formulation w e p ming and p zhang j amer math soc 18 2005 no 1 121 156 we obtain optimal convergence results for dimension d leq3 and for a fully discrete method which takes into account the microscale discretization in addition our results are also valid for quadrilateral finite elements optimal a priori error estimates are obtained for the h 1 and l 2 norms improved estimates are obtained for the resonance error and the newton method used to compute the solution is shown to converge numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10