a unified bayesian semiparametric approach to assess discrimination ability in survival analysis the discriminatory ability of a marker for censored survival data is routinely assessed by the time dependent roc curve and the c index the time dependent roc curve evaluates the ability of a biomarker to predict whether a patient lives past a particular time t the c index measures the global concordance of the marker and the survival time regardless of the time point we propose a bayesian semiparametric approach to estimate these two measures the proposed estimators are based on the conditional distribution of the survival time given the biomarker and the empirical biomarker distribution the conditional distribution is estimated by a linear dependent dirichlet process mixture model the resulting roc curve is smooth as it is estimated by a mixture of parametric functions the proposed c index estimator is shown to be more efficient than the commonly used harrell s c index since it uses all pairs of data rather than only informative pairs the proposed estimators are evaluated through simulations and illustrated using a lung cancer dataset

on spectral multigrid methods for the time dependent navier stokes equations a new splitting scheme is proposed for the numerical solution of the time dependent incompressible navier stokes equations by spectral methods a staggered grid is used for the pressure improved intermediate boundary conditions are employed in the split step for the velocity and spectral multigrid techniques are used for the solution of the implicit equations

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

critical blowup and global existence numbers for a weakly coupled system of reaction diffusion equations let d rn be either all of rn or else a cone in rn whose vertex we may take to be at the origin without loss of generality let pi qj i 1 2 be nonnegative with 0 p1 q1 p2 q2 we consider the long time behavior of nonnegative solutions of the system u t delta u u p 1 v q 1 v t delta v u p 2 v q 2 s in d 0 with u0 v0 0 on d u v t x 0 0 0 t x u0 0 0 u0 0 e l d we obtain fujita type global existence global nonexistence theorems for s analogous to the classical result of fujita and others for the initial value problem for ut u up u x 0 u0 x 0 the principal result in the case d rn and p2q1 0 is that when p1 1 the system behaves like the single equation ut u up1vq1 with respect to fujita type blowup theorems whereas if p1 1 the behavior of the system is more complicated some of the results extend those of escobedo herrero when d rn and of levine when d is a cone these authors considered s in the case of p1 q2 0 an example of nonuniqueness is also given

estimating normal means with a dirichlet process prior abstract in this article the dirichlet process prior is used to provide a nonparametric bayesian estimate of a vector of normal means in the past there have been computational difficulties with this model this article solves the computational difficulties by developing a gibbs sampler algorithm the estimator developed in this article is then compared to parametric empirical bayes estimators peb and nonparametric empirical bayes estimators npeb in a monte carlo study the monte carlo study demonstrates that in some conditions the peb is better than the npeb and in other conditions the npeb is better than the peb the monte carlo study also shows that the estimator developed in this article produces estimates that are about as good as the peb when the peb is better and produces estimates that are as good as the npeb estimator when that method is better

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

multidomain local fourier method for pdes in complex geometries abstract a low communication parallel algorithm is developed for the solution of time dependent nonlinear pdes the parallelization is achieved by domain decomposition the discretization in time is performed via a third order semi implicit stiffly stable scheme the elemental solutions in the subdomains are constructed using a high order method with the local fourier basis lfb the continuity of the global solution is accomplished by a point wise matching of the local subsolutions on the interfaces the matching relations are derived in terms of the jumps on the interfaces the lfb method enables splitting a two dimensional problem with global coupling of the interface unknowns into a set of uncoupled one dimensional differential equations localization properties of an elliptic operator resulting from the discretization in time of a time dependent problem are utilized in order to simplify the matching relations in effect only local neighbor to neighbor communication between the processors becomes necessary the present method allows the treatment of problems in various complex geometries by the mapping of curvilinear domains into simpler rectangular or circular regions with subsequent matching of local solutions the operator with nonconstant coefficients obtained in the transformed domain is preconditioned by an appropriate constant coefficient operator easily inverted by the lfb the problem is then solved with spectral accuracy by a rapidly convergent conjugate gradient iteration

a rotational velocity correction projection method for unsteady incompressible magnetohydrodynamics equations abstract this paper provides a rotational velocity correction projection method for the unsteady incompressible magnetohydrodynamics mhd equations in this method the electrical field variable e was eliminated in this way the gauss s law can be conserved the velocity variable u and the pressure field p are solved by a rotational velocity correction projection method the optimal error estimates of the l 2 norm will be given finally we shall present some numerical results to confirm our analysis it shows that our method is effective and can conserve the gauss s law very well

using smoothed receiver operating characteristic curves to summarize and compare diagnostic systems abstract receiver operating characteristic roc curves are used for summarizing the performance of imperfect diagnostic systems especially in biomedical research these curves are also appropriate for summarizing the performance of a discriminant analysis but are under utilized by statisticians this article is illustrates the use of these curves for comparing competing diagnostic systems develops new estimation methods based on kernel density estimation and studies the statistical performance of the new method a transform of the roc curve is further suggested based on the idea of local population separation this graphic is quite generally useful for displaying the differences between two populations the methods are applied to a dataset comprising the results of seven diagnostics for predicting cancer activity on 353 patients the distributions of these diagnostics are not well modeled parametrically and so either completely nonparametric or kernel density estimation seems appropriate construct

computational study of subcritical response in flow past a circular cylinder flow past a circular cylinder is investigated in the subcritical regime below the onset of benard von karman vortex shedding at reynolds number re c similar or equal to 47 the transient response of infinitesimal perturbations is computed the domain requirements for obtaining converged results is discussed at length it is shown that energy amplification occurs as low as re 2 2 throughout much of the subcritical regime the maximum energy amplification increases approximately exponentially in the square of re reaching 6800 at re c the spatiotemporal structure of the optimal transient dynamics is shown to be transitory benard von karman vortex streets at re similar or equal to 42 the long time structure switches from exponentially increasing downstream to exponentially decaying downstream three dimensional computations show that two dimensional structures dominate the energy growth except at short times

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

spectral distributed lagrange multiplier method algorithm and benchmark tests we extend the formulation of the distributed lagrange multiplier dlm approach for particulate flows to high order methods within the spectral hp element framework we implement the rigid body motion constraint inside the particle via a penalty method the high order dlm method demonstrates spectral convergence rate i e discretization errors decrease exponentially as the order of spectral polynomials increases we provide detailed comparisons between the spectral dlm method direct numerical simulations and the force coupling method for a number of 2d and 3d benchmark flow problems we also validate the spectral dlm method with available experimental data for a transient problem the new dlm method can potentially be very effective in many moving body problems where a smaller number of grid points is required in comparison with low order methods

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a new domain decomposition method with overlapping patches for ultrascale simulations application to biological flows we address the failure in scalability of large scale parallel simulations that are based on semi implicit time stepping and hence on the solution of linear systems on thousands of processors we develop a general algorithmic framework based on domain decomposition that removes the scalability limitations and leads to optimal allocation of available computational resources it is a non intrusive approach as it does not require modification of existing codes specifically we present here a two stage domain decomposition method for the navier stokes equations that combines features of discontinuous and continuous galerkin formulations at the first stage the domain is subdivided into overlapping patches and within each patch a c 0 spectral element discretization second stage is employed solution within each patch is obtained separately by applying an efficient parallel solver proper inter patch boundary conditions are developed to provide solution continuity while a multilevel communicating interface mci is developed to provide efficient communication between the non overlapping groups of processors of each patch the overall strong scaling of the method depends on the number of patches and on the scalability of the standard solver within each patch this dual path to scalability provides great flexibility in balancing accuracy with parallel efficiency the accuracy of the method has been evaluated in solutions of steady and unsteady 3d flow problems including blood flow in the human intracranial arterial tree benchmarks on bluegene p cray xt5 and sun constellation linux cluster have demonstrated good performance on up to 96 000 cores solving up to 8 21b degrees of freedom in unsteady flow problem the proposed method is general and can be potentially used with other discretization methods or in other applications

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

bayesian semi parametric roc analysis this paper describes a semi parametric bayesian approach for estimating receiver operating characteristic roc curves based on mixtures of dirichlet process priors mdp we address difficulties in modelling the underlying distribution of screening scores due to non normality that may lead to incorrect choices of diagnostic cut offs and unreliable estimates of prevalence of the disease mdp is a robust tool for modelling non standard diagnostic distributions associated with imperfect classification of an underlying diseased population for example when a diagnostic test is not a gold standard for posterior computations we propose an efficient gibbs sampling framework based on a finite dimensional approximation to mdp we show using both simulated and real data sets that mdp modelling for roc curve estimation closely parallels the frequentist kernel density estimation kde approach

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

an integrated bayesian nonparametric approach for stochastic and variability orders in roc curve estimation an application to endometriosis diagnosis in estimating roc curves of multiple tests some a priori constraints may exist either between the healthy and diseased populations within a test or between tests within a population in this article we proposed an integrated modeling approach for roc curves that jointly accounts for stochastic and variability orders the stochastic order constrains the distributional centers of the diseased and healthy populations within a test while the variability order constrains the distributional spreads of the tests within each of the populations under a bayesian nonparametric framework we used features of the dirichlet process mixture to incorporate these order constraints in a natural way we applied the proposed approach to data from the physician reliability study that investigated the accuracy of diagnosing endometriosis using different clinical information to address the issue of no gold standard in the real data we used a sensitivity analysis approach that exploited diagnosis from a panel of experts

dealiasing techniques for high order spectral element methods on regular and irregular grids high order methods are becoming increasingly attractive in both academia and industry especially in the context of computational fluid dynamics however before they can be more widely adopted issues such as lack of robustness in terms of numerical stability need to be addressed particularly when treating industrial type problems where challenging geometries and a wide range of physical scales typically due to high reynolds numbers need to be taken into account one source of instability is aliasing effects which arise from the nonlinearity of the underlying problem in this work we detail two dealiasing strategies based on the concept of consistent integration the first uses a localised approach which is useful when the nonlinearities only arise in parts of the problem the second is based on the more traditional approach of using a higher quadrature the main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods thereby reducing aliasing driven instabilities we demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations where in the latter both volumetric and interface approximations must be considered we show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

a fourier legendre spectral element method in polar coordinates in this paper a new fourier legendre spectral element method based on the galerkin formulation is proposed to solve the poisson type equations in polar coordinates the 1 r singularity at r 0 is avoided by using gauss radau type quadrature points in order to break the time step restriction in the time dependent problems the clustering of collocation points near the pole is prevented through the technique of domain decomposition in the radial direction a number of poisson type equations subject to the dirichlet or neumann boundary condition are computed and compared with the results in literature which reveals a desirable result

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

nonparametric hierarchical bayes via sequential imputations we consider the empirical bayes estimation of a distribution using binary data via the dirichlet process let d denote a dirichlet process with being a finite measure on 0 1 instead of having direct samples from an unknown random distribution f from d we assume that only indirect binomial data are observable this paper presents a new interpretation of lo s formula and thereby relates the predictive density of the observations based on a dirichlet process model to likelihoods of much simpler models as a consequence the log likelihood surface as well as the maximum likelihood estimate of c 0 1 is found when the shape of is assumed known together with a formula for the fisher information evaluated at the estimate the sequential imputation method of kong liu and wong is recommended for overcoming computational difficulties commonly encountered in this area the related approximation formulas are provided an analysis of the tack data of beckett and diaconis which motivated this study is supplemented to illustrate our methods

nonparametric bayesian covariate adjusted estimation of the youden index a novel nonparametric regression model is developed for evaluating the covariate specific accuracy of a continuous biological marker accurately screening diseased from nondiseased individuals and correctly diagnosing disease stage are critically important to health care on several fronts including guiding recommendations about combinations of treatments and their intensities the accuracy of a continuous medical test or biomarker varies by the cutoff threshold c used to infer disease status accuracy can be measured by the probability of testing positive for diseased individuals the true positive probability or sensitivity se c of the test and the true negative probability specificity sp c of the test a commonly used summary measure of test accuracy is the youden index yi max se c sp c 1 c which is popular due in part to its ease of interpretation and relevance to population health research in addition clinical practitioners benefit from having an estimate of the optimal cutoff that maximizes sensitivity plus specificity available as a byproduct of estimating yi we develop a highly flexible nonparametric model to estimate yi and its associated optimal cutoff that can respond to unanticipated skewness multimodality and other complexities because data distributions are modeled using dependent dirichlet process mixtures important theoretical results on the support properties of the model are detailed inferences are available for the covariate specific youden index and its corresponding optimal cutoff threshold the value of our nonparametric regression model is illustrated using multiple simulation studies and data on the age specific accuracy of glucose as a biomarker of diabetes

a fast 3d poisson solver of arbitrary order accuracy we present a direct solver for the poisson and laplace equations in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the gibbs phenomenon and achieving any prescribed rate of convergence the algorithm requireso n3logn operations wherenis the number of grid points in each direction we show that our approach allows accurate treatment of singular cases which arise when the boundary function is discontinuous or incompatible with the differential equation

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

bayesian semiparametric roc curve estimation and disease diagnosis we develop a novel semiparametric modeling framework involving mixtures of polya trees for screening data with the dual purpose of diagnosing infection or disease status and of assessing the accuracy of continuous diagnostic measures in this framework we obtain i predictive probabilities of disease based on continuous diagnostic test outcomes in conjunction with other information including relevant covariates and results from one or more independent binary diagnostic tests an example would be the modeling of a serum enzyme linked immunosorbent assay elisa procedure for detecting antibodies to an infectious agent when used in conjunction with culture for antigen detection our second goal is to ii characterize measures of diagnostic performance of continuous tests by estimating receiver operating characteristic curves and area under the curve primarily when such extra information is available when true disease status is unknown parametric and nonparametric analyses require sufficient separation between the distributions of outcome values for the diseased and nondiseased populations however this overlap becomes less problematic when additional information in the form of either an informative prior that is based on real preferably data based scientific input or when additional information or both are available the additional information can be used to distinguish diseased from nondiseased individuals we present an example using simulated data that illustrates this point we also present an example involving data from an animal health survey for johne s disease where the performance of a serum elisa is evaluated using additional information obtained from fecal culture issues related to identifiability and partial identifiability are also discussed

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

a high order discontinuous galerkin fourier incompressible 3d navier stokes solver with rotating sliding meshes we present the development of a sliding mesh capability for an unsteady high order order 3 h p discontinuous galerkin solver for the three dimensional incompressible navier stokes equations a high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh through the use of curved boundary elements and mixed triangular quadrilateral meshes a second order stiffly stable method is used to discretise in time the arbitrary lagrangian eulerian form of the incompressible navier stokes equations spatial discretisation is provided by the symmetric interior penalty galerkin formulation with modal basis functions in the x y plane allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques spatial discretisation in the z direction is provided by a purely spectral method that uses fourier series and allows computation of spanwise periodic three dimensional flows the developed solver is shown to provide high order solutions second order in time convergence rates and spectral convergence when solving the incompressible navier stokes equations on meshes where fixed and rotating elements coexist in addition an exact implementation of the no slip boundary condition is included for curved edges circular arcs and naca 4 digit airfoils where analytic expressions for the geometry are used to compute the required metrics the solver capabilities are tested for a number of two dimensional problems governed by the incompressible navier stokes equations on static and rotating meshes the taylor vortex problem a static and rotating symmetric naca0015 airfoil and flows through three bladed cross flow turbines in addition three dimensional flow solutions are demonstrated for a three bladed cross flow turbine and a circular cylinder shadowed by a pitching naca0012 airfoil

a semilinear parabolic system with generalized wentzell boundary condition abstract in this paper we consider a weak coupled semilinear parabolic system with general wentzell boundary condition we prove the well posedness of the problem and derive different conditions in terms of the powers of the nonlinear terms under which the global solution exists and finite time blow up occurs

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

long time stability of high order multistep numerical schemes for two dimensional incompressible navier stokes equations the long time stability properties of a few multistep numerical schemes for the two dimensional incompressible navier stokes equations formulated in vorticity stream function are investigated in this article these semi implicit numerical schemes use a combination of explicit adams bashforth extrapolation for the nonlinear convection term and implicit adams moulton interpolation for the viscous diffusion term up to fourth order accuracy in time as a result only two poisson solvers are needed at each time step to achieve the desired temporal accuracy the fully discrete schemes with fourier pseudospectral approximation in space are analyzed in detail with the help of a priori analysis and aliasing error control techniques we prove uniform in time bounds for these high order schemes in both l 2 and h m norms for m ge 1 provided that the time step is bounded by a given constant such a long time stability is also demonstrated by the numerical experiments

analysis and application of fourier gegenbauer method to stiff differential equations the fourier gegenbauer fg method introduced by gottlieb shu solomonoff and vandeven icase report 92 4 hampton va 1992 is aimed at removing the gibbs phenomenon that is recovering the point values of a nonperiodic function from its fourier coefficients in this paper we discuss some numerical aspects of the fg method related to its em pseudospectral implementation in particular we analyze the behavior of the gegenbauer series with a moderate several hundred number of terms suitable for computations we also demonstrate the ability of the fg method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles bearing on the previous analysis we suggest a high order spectral fourier method for the solution of nonperiodic differential equations it includes a polynomial subtraction technique to accelerate the convergence of the fourier series and the fg algorithm to evaluate derivatives on the boundaries of nonperiodic functions the present hybrid fourier gegenbauer hfg method possesses better resolution properties than the original fg method the precision of this method is demonstrated by solving stiff elliptic problems with steep solutions

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

unsteady two dimensional flows in complex geometries comparative bifurcation studies with global eigenfunction expansions we present a bifurcation study of the incompressible navier stokes equations in a model complex geometry a spatially periodic array of cylinders in a channel the dynamics of the flow include a hopf bifurcation from steady to oscillatory flow at an approximate reynolds number r of 350 and the appearance of a second frequency at approximately r simeq 890 the multiple frequency dynamics include a substantial increase in spatial and temporal scales with reynolds number as compared with the simple limit cycle oscillation present close to r 350 numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions the first basis set is derived through principal factor analysis karhunen loeve expansion of snapshots from accurate direct spectral element numerical solutions of the navier stokes equations the second set is obtained from the eigenfunctions of the stokes operator for this geometry finally eigenfunctions are derived from a singular stokes operator i e the stokes operator modified to include a variable coefficient which vanishes at the domain boundaries truncated systems of sim 100 odes are obtained through projection of the navier stokes equations onto the basis sets and a comparative study of the resulting dynamical models is performed

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

a new parametric method based on s distributions for computing receiver operating characteristic curves for continuous diagnostic tests receiver operating characteristic roc curves provides a method for evaluating the performance of a diagnostic test these curves represent the true positive ratio that is the true positives among those affected by the disease as a function of the false positive ratio that is the false positives among the healthy corresponding to each possible value of the diagnostic variable when the diagnostic variable is continuous the corresponding roc curve is also continuous however estimation of such curve through the analysis of sample data yields a step line unless some assumption is made on the underlying distribution of the considered variable since the actual distribution of the diagnostic test is seldom known it is difficult to select an appropriate distribution for practical use data transformation may offer a solution but also may introduce a distortion on the evaluation of the diagnostic test in this paper we show that the distribution family known as the s distribution can be used to solve this problem the s distribution is defined as a differential equation in which the dependent variable is the cumulative this special form provides a highly flexible family of distributions that can be used as models for unknown distributions it has been shown that classical statistical distributions can be represented accurately as s distributions and that they occur in a definite subspace of the parameter space corresponding to the whole s distribution family consequently many other distributional forms that do not correspond to known distributions are provided by the s distribution this property can be used to model observed data for unknown distributions and is very useful in constructing parametric roc curves in those cases after fitting an s distribution to the observed samples of diseased and healthy populations roc curve computation is straightforward a roc curve can be considered as the solution of a differential equation in which the dependent variable is the ratio of true positives and the independent variable is the ratio of false positives this equation can be easily obtained from the s distributions fitted to observed data using these results we can compute pointwise confidence bands for the roc curve and the corresponding area under the curve we shall compare this approach with the empirical and the binormal methods for estimating a roc curve to show that the s distribution based method is a useful parametric procedure

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

on mass conservation and solvability of the discretized variable density zero mach navier stokes equations abstract this paper presents a study of a variable density zero mach navier stokes discretization and subtle difficulties associated with the pressure a consistent compatibility condition between the temperature and velocity is first derived it is shown that one can accurately and efficiently solve all primitive variables with the uniform order of accuracy of the specified discretization and time marching schemes it is shown that higher density ratios than those commonly reported are achievable and it is in part traced to the solvability of the pressure poisson equation numerical properties are assessed with the method of manufactured solutions and several classical testing cases are carried out to validate the code ranging from variable density mixing problems to premixed flames with forcing

modeling continuous diagnostic test data using approximate dirichlet process distributions there is now a large literature on the analysis of diagnostic test data in the absence of a gold standard test latent class analysis is most often used to estimate the prevalence of the condition of interest and the properties of the diagnostic tests when test results are measured on a continuous scale both parametric and nonparametric models have been proposed parametric methods such as the commonly used bi normal model may not fit the data well nonparametric methods developed to date have been relatively complex to apply in practice and their properties have not been carefully evaluated in the diagnostic testing context in this paper we propose a simple yet flexible bayesian nonparametric model which approximates a dirichlet process for continuous data we compare results from the nonparametric model with those from the bi normal model via simulations investigating both how much is lost in using a nonparametric model when the bi normal model is correct and how much can be gained in using a nonparametric model when normality does not hold we also carefully investigate the trade offs that occur between flexibility and identifiability of the model as different dirichlet process prior distributions are used motivated by an application to tuberculosis clustering we extend our nonparametric model to accommodate two additional dichotomous tests and proceed to analyze these data using both the continuous test alone as well as all three tests together copyright 2011 john wiley sons ltd

a fast poisson solver of arbitrary order accuracy in rectangular regions in this paper we propose a direct method for the solution of the poisson equation in rectangular regions it has an arbitrary order accuracy and low cpu requirements which makes it practical for treating large scale problems the method is based on a pseudospectral fourier approximation and a polynomial subtraction technique fast convergence of the fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions these functions have the same discontinuities at the corner points as the sought solution in addition to this they satisfy the laplace equation so that the subtraction procedure does not generate nonperiodic nonhomogeneous terms the solution of a boundary value problem is obtained in a series form in o n log n floating point operations where n2 is the number of grid nodes evaluating the solution at all n2 interior points requires o n2 log n operations

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

spectral element formulations on non conforming grids a comparative study of pointwise matching and integral projection methods pointwise matching pm and integral projection ip methods are two widely used techniques to extend the classical weak formulations to include non conforming grids we present spectral element formulations on polynomial p type and geometric h type non conforming grids using both the pm also known as the constrained approximation and ip also known as the mortar element methods we systematically compare the convergence characteristics of pm and ip methods for diffusion convection and convection diffusion equations consistency errors due to the non conforming formulations of the diffusion equation result in convergence problems for the pm method using the maximum rule both non conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values causing convergence problems due to the consistency errors however small physical diffusion in the convection diffusion equation eliminates these problems resulting in spectral convergence for both methods encouraged by this we present spectral element formulations for incompressible navier stokes equations using pm and ip methods on p type and h type non conforming grids and demonstrate spectral convergence for unsteady and steady test cases results for two dimensional lid driven cavity flow at re 1000 are also presented

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

a semilinear equation with generalized wentzell boundary condition abstract in this paper we consider a semilinear equation with a generalized wentzell boundary condition we prove the local well posedness of the problem and derive the conditions of the global existence of the solution and the conditions for finite time blow up we also derive an estimate for the blow up time

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

a spectral deferred correction method for incompressible flow with variable viscosity abstract this paper presents a semi implicit spectral deferred correction sdc method for incompressible navier stokes problems with variable viscosity and time dependent boundary conditions the proposed method integrates elements of velocity and pressure correction schemes which yields a simpler pressure handling and a smaller splitting error than the sdpc method of minion and saye j comput phys 375 797 822 2018 combined with the discontinuous galerkin spectral element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant as well as varying solution dependent viscosity compared to sdpc the present method yields a substantial improvement of accuracy and robustness against order reduction caused by time dependent boundary conditions

velocity correction schemes for the incompressible navier stokes equations in general coordinate systems this paper presents methods of including coordinate transformations into the solution of the incompressible navier stokes equations using the velocity correction scheme which is commonly used in the numerical solution of unsteady incompressible flows this is important when the transformation leads to symmetries that allow the use of more efficient numerical techniques like employing a fourier expansion to discretize a homogeneous direction two different approaches are presented in the first approach all the influence of the mapping is treated explicitly while in the second the mapping terms related to convection are treated explicitly with the pressure and viscous terms treated implicitly through numerical results we demonstrate how these methods maintain the accuracy of the underlying high order method and further apply the discretisation strategy to problems where mixed fourier spectral hp element discretisations can be applied thereby extending the usefulness of this discretisation technique

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

spectral elements on infinite domain a spectral element technique to approximate partial differential equations pdes on an infinite domain is examined the method is based on boyd s mapping of a semiinfinite interval to a finite interval and it is extended to a variational setting which allows for an implementation using a spectral element method by extending the method to a variational form a straightforward implementation allows for high order approximations over an infinite computational domain methods to find a high order approximation have been proposed by funaro maday mavriplis and many others the proposed methods all have drawbacks including restrictions on the behavior of the approximation at infinity because the proposed method is based on a mapping to a finite domain a much wider variety of boundary conditions can be enforced on the approximation

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

bayesian curve fitting using multivariate normal mixtures summary problems of regression smoothing and curve fitting are addressed via predictive inference in a flexible class of mixture models multidimensional density estimation using dirichlet mixture models provides the theoretical basis for semi parametric regression methods in which fitted regression functions may be deduced as means of conditional predictive distributions these bayesian regression functions have features similar to generalised kernel regression estimates but the formal analysis addresses problems of multivariate smoothing parameter estimation and the assessment of uncertainties about regression functions naturally computations are based on multidimensional versions of existing markov chain simulation analysis of univariate dirichlet mixture models

smooth non parametric receiver operating characteristic roc curves for continuous diagnostic tests receiver operating characteristic roc curves have seen increasing use to assess the accuracy of diagnostic tests that yield continuous test results we can calculate a fully non parametric roc curve from the empirical false positive rate fpr and true positive rate tpr at each possible decision threshold but it may be quite jagged we can obtain a smooth roc curve by directly fitting a parametric model for example binormal or bilogistic to the actual test results but substantial lack of fit may result if the distributional assumptions are not valid a recently proposed alternative algorithm labroc4 is insensitive to such departures but is not fully flexible in that the form of the resulting roc curve is restricted to be binormal we propose a smooth non parametric roc curve derived from kernel density estimates of the two test result distributions we obtain pointwise standard errors for the estimated tpr at each fpr and use them to construct pointwise confidence intervals for the roc curve using a logit transformation we also obtain standard errors for the estimated tpr and fpr at given thresholds and use them to construct confidence rectangles in fpr tpr space that correspond to a specific threshold we adapt existing methods for the unsmoothed non parametric roc curve to obtain the area under the roc curve and its standard error we also give partial areas and corresponding standard errors we have created a fortran algorithm roc rol available upon request we compare roc curves and areas obtained by applying our methods and its competitors to two data sets one of which is fit well by parametric methods and labroc4 the other of which is not 1997 john wiley sons ltd

solution of time dependent diffusion equations with variable coefficients using multiwavelets a new numerical algorithm is developed for the solution of time dependent differential equations of diffusion type it allows for an accurate and efficient treatment of multidimensional problems with variable coefficients nonlinearities and general boundary conditions for space discretization we use the multiwavelet bases introduced by alpert 1993 siam j math anal 24 246 262 and then applied to the representation of differential operators and functions of operators presented by alpert beylkin and vozovoi representation of operators in the multiwavelet basis in preparation an important advantage of multiwavelet basis functions is the fact that they are supported only on non overlapping subdomains thus multiwavelet bases are attractive for solving problems in finite non periodic domains boundary conditions are imposed with a penalty technique of hesthaven and gottlieb 1996 siam j sci comput 579 612 which can be used to impose rather general boundary conditions the penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem this penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains consequently the matrices representing the differential operators on the finest scale have block diagonal structure for a fixed order of multiwavelets i e a fixed number of vanishing moments the computational complexity of the present algorithm is proportional to the number of subdomains the time discretization method of beylkin keiser and vozovoi 1998 pam report 347 is used in view of its favorable stability properties numerical results are presented for evolution equations with variable coefficients in one and two dimensions