principal stratification analysis using principal scores summary practitioners are interested in not only the average causal effect of a treatment on the outcome but also the underlying causal mechanism in the presence of an intermediate variable between the treatment and outcome however in many cases we cannot randomize the intermediate variable resulting in sample selection problems even in randomized experiments therefore we view randomized experiments with intermediate variables as semiobservational studies in parallel with the analysis of observational studies we provide a theoretical foundation for conducting objective causal inference with an intermediate variable under the principal stratification framework with principal strata defined as the joint potential values of the intermediate variable our strategy constructs weighted samples based on principal scores defined as the conditional probabilities of the latent principal strata given covariates without access to any outcome data this principal stratification analysis yields robust causal inference without relying on any model assumptions on the outcome distributions we also propose approaches to conducting sensitivity analysis for violations of the ignorability and monotonicity assumptions the very crucial but untestable identification assumptions in our theory when the assumptions required by the classical instrumental variable analysis cannot be justified by background knowledge or cannot be made because of scientific questions of interest our strategy serves as a useful alternative tool to deal with intermediate variables we illustrate our methodologies by using two real data examples and find scientifically meaningful conclusions

incremental multigrid strategy for the fluid dynamic equations abstract this paper provides a novel incremental multigrid strategy for the equations of fluid dynamics the time dependent governing equations are discretized in time by means of a two level implicit euler scheme and linearized using taylor series and the incremental delta form of beam and warming the coefficients and the right hand side of the resulting linear systems are evaluated always at the finest grid level whereas the delta unknowns are computed approximately by a single relaxation sweep on a sequence of coarser meshes at every grid level the computed deltas are interpolated up to the finest grid level and used to update the solution as well as the coefficients and the right hand side of the linear systems this process is repeated sweeping all grid levels successively until a satisfactory convergence criterion is met the validity of the proposed approach is demonstrated by solving a simple linear problem and the vorticity stream function navier stokes equations using line relaxation methods as smoothers and the lambda formulation euler equations in conjunction with a simple explicit smoother in all cases the proposed multigrid strategy provides a considerable efficiency gain over the corresponding single grid methods

application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

edges stars superedges and chains abstract several possibilities for the reduction of indirect addressing operations within edge based solvers are discussed it is shown that indirect addressing operations may be reduced by a factor in excess of 1 4 if edges are grouped and linked together tests of benchmark loops on two common engineering computer platforms reveal that cpu saving factors ranging from 1 3 to 3 5 may be realized when switching from edge based to superedge or chain based loops

multigrid solution of the two dimensional euler equations on unstructured triangular meshes a multigrid algorithm has been developed to accelerate the convergence of the euler equations to a steady state on unstructured triangular meshes the method operates on a sequence of coarse and fine unstructured meshes and assumes no relation exists between the various meshes a tree search algorithm is used to efficiently identify regions of overlap between coarse and fine grid cells the multigrid method is applied to two different discretizations of the euler equations and results about multi element airfoils are presented for both discretization schemes convergence is accelerated by an order of magnitude resulting in convergence rates comparable to those obtained with structured multigrid algorithms

principal causal effect identification and surrogate end point evaluation by multiple trials summary principal stratification is a causal framework to analyse randomized experiments with a post treatment variable between the treatment and end point variables because the principal strata defined by the potential outcomes of the post treatment variable are not observable we generally cannot identify the causal effects within principal strata motivated by a real data set of phase iii adjuvant colon cancer clinical trials we propose approaches to identifying and estimating the principal causal effects via multiple trials for the identifiability we remove the commonly used exclusion restriction assumption by stipulating that the principal causal effects are homogeneous across these trials to remove another commonly used monotonicity assumption we give a necessary condition for the local identifiability which requires at least three trials applying our approaches to the data from adjuvant colon cancer clinical trials we find that the commonly used monotonicity assumption is untenable and disease free survival with 3 year follow up is a valid surrogate end point for overall survival with 5 year follow up which satisfies both causal necessity and causal sufficiency we also propose a sensitivity analysis approach based on bayesian hierarchical models to investigate the effect of the deviation from the homogeneity assumption

optimization of convergence acceleration in multigrid numerical solutions of conductive convective problems the present work investigates the existence of optimal algorithm parameters for multigrid numerical solutions of a two dimensional steady state conductive convective problem the velocity field inside the rectangular domain and the temperature distribution at its four boundaries are known and kept constant the numerical method includes finite volume discretization and weighted upstream differencing scheme wuds interpolation on structured orthogonal and regular meshes multigrid is implemented according to the correction storage cs formulation minimum computational effort is sought as a function of control volume peclet number different numbers of grids number of smoothing sweeps in each level and distinct combinations of iterative solution algorithm

linearized stability of partial differential equations with application to stabilization of the kuramoto sivashinsky equation linearization is a useful tool for analyzing the stability of nonlinear differential equations unfortunately the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations general results giving conditions for when stability or instability of the linearized equation implies the same for the nonlinear equation are given here these results are applied to stability and stabilization of the kuramoto sivashinsky equation a nonlinear partial differential equation that models reaction diffusion systems the stability of the equilibrium solutions depends on the value of a positive parameter nu it is shown that if nu 1 then the set of constant equilibrium solutions is globally asymptotically stable if nu 1 then the equilibria are unstable it is also shown that stabilizing the linearized equation implies local exponential stability of the equation stabilization of the kuramoto sivashinsky equation using a sin

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

inertial manifolds the non self adjoint case abstract in contrast with the existing theories of inertial manifolds which are based on the self adjoint assumption of the principal differential operator in this paper we show that for general dissipative evolutionary systems described by semilinear parabolic equations with principal differential operator being sectorial and having compact resolvent there exists an inertial manifold provided that certain gap conditions hold we also show that by using an elliptic regularization this theory can be extended to a class of kdv equations where the principal differential operator is not sectorial

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

nonlinear feedback control of highly manoeuvrable aircraft abstract this paper describes the application of nonlinear quadratic regulator nlqr theory to the design of control laws for a typical high performance aircraft the nlqr controller design is performed using truncated solutions of the hamil ton jacobi bellman equation of optimal control theory the performance of the nlqr controller is compared with the performance of a conventional p i gain scheduled controller designed by applying standard frequency response techniques to the equations of motion of the aircraft linearized at various angles of attack both techniques result in control laws which are very similar in structure to one another and which yield similar performance the results of applying both control laws to a high g vertical turn are illustrated by nonlinear simulation

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

periodic unsteady flow analysis using a diagonally implicit harmonic balance method an implicit harmonic balance method is applied to periodic unsteady flow problems diagonal terms of the harmonic source vector are treated implicitly and easily implemented on a diagonalized implicit solver that is very similar to a flow solver the multi grid performance and solution convergence of the present implicit harmonic balance method are investigated with a 2 d oscillating airfoil and a 3 d wing with pitching motion computational results are compared with results from explicit trials and a dual timestepping method in the time domain the present harmonic balance method provides fast and stable convergence characteristics regardless of the number of harmonics results show that the present method can largely reduce the computing time for unsteady flow problems compared to that necessary in the dual time stepping and explicit harmonic balance methods

an adaptive multigrid technique for the incompressible navier stokes equations an automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady state incompressible navier stokes equations solutions have been obtained for the driven cavity and flow over a backward facing step for reynolds numbers up to 5000 and 800 respectively the refinement criterion is based on the local truncation error the solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level for driven cavity flow at re 1000 the adaptive refinement approach reduced the computer memory and cpu time to 20 and 40 of the requirements of the pure multigrid method the primitive variable formulation of the navier stokes equations is used so the method can be extended easily to three dimensions application to other nonlinear elliptic problems is equally possible

a nonlinear multigrid method for inverse problem in the multiphase porous media flow abstract in this paper we consider a parameter identification problem for the nonlinear convection diffusion equation in the multiphase porous media flow a nonlinear multigrid method is proposed for the recovery of permeability this method works by dynamically adjusting the objective functionals at different grids so that they are consistent with each other and ultimately reduce the finest grid objective functional in this manner the nonlinear multigrid method can efficiently compute the solution to a desired fine grid inverse problem numerical results illustrate that the proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

c1approximations of inertial manifolds for dissipative nonlinear equations abstract in this paper we study a class of nonlinear dissipative partial differential equations that have inertial manifolds this means that the long time behavior is equivalent to a certain finite system of ordinary differential equations we investigate ways in which these finite systems can be approximated in thec1sense geometrically this may be interpreted as constructing manifolds in phase space that arec1close to the inertial manifold of the partial differential equation under such approximations the invariant hyperbolic sets of the global attractor persist

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

a multigrid homotopy method for nonlinear inverse problems abstract in the present contribution we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems in which the forward problems are modeled by some forms of partial differential equations the method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest and then carries out the inversion on the coarsest grid by the homotopy method the strategy may give a rapidly and globally convergent method as a practical application this method is used to solve the nonlinear inverse problem of a nonlinear convection diffusion equation which is the saturation equation within the two phase porous media flow we demonstrate the effectiveness and merits of the multigrid homotopy method on two actual model problems

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

criteria for surrogate end points based on causal distributions summary when a treatment has a positive average causal effect ace on an intermediate variable or surrogate end point which in turn has a positive ace on a true end point the treatment may have a negative ace on the true end point due to the presence of unobserved confounders which is called the surrogate paradox a criterion for surrogate end points based on aces has recently been proposed to avoid the surrogate paradox for a continuous or ordinal discrete end point the distributional causal effect dce may be a more appropriate measure for a causal effect than the ace we discuss criteria for surrogate end points based on dces we show that commonly used models such as generalized linear models and cox s proportional hazard models can make the sign of the dce of the treatment on the true end point determinable by the sign of the dce of the treatment on the surrogate even if the models include unobserved confounders furthermore for a general distribution without any assumption of parametric models we give a sufficient condition for a distributionally consistent surrogate and prove that it is almost necessary

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

an approximate inertial manifold for computing burgers equation abstract we present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals our algorithm is motivated from the dynamical systems point of view in particular we adapt the methodology of approximate inertial manifolds to a finite difference scheme this leads to a differential treatment in which the higher i e unresolved modes are expressed in terms of the lower modes as a particular example we derive an approximate inertial manifold for burgers equation and develop a numerical algorithm suitable for computing we perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme for all values of the parameters which are the coefficient of viscosity and the cell size we obtain a decrease in the numerical error by at least a factor of 2 0 with the modified scheme the decrease in error is substantially greater over large regions of the parameters space

some remarks on multivariate chebyshev polynomials abstract several generalizations of the chebyshev polynomials in one variable to chebyshev polynomials in several variables are presented

reduced order based feedback control of the kuramoto sivashinsky equation in this paper we consider the kuramoto sivashinsky equation kse which describes the long wave motions of a thin film over a vertical plane solution procedures for the kse often yield a large or infinite dimensional nonlinear system we first discuss two reduced order methods the approximate inertial manifold and the proper orthogonal decomposition and show that these methods can be used to obtain a reduced order system that can accurately describe the dynamics of the kse moreover from this resulting reduced order system the feedback controller can readily be designed and synthesized for our control techniques we use the linear and nonlinear quadratic regulator methods which are the first and second order approximated solutions of the hamilton jacobi bellman equation respectively numerical simulations comparing the performance of the reduced order based linear and nonlinear controllers are presented

hdg pod reduced order model of the heat equation abstract we propose a new hybridizable discontinuous galerkin hdg model order reduction technique based on proper orthogonal decomposition pod we consider the heat equation as a test problem and prove error bounds that converge to zero as the number of pod modes increases we present 2d and 3d numerical results to illustrate the convergence analysis

nonintrusive continuum sensitivity analysis for fluid applications abstract continuum sensitivity analysis csa provides an analytic method of computing derivatives for structures fluids and fluid structure interaction problems with respect to shape or value parameters csa does not require mesh sensitivity to calculate local shape sensitivity thereby contributing to its computational efficiency csa involves solving linear sensitivity equations with the corresponding sensitivity boundary conditions spatial gradients of flow variables are required to construct the sensitivity boundary conditions the accuracy of which significantly affects the flow shape derivatives a method called spatial gradient reconstruction sgr is used to accurately compute the flow spatial gradients that are further used to set the sensitivity boundary conditions here we present a nonintrusive csa formulation for calculating the material derivatives of flow variables with respect to shape design parameters nonintrusive csa means csa without any modification to the black box program used for the cfd flow analysis moreover nonintrusive csa is flow solver agnostic in that knowledge of the solver s discretization is not needed an example of two dimensional flow over a naca0012 airfoil demonstrates the application of nonintrusive csa to steady flow problems involving euler compressible inviscid equations over unstructured grids with finite volume spatial discretization accuracy was studied by investigating convergence for a family of six high quality grids ranging from four thousand to four million cells factors such as the accuracy of the flow spatial gradients flow sensitivity boundary conditions weak implementation of the boundary conditions in the finite volume framework and the use of an approximate flux jacobian matrix all of which affect the accuracy of the material derivatives are discussed a novelty of this work is that the sensitivity analysis is done nonintrusively using fluent and su2 software to establish the use of black box codes for obtaining flow derivatives using the csa approach

instrumental variables as bias amplifiers with general outcome and confounding drawing causal inference with observational studies is the central pillar of many disciplines one sufficient condition for identifying the causal effect is that the treatment outcome relationship is unconfounded conditional on the observed covariates it is often believed that the more covariates we condition on the more plausible this unconfoundedness assumption is this belief has had a huge impact on practical causal inference suggesting that we should adjust for all pretreatment covariates however when there is unmeasured confounding between the treatment and outcome estimators adjusting for some pretreatment covariate might have greater bias than estimators without adjusting for this covariate this kind of covariate is called a bias amplifier and includes instrumental variables that are independent of the confounder and affect the outcome only through the treatment previously theoretical results for this phenomenon have been established only for linear models we fill in this gap in the literature by providing a general theory showing that this phenomenon happens under a wide class of models satisfying certain monotonicity assumptions we further show that when the treatment follows an additive or multiplicative model conditional on the instrumental variable and the confounder these monotonicity assumptions can be interpreted as the signs of the arrows of the causal diagrams

convergence analysis of the gauss seidel preconditioner for discretized one dimensional euler equations we consider the nonlinear system of equations that results from the van leer flux vector splitting discretization of the one dimensional euler equations this nonlinear system is linearized at the discrete solution the main topic of this paper is a convergence analysis of block gauss seidel methods applied to this linear system of equations both the lexicographic and the symmetric block gauss seidel method are considered we derive results which quantify the quality of these methods as preconditioners these results show for example that for the subsonic case the symmetric gauss seidel method can be expected to be a much better preconditioner than the lexicographic variant sharp bounds for the condition number of the preconditioned matrix are derived

sharp bounds on causal effects using a surrogate endpoint this paper considers a problem of evaluating the causal effect of a treatment x on a true endpoint y using a surrogate endpoint s in the presence of unmeasured confounders between s and y such confounders render the causal effect of x on y unidentifiable from the causal effect of x on s and the joint probability of s and y to evaluate the causal effect of x on y in such a situation this paper derives closed form formulas for the sharp bounds on the causal effect of x on y based on both the causal effect of x on s and the joint probability of s and y under various assumptions in addition we show that it is not always necessary to observe y to test the null causal effect of x on y under the monotonicity assumption between x and s these bounds enable clinical practitioners and researchers to assess the causal effect of a treatment on a true endpoint using a surrogate endpoint with minimum computational effort

acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure abstract the paper presents the development and investigation of an adaptive smoothing as procedure in conjunction with the full multigrid full approximation storage method the latter has been previously implemented by the authors 1 for the incompressible navier stokes equations in conjunction with the artificial compressibility method and forms the basis for investigating the current as approach the principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and subsequently to reduce the total computing time the implementation of the as approach is investigated in conjunction with three different adaptivity criteria for two and three dimensional incompressible flows furthermore a dynamic procedure henceforth labeled dynamic adaptivity for defining variably the as parameters during the computation is also proposed and its performance is investigated in contrast to as with constant parameters henceforth labeled static adaptivity both static and dynamic adaptivity can provide similar acceleration but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method numerical experiments are presented for attached and separated flows around airfoils as well as for three dimensional flow in a curved channel for external flows the as performs better when it is applied in all grid levels of the multigrid method while for internal flows the best performance is achieved when as is applied in the finest grid only overall the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing

multi grid calculation of three dimensional transonic potential flows a multi grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing fuselage combinations the method is based upon a fully conservative finite volume approximation to the steady potential equation which is second order accurate everywhere in the flow field except near shock waves the multi grid scheme is incorporated within the framework of an alternating successive line overrelaxation slor solver of the difference equations computed results confirm the second order accuracy of the scheme and demonstrate the effectiveness of the multi grid procedure

the kuramoto sivashinsky equation a bridge between pde s and dynamical systems abstract through extensive numerical simulation we have characterized the transition to chaos of the solutions to the kuramoto sivashinsky equation the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points invariant tori traveling wave trains and homoclinic orbits we relate this behavior to earlier work where the kuramoto sivashinsky equation was shown to behave as a finite dimensional dynamical system of ordinary differential equations

reconstruction of a permeability field with the wavelet multiscale homotopy method for a nonlinear convection diffusion equation this paper deals with the reconstruction of a piecewise constant permeability field for a nonlinear convection diffusion equation which arises as the saturation equation in the fractional flow formulation of the two phase porous media flow equations this permeability identification problem is solved through the minimization of a cost functional which depends on the discrepancy in a least square sense between some measurements and associated predictions in order to cope with the local convergence property of the optimizer the presence of numerous local minima in the cost functional and the large computational cost a wavelet multiscale homotopy is developed implemented and validated this method combines the wavelet multiscale inversion idea with the homotopy method numerical results illustrate the global convergence computational efficiency anti noise ability of the proposed method

sufficient conditions for concluding surrogacy based on observed data in this paper we show that prentice s criteria for surrogates can be strengthened to remedy some weaknesses of the criteria and we also propose some sufficient conditions under which the treatment effects on a surrogate and on the true endpoint have qualitative implication and equivalence relations with or without requiring prentice s criteria we discuss what conditions are required to qualitatively assess the causal effect of treatment on an unobserved endpoint in terms of an observed surrogate rather than a correlation between a surrogate and an endpoint we require stricter measurements of association for the qualitative assessment further we show that these conditions can be satisfied by commonly used models such as generalized linear models and cox s proportional hazard models copyright 2011 john wiley sons ltd

euler calculations with embedded cartesian grids and small perturbation boundary conditions this study examines the use of stationary cartesian mesh for steady and unsteady flow computations the surface boundary conditions are imposed by reflected points a cloud of nodes in the vicinity of the surface is used to get a weighted average of the flow properties via a gridless least squares technique if the displacement of the moving surface from the original position is typically small a small perturbation boundary condition method can be used to ensure computational efficiency multigrid solution is made via a framework of embedded grids for local grid refinement computations of airfoil wing and wing body test cases show the practical usefulness of the embedded cartesian grids with the small perturbation boundary conditions approach

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

multilevel ensemble kalman filtering this work embeds a multilevel monte carlo sampling strategy into the monte carlo step of the ensemble kalman filter enkf in the setting of finite dimensional signal evolution and noisy discrete time observations the signal dynamics is assumed to be governed by a stochastic differential equation sde and a hierarchy of time grids is introduced for multilevel numerical integration of that sde the resulting multilevel enkf is proved to asymptotically outperform enkf in terms of computational cost versus approximation accuracy the theoretical results are illustrated numerically

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

identification and estimation of local average treatment effects we investigate conditions sufficient for identification of average treatment effects using instrumental variables first we show that the existence of valid instruments is not sufficient to identify any meaningful average treatment effect we then establish that the combination of an instrument and a condition on the relation between the instrument and the participation status is sufficient for identification of a local average treatment effect for those who can be induced to change their participation status by changing the value of the instrument finally we derive the probability limit of the standard iv estimator under these conditions it is seen to be a weighted average of local average treatment effects this abstract was borrowed from another version of this item

a multigrid solution for the cahn hilliard equation on nonuniform grids we present a nonlinear multigrid method to solve the cahn hilliard ch equation on nonuniform grids the ch equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys the model has the characteristics of thin diffusive interfaces to resolve the sharp interfacial transition we need a very fine grid which is computationally expensive to reduce the cost we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region the ch equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time we use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels various numerical results on one two and three dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the ch equation

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

criteria for surrogate end points summary a surrogate end point is often used to evaluate the effects of treatments or exposures on the true end point in medical researches various criteria for the statistical surrogate principal surrogate and strong surrogate have been proposed we first illustrate that with a surrogate end point that is defined by these criteria it is possible that a treatment has a positive effect on the surrogate which in turn has a positive effect on the true end point but the treatment has a negative effect on the true end point we define such a phenomenon as a surrogate paradox the surrogate paradox also means that the sign of the treatment effect on the true end point is unpredictable by the effect signs of both the treatment on the surrogate and the surrogate on the true end point then we propose two notions for a consistent surrogate and a strictly consistent surrogate to avoid the surrogate paradox with the causal network that was presented by lauritzen we discuss the conditions for a strong surrogate to be a consistent surrogate and a strictly consistent surrogate

order and complexity in the kuramoto sivashinsky model of weakly turbulent interfaces abstract we present a large number of new geometric ergodic and statistical properties of the kuramoto sivashinsky equation modeling interfacial turbulence in various physical contexts in addition this equation has the remarkable property of inertial manifolds where some finite dimensional dynamical system is rigorously equivalent to this infinite dimensional partial differential equation in moderate size domains up to ten periods in length a low dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold the extreme numerical sensitivity of chaos in this dissipative pde requires very high precision methods despite the geometrical complexities of the bifurcation structure some statistical properties remain remarkably simple there is overwhelming evidence that for some parameter values a permanent unsteady state exists an unexpectedly simple diffusive relaxation of the large scale fluctuations is extracted from extensive numerical simulations in these calculations we observe long time tails for the correlation functions of relevant quantities we propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence

multigrid one shot method for state constrained aerodynamic shape optimization this paper presents applications of an optimization based multigrid method to state constrained aerodynamic shape optimization problems the one shot simultaneous pseudo time stepping method has been applied successfully to such problems reducing the cost of computation up to 70 from that of a traditional gradient method however the number of optimization iterations is comparatively large in this method since we update the design parameters in each time step of state and costate runs in this paper we use an optimization based multigrid strategy to reduce the total number of optimization iterations while keeping the idea of the one shot pseudo time stepping method design examples of drag reduction with constant lift together with geometrical constraints for an rae2822 airfoil are included the total effort required for convergence of the optimization problem is between four and six times that of forward simulation runs depending on computational grid size

optimal multigrid solutions of two dimensional convection conduction problems the present work investigates the efficiency of the multigrid numerical method when used to solve two dimensional laminar velocity and temperature fields inside a rectangular domain numerical analysis is based on the finite volume discretization scheme applied to structured orthogonal regular meshes performance of the correction storage cs multigrid algorithm is compared for different inlet reynolds number re i n and number of grids up to four grids were used for both v and w cycles simultaneous and uncoupled temperature velocity solution schemes were investigated advantages in using more than one grid are discussed for simultaneous solution results further indicate an increase in the computational effort for higher inlet reynolds number re i n optimal number of intermediate relaxation sweeps for within both v and w cycles is discussed upon

fourier analysis of multigrid methods for general systems of pdes most iteration methods for solving boundary value problems can be viewed as approximations of a time dependent differential equation in this paper we show that the multigrid method has the effect of increasing the timestep for the smooth part of the solution leading back to an increase of the convergence rate for the nonsmooth part the convergence is an effect of damping fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods the analysis is performed for general partial differential equations and an arbitrary number of grids the difference in the behavior of the iterations between firstand second order equations is discussed the theoretical results are confirmed in simple numerical experiments

nonlinear multigrid algorithms for bayesian optical diffusion tomography optical diffusion tomography is a technique for imaging a highly scattering medium using measurements of transmitted modulated light reconstruction of the spatial distribution of the optical properties of the medium from such data is a difficult nonlinear inverse problem bayesian approaches are effective but are computationally expensive especially for three dimensional 3 d imaging this paper presents a general nonlinear multigrid optimization technique suitable for reducing the computational burden in a range of nonquadratic optimization problems this multigrid method is applied to compute the maximum a posteriori map estimate of the reconstructed image in the optical diffusion tomography problem the proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality