spherical prediction variance properties of central composite and box behnken designs for quadratic regression on the hypercube a single number criterion such as a g efficiency that is based on the prediction variance is often included as one of the criteria when selecting a response surface design as an alternative to the single number criterion approach the variance dispersion graph presented by giovannitti jensen and myers is a graphical technique for evaluating prediction variance properties throughout the experimental region three properties of interest are the maximum minimum and average spherical prediction variances given the spherical radius as an alternative to the computer based approach requiring an optimization algorithm to evaluate these properties the maximum minimum and spherical prediction variances for central composite and box behnken designs can be determined analytically and are functions only of the radius and the design parameters these functions yield the exact values of the spherical prediction variance properties of central composite and box behnke

new formulations for interface problems in polar coordinates in this paper numerical methods are proposed for some interface problems in polar or cartesian coordinates the new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution the new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution in conjunction with the fast immersed interface method a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system

a large sample confidence region useful in characterizing the stationary point of a quadratic response surface a second order response surface model is often used to approximate the relationship between a response variable and a set of explanatory variables the nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model s second order terms since the elements of this matrix are estimated from the data however it follows that the eigenvalues are random variables hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point in this article it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface the delta method is used to construct an approximate confidence region for these eigenvalues box and draper 1987 gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs a simulation study was performed to

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

a measure of rotatability for response surface designs a measure that quantifies the amount of rotatability in a given response surface design is introduced in this article the measure which is expressible as a percentage takes the value 100 if and only if the design is rotatable one of the main advantages of this measure is that it can be used to repair a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest four numerical examples are given to illustrate the applications of this measure

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described in l adams and z li siam j sci comput 24 2002 pp 463 479 and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices this paper compares the immersed interface multigrid methods introduced in l adams and z li siam j sci comput 24 2002 pp 463 479 and l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 with algebraic multigrid which uses no geometric information to set up the multigrid components for coarse grid correction we show that algebraic multigrid is a robust solver for our test problems it outperforms the method in l adams and z li siam j sci comput 24 2002 pp 463 479 and performs nearly as well as the method in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 which is shown to be the most efficient for all problem parameters and sizes

i optimal design of mixture experiments abstractin mixture experiments the factors under study are proportions of the ingredients of a mixture the special nature of the factors necessitates specific types of regression models and specific types of experimental designs although mixture experiments usually are intended to predict the response s for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients little research has been done concerning their i optimal design this is surprising given that i optimal designs minimize the average variance of prediction and therefore seem more appropriate for mixture experiments than the commonly used d optimal designs which focus on a precise model estimation rather than precise predictions in this article we provide the first detailed overview of the literature on the i optimal design of mixture experiments and identify several contradictions for the second order and the special cubic model we present continuous i optimal designs and contrast the

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

higher order cartesian grid based finite difference schemes for elliptic equations on irregular domains second and fourth order cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations and associated eigenvalue problems on irregular domains with general boundary conditions our methods are based on the continuation of a solution idea using multivariable taylor s expansion of the solution about selected boundary points and the core ideas of the immersed interface method the methods offer systematic treatment of the general boundary conditions in two and three dimensional domains and are directly applied to semi discretize heat equations on irregular domains convergence analysis and numerical examples are presented the validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem

a decomposed immersed interface method for variable coefficient elliptic equations with non smooth and discontinuous solutions a second order accurate finite difference method is presented for solving two dimensional variable coefficient elliptic equations on cartesian grids in which the coefficients the source term the solution and its derivatives may be nonsmooth or discontinuous across an interface a correction term is introduced to the standard central difference stencil so that the numerical discretization is well defined across the interface we also propose a new method to approximate the correction term as part of the iterative procedure the method is easy to implement since the correction term only needs to be added to the right hand side of the system therefore the coefficient matrix remains symmetric and diagonally dominant allowing for most standard solvers to be used numerical examples show good agreements with exact solutions and the order of accuracy is comparable with other immersed interface methods

graphical tools for assessing the sensitivity of response surface designs to model misspecification when comparing different designs for an experiment optimality criteria and other measures often depend on the correctness of the assumed model in this article we develop and illustrate an approach for comparing designs given the potential effect of bias due to an underspecified model we illustrate this approach using graphical summaries of the expected mean squared error emse that allow assessment of the robustness of designs to model misspecification for response surface designs in cuboidal regions the excellent performance of the central composite designs when the quadratic model is correct is tempered when bias is present the box behnken designs demonstrate superiority in the presence of missing cubic terms for a screening design scenario the amount of bias present leads to different conclusions as to which design is best

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

a optimality for regression designs abstract use is made of a result of horn amer j math 76 1954 620 630 on the existence of a symmetric matrix with prescribed diagonal elements and eigenvalues a necessary and sufficient condition is then given for the existence of an a optimal design for a regression experiment in the dorogovcev selected transl math statist probab 10 1971 35 41 setting

on the outflow boundary condition for external incompressible flows a new approach this work presents a new procedure for extrapolating velocities at the outflow boundary in the computations of incompressible flows around rigid bodies the extrapolation procedure is based on the radial variation of the velocity field at large distances from the rigid body which can be inferred from mass conservation and vorticity considerations since the extrapolation is based on these physical considerations the proposed boundary condition is considered to be physically consistent it has been demonstrated via numerical simulations of 2d laminar incompressible viscous flow past a circular and a square cylinder at re 100 that the application of the proposed boundary condition allows one to limit the unbounded domain to a small size 6 8 times the characteristic size of the body without any significant change in the flow characteristics like the lift coefficient c l and the strouhal number st thus the proposed boundary conditions can enhance the computational efficiency of this class of flows

convergence of the ghost fluid method for elliptic equations with interfaces this paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps a weak formulation of the problem is first presented which then yields the existence and uniqueness of a solution to the problem by classical methods it is shown that the application of the ghost fluid method by fedkiw kang and liu to this problem can be obtained in a natural way through discretization of the weak formulation an abstract framework is given for proving the convergence of finite difference methods derived from a weak problem and as a consequence the ghost fluid method is proved to be convergent

graphical assessment of the prediction capability of response surface designs measures of the quality of prediction at locations on the surface of a hyperspherc are presented these measures are used to form a graphical method of assessing the overall prediction capability of an experimental design throughout the region of interest a plot of the spherical variance and the maximum and minimum prediction variances for locations on a sphere against the radius of the sphere a vuriance dispersion graph is used to give a comprehensive picture of the behavior of the prediction variances throughout a region and hence of the quality of the predicted responses obtained with a particular design such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher

comparison of design optimality criteria of reduced models for response surface designs in the hypercube results from the comparison of d g and a efficiencies and the scaled average prediction variance iv criterion are presented for the central composite small composite notz hoke box draper and computer generated designs these design optimality criteria are evaluated over the cuboidal design region for three four and five design variables

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction one of the fundamental purposes of response surface methodology is to develop a relatively simple linear model to explain a more complex phenomenon since the proposed model is only an approximation analysts almost always must address the potential of bias due to model misspecification this article proposes a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction in particular this article develops measures of the maximum and minimum mean squared error of prediction over the surface of spheres through a region of interest these measures are then plotted against radius the resulting plots reveal at a glance how a response surface design performs in the presence of model misspecification these plots are used to investigate the performances of several common response surface designs

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

new geometric immersed interface multigrid solvers in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described therein and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in this paper we improve on this method by giving a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices we compare this new restrictor to injection and the transpose of interpolation we show that the number of v cycles is constant as the mesh size decreases and increases only slightly as the ratio of the discontinuous problem coefficient grows at the interface only when this new restrictor is used

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

a jump condition capturing finite difference scheme for elliptic interface problems we propose a simple finite difference scheme for the elliptic interface problem with a discontinuous diffusion coefficient using a body fitted curvilinear coordinate system the resulting matrix is symmetric and positive definite standard techniques of acceleration such as pcg and multigrid can be used to invert the matrix the main advantage of the scheme is its simplicity the entries of the matrix are simply the centered difference second order approximation of the metric tensor g alpha beta in addition the interface jump conditions are naturally built into the finite difference discretization no interpolation extrapolation process is involved in the derivation of the scheme both the solution and the flux are observed to have second order accuracy

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

three dimensional elliptic solvers for interface problems and applications second order accurate elliptic solvers using cartesian grids are presented for three dimensional interface problems in which the coefficients the source term the solution and its normal flux may be discontinuous across an interface one of our methods is designed for general interface problems with variable but discontinuous coefficient the scheme preserves the discrete maximum principle using constrained optimization techniques an algebraic multigrid solver is applied to solve the discrete system the second method is designed for interface problems with piecewise constant coefficient the method is based on the fast immersed interface method and a fast 3d poisson solver the second method has been modified to solve helmholtz poisson equations on irregular domains an application of our method to an inverse interface problem of shape identification is also presented in this application the level set method is applied to find the unknown surface iteratively

mixed response surface models with heterogeneous within block error variances the experimental error variance in a response surface model with a block effect has traditionally been assumed to be constant in some experimental situations however this variance may be different for the different blocks that compose the associated design this article discusses the analysis of a response surface model with a random block effect and heterogeneous error variances among the blocks the model includes interactions between the fixed polynomial effects and the block effect procedures for estimating and testing the fixed effects and the random effects in the model are described in particular anova based procedures for testing equality of the error variances and testing the random effects are introduced

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim