variable selection for gaussian process models in computer experiments in many situations simulation of complex phenomena requires a large number of inputs and is computationally expensive identifying the inputs that most impact the system so that these factors can be further investigated can be a critical step in the scientific endeavor in computer experiments it is common to use a gaussian spatial process to model the output of the simulator in this article we introduce a new simple method for identifying active factors in computer screening experiments the approach is bayesian and only requires the generation of a new inert variable in the analysis however in the spirit of frequentist hypothesis testing the posterior distribution of the inert factor is used as a reference distribution against which the importance of the experimental factors can be assessed the methodology is demonstrated on an application in material science a computer experiment from the literature and simulated examples

latin hypercube designs with controlled correlations and multi dimensional stratification various methods have been proposed to construct latin hypercube designs with small correlations orthogonal arrays have been used to construct latin hypercube designs with multi dimensional stratification to integrate these two ideas we propose a method to construct latin hypercube designs with both controlled correlations and multi dimensional stratification for numerical integration the constructed designs not only filter out lower dimensional variance components as effectively as ordinary orthogonal array based latin hypercube designs but also filter out bilinear terms more effectively the proposed construction method entails no iterative searches sampling properties of the constructed designs are derived examples are given to illustrate the proposed construction method and the theoretical results

a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

jensen s inequality for polynomials with concentration at low degrees we give a generalization of jensen s inequality valid for polynomials having some concentration at low degrees we investigate the constants involved both from a theoretical and a numerical point of view

controlling correlations in latin hypercube samples abstract monte carlo integration is competitive for high dimensional integrands latin hypercube sampling is a stratification technique that reduces the variance of the integral previous work has shown that the additive part of the integrand is integrated with error op n 1 2 under latin hypercube sampling with n integrand evaluations a bilinear part of the integrand is more accurately estimated if the sample correlations among input variables are negligible other authors have proposed an algorithm for controlling these correlations we show that their method reduces the correlations by roughly a factor of 3 for 10 n 500 we propose a method that based on simulations appears to produce correlations of order op n 3 2 an analysis of the algorithm indicates that it cannot be expected to do better than n 3 2

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

minimum aberration designs with two level and four level factors summary for optimal assignment of factors to columns of a 4 2 n design we consider extensions of the minimum aberration criterion for the 2 level designs minimum aberration 412n and 422n designs in 16 and 32 runs are obtained and tabulated some results on the wordlength patterns and design classification are obtained

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

a construction method for orthogonal latin hypercube designs summary the latin hypercube design is a popular choice of experimental design when computer simulation is used to study a physical process these designs guarantee uniform samples for the marginal distribution of each single input a number of methods have been pro posed for extending the uniform sampling to higher dimensions we show how to construct latin hypercube designs in which all main effects are orthogonal our method can also be used to construct latin hypercube designs with low correlation of first order and second order terms our method generates orthogonal latin hypercube designs that can include many more factors than those proposed by ye 1998

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

energy norm a posteriori error estimates for mixed finite element methods this paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations it is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method the analysis is performed in two different ways under a saturation assumption and using a helmholtz decomposition for vector fields

blocking schemes for definitive screening designs in earlier work jones and nachtsheim proposed a new class of screening designs called definitive screening designs as originally presented these designs are three level designs for quantitative factors that provide estimates of main effects that are unbiased by any second order effect and require only one more than twice as many runs as there are factors definitive screening designs avoid direct confounding of any pair of second order effects and for designs that have more than five factors project to efficient response surface designs for any two or three factors recently jones and nachtsheim expanded the applicability of these designs by showing how to include any number of two level categorical factors however methods for blocking definitive screening designs have not been addressed in this article we develop orthogonal blocking schemes for definitive screening designs we separately consider the cases where all of the factors are quantitative and where there is a mix of quantitative and tw

an explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems abstract this paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems it is shown that this estimator is reliable with respect to an energy type error numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

a generalization of jensen s inequality for polynomials having concentration at low degrees in this paper we give some asymptotic estimates for the best lower bound c d k p of the jensen s functional j f email protected 0 2 plog f e i q email protected p for when the polynomial f has a concentration d at low degrees k measured by the l p norm under the l p norm p 2

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

spline based emulators for radiative shock experiments with measurement error radiation hydrodynamics and radiative shocks are of fundamental interest in the high energy density physics research due to their importance in understanding astrophysical phenomena such as supernovae in the laboratory experiments can produce shocks with fundamentally similar physics on reduced scales however the cost and time constraints of the experiment necessitate use of a computer algorithm to generate a reasonable number of outputs for making valid inference we focus on modeling emulators that can efficiently assimilate these two sources of information accounting for their intrinsic differences the goal is to learn how to predict the breakout time of the shock given the information on associated parameters such as pressure and energy under the framework of the kennedy o hagan model we introduce an emulator based on adaptive splines depending on the preference of having an interpolator for the computer code output or a computationally fast model a couple of different variants are proposed

a bayesian variable selection approach for analyzing designed experiments with complex aliasing experiments using designs with complex aliasing patterns are often performed for example twolevel nongeometric plackett burman designs multilevel and mixed level fractional factorial designs two level fractional factorial designs with hard to control factors and supersaturated designs hamada and wu proposed an iterative guided stepwise regression strategy for analyzing the data from such designs that allows entertainment of interactions their strategy provides a restricted search in a rather large model space however this article provides an efficient methodology based on a bayesian variable selection algorithm for searching the model space more thoroughly we show how the use of hierarchical priors provides a flexible and powerful way to focus the search on a reasonable class of models the proposed methodology is demonstrated with four examples three of which come from actual industrial experiments

on latin hypercube sampling this paper contains a collection of results on latin hypercube sampling the first result is a berry esseen type bound for the multivariate central limit theorem of the sample mean hat mu n based on a latin hypercube sample the second establishes sufficient conditions on the convergence rate in the strong law for hat mu n finally motivated by the concept of empirical likelihood a way of constructing nonparametric confidence regions based on latin hypercube samples is proposed for vector means

sequentially refined latin hypercube designs reusing every point the use of iteratively enlarged latin hypercube designs for running computer experiments has recently gained popularity in practice this approach conducts an initial experiment with a computer code using a latin hypercube design and then runs a follow up experiment with additional runs elaborately chosen so that the combined design set for the two experiments forms a larger latin hypercube design this augmenting process can be repeated multiple stages where in each stage the augmented design set is guaranteed to be a latin hypercube design we provide a theoretical framework to put this approach on a firm footing numerical examples are given to corroborate the derived theoretical results supplementary materials for this article are available online

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

modification of the maximin and ϕp phi criteria to achieve statistically uniform distribution of sampling points this article proposes a sampling technique that delivers robust designs that is point sets selected from a design domain in the shape of a unit hypercube the designs are guaranteed to provide a

a robust criterion for experimental designs for serially correlated observations a new robust criterion for experimental designs is proposed for a given design the criterion is to minimize over all possible run orders the absolute value of the change of the variance function due to possible correlation between the observations the resulting design is robust against possible auto correlation among the observations in the sense that confidence interval coverage levels are maintained accurately applications are given for 2k factorial designs and one mixed level fractional experiment and robust runs are obtained computational strategies are also discussed and a simulated annealing algorithm is developed to search for robust designs

main effect plans orthogonal through the block factor many industrial experiments require different number of levels for different factors orthogonal main effect plans omeps for such asymmetrical experiments often require large run size if blocking is needed then run size usually becomes even larger in this article we show that there are situations when use of blocks may in fact be helpful in finding an omep with small run size explicitly we obtained saturated omeps for a 33n 23n experiment in 3n blocks of size 4 each for each hadamard number n in each of these plans the three level factors are nonorthogonal to the block factor but are pairwise orthogonal through the block factor the two level factors are orthogonal to the block factor the data analysis for such a plan is illustrated with the help of examples this article has supplementary material online

asymmetric nested lattice samples we propose an approach for constructing a new class of space filling designs called asymmetric nested lattice samples as a stepping stone to the proposed approach we introduce two methods for generating asymmetric nested orthogonal arrays we then present an elaborate randomization method to shuffle the levels of an asymmetric nested orthogonal array to produce a pair of asymmetric nested lattice samples the constructed designs have a desirable nested structure possess attractive space filling properties and allow different axes to be divided at different scales of fineness for multifidelity computer experiments sequential evaluations and multistep functional fitting and linking parameters the asymmetric feature is appealing for situations where some factors are believed to be more important or deserve more attention than others the proposed designs are also useful in problems where different factors by nature require dividing their axes at different levels of fineness

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

nearly orthogonal arrays with mixed levels and small runs in running a factorial experiment it may be desirable to use an orthogonal array with different mixed numbers of factor levels because of the orthogonality requirement such arrays may have a large run size by slightly sacrificing the orthogonality requirement we can obtain nearly orthogonal arrays with economic run size some general methods for constructing such arrays are given for 12 18 20 and 24 runs many orthogonal arrays and nearly orthogonal arrays with mixed levels are constructed and tabulated effects of near orthogonality on estimation efficiency and analysis are studied

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

columnwise pairwise algorithms with applications to the construction of supersaturated designs motivated by the construction of supersaturated designs we develop a class of algorithms called columnwise pairwise exchange algorithms they differ from the k exchange algorithms in two respects 1 they exchange columns instead of rows of the design matrix and 2 they employ a pairwise adjustment in the search for a better column the proposed algorithms perform very well in the construction of supersaturated designs both for a single criterion and for multiple criteria they are also applicable to the construction of designs that are not supersaturated

sudoku based space filling designs sudoku is played by millions of people across the globe it has simple rules and is very addictive the game board is a nine by nine grid of numbers from one to nine several entries within the grid are provided and the remaining entries must be filled in subject to no row column or three by three subsquare containing duplicate numbers by exploiting these three types of uniformity we propose an approach to constructing a new type of design called a sudoku based space filling design such a design can be divided into groups of subdesigns so that the complete design and each subdesign achieve maximum uniformity in univariate and bivariate margins examples are given illustrating the proposed construction method applications of such designs include computer experiments with qualitative and quantitative factors linking parameters in engineering and crossvalidation copyright 2011 oxford university press

k circulant supersaturated designs a class of supersaturated designs called k circulant designs is explored these designs are constructed from cyclic generators by cycling k elements at a time the class of designs includes many es2 optimal designs some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity generators for the most efficient designs are listed and projection properties of some of the designs are explored we also illustrate that some k circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows this paper develops an posteriori error estimate for lowest order locally conservative methods on meshes consisting of very general polytopal elements we focus on the ease of implementation of the methodology based on h1 conforming potential reconstruction and h div conforming flux reconstruction in particular the evaluation of our estimates merely consists in some local matrix vector multiplications where on each mesh element the matrices are either directly inherited from the given numerical method or trivially constructed from the element geometry while the vectors are the degrees of freedom on the given element we then apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media here on each step of the time marching scheme linearization procedure and linear algebraic solver we distinguish the corresponding error components this leads to an easy to implement and fast to run adaptive algorithm with simultaneously guaranteed overall precision and optimal efficiency ensured through the use of adaptive stopping criteria together with adaptive space and time mesh refinements numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology

convergence of the mimetic finite difference method for eigenvalue problems in mixed form optimal convergence rates for the mimetic finite difference method applied to eigenvalue problems in mixed form are proved the analysis is based on a new a priori error bound for the source problem and relies on the existence of an appropriate elemental lifting of the mimetic discrete solution compared to the original convergence analysis of the method the new a priori estimate does not require any extra regularity assumption on the right hand side of the source problem numerical results confirming the optimal behavior of the method are presented

on the existence of saturated and nearly saturated asymmetrical orthogonal arrays we develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2 this condition limits the choice of integral solutions to the system of equations in the bose bush approach and can thus strengthen considerably the bose bush approach as applied to a symmetrical part of such an array as a consequence several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2 one of these leads to a partial settlement of an issue left open in a paper by wu zhang and wang nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed

a higher order formulation of the mimetic finite difference method a new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes this approach provides a higher order accurate approximation to the flux of the exact solution in analogy with the original formulation a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability the scalar solution field is approximated by a piecewise constant function a computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable finally optimal convergence rates and accuracy improvement with respect to the lower order formulation are shown by numerical examples

construction of 2 m4 n designs via a grouping scheme on developpe une methode pour grouper les 2 k 1 effets factoriels dans un plan factoriel a deux niveaux dans des ensembles disjoints mutuellement s t st ou st est l interaction generalisee des effets s et t comme application on construit des tableaux orthogonaux oa 2 k 2 m 4 n 2 de taille 2 k m contraintes avec 2 niveaux et n contraintes avec 4 niveaux satisfaisant m 3n 2 k 1 et de puissance 2 le nombre maximum de contraintes avec 4 niveaux dans la construction ne peut plus etre ameliore dans ce sens le schema de groupement qu on donne est optimal on discute les avantages de la presente approche par rapport a d autres methodes de construction

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

a mimetic discretization of the reissner mindlin plate bending problem we present a mimetic approximation of the reissner mindlin plate bending problem which uses deflections and rotations as discrete variables the method applies to very general polygonal meshes even with non matching or non convex elements we prove linear convergence for the method uniformly in the plate thickness

the idle column method design construction properties and comparisons taguchi s idle column method for constructing arrays is investigated and generalized a general construction method is proposed via a necessary and sufficient condition on two level orthogonal arrays to which the method is applicable a method is developed for identifying the idle column and pairs of columns in the arrays mixed two and three level idle column arrays with 64 or fewer runs are tabulated some guidelines are offered on when the idle column arrays should be used and when they should be replaced by orthogonal main effect plans or d optimal designs

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

sliced full factorial based latin hypercube designs as a framework for a batch sequential design algorithm when fitting complex models such as finite element or discrete event simulations the experiment design should exhibit desirable properties of both projectivity and orthogonality to reduce experimental effort sequential design strategies allow experimenters to collect data only until some measure of prediction precision is reached in this article we present a batch sequential experiment design method that uses sliced full factorial based latin hypercube designs sfflhds which are an extension to the concept of sliced orthogonal array based latin hypercube designs oalhds at all stages of the sequential design good univariate stratification is achieved the structure of the fflhds also tends to produce uniformity in higher dimensions especially at certain stages of the design we show that our batch sequential design approach has good sampling and fitting qualities through both empirical studies and theoretical arguments supplementary materials are available online

noncollapsing space filling designs for bounded nonrectangular regions many researchers use computer simulators as experimental tools especially when physical experiments are infeasible when computer codes are computationally intensive nonparametric predictors can be fitted to training data for detailed exploration of the input output relationship the accuracy of such flexible predictors is enhanced by taking training inputs to be space filling if there are inputs that have little or no effect on the response it is desirable that the design be noncollapsing in the sense of having space filling lower dimensional projections this article describes an algorithm for constructing noncollapsing space filling designs for bounded input regions that are of possibly high dimension online supplementary materials provide the code for the algorithm examples of its use and show its performance in multiple settings

maximum projection designs for computer experiments summary space filling properties are important in designing computer experiments the traditional maximinandminimaxdistancedesignsconsideronlyspace fillinginthefull dimensionalspace thiscanresultinpoorprojectionsontolower dimensionalspaces whichisundesirablewhenonly af ew factors are active restricting maximin distance design to the class of latin hypercubes can improveone dimensionalprojectionsbutcannotguaranteegoodspace fillingpropertiesinlarger subspaces we propose designs that maximize space filling properties on projections to all subsets of factors we call our designs maximum projection designs our design criterion can be computed at no more cost than a design criterion that ignores projection properties

orthogonal and nearly orthogonal designs for computer experiments we introduce a method for constructing a rich class of designs that are suitable for use in computer experiments the designs include latin hypercube designs and two level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs the basic construction method is simple building a series of larger designs based on a given small design if the base design is orthogonal the resulting designs are orthogonal likewise if the base design is nearly orthogonal the resulting designs are nearly orthogonal we present two generalizations of our basic construction method the first generalization improves the projection properties of the basic method the second generalization gives rise to designs that have smaller correlations sample constructions are presented and properties of these designs are discussed

strong orthogonal arrays of strength two plus strong orthogonal arrays were recently introduced and studied in he and tang biometrika100 2013 254 260 as a class of space filling designs for computer experiments to enjoy the benefits of better space filling properties when compared to ordinary orthogonal arrays strong orthogonal arrays need to have strength three or higher which may require run sizes that are too large for experimenters to afford to address this problem we introduce a new class of arrays called strong orthogonal arrays of strength two plus these arrays while being more economical than strong orthogonal arrays of strength three still enjoy the better two dimensional space filling property of the latter among the many results we have obtained on the characterizations and constructions of strong orthogonal arrays of strength two plus worth special mention is their intimate connection with second order saturated designs

theory of surround system and associated inequalities in this paper we define the surround system s m c l and the surround system with a center s 2 p c l under the proper hypotheses we establish several geometric inequalities for s 2 p c l as applications we give the background of the surround system with a center s 2 p c l in space science

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

sliced space filling designs we propose an approach to constructing a new type of design a sliced space filling design intended for computer experiments with qualitative and quantitative factors the approach starts with constructing a latin hypercube design based on a special orthogonal array for the quantitative factors and then partitions the design into groups corresponding to different level combinations of the qualitative factors the points in each group have good space filling properties some illustrative examples are given

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems the maximum principle is one of the most important properties of solutions of partial differential equations its numerical analog the discrete maximum principle dmp is one of the most difficult properties to achieve in numerical methods especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic violation of the dmp may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one in this work we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference mfd method on two and three dimensional meshes these conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element efficient solutions are devised for meshes consisting of simplexes parallelograms and parallelepipeds and orthogonal locally refined elements as those used in the amr methodology on simplicial meshes it turns out that the mfd method coincides with the mixed hybrid finite element methods based on the low order raviart thomas vector space thus in this case we recover the well established conventional angle conditions of such approximations instead in the other cases a suitable design of the mfd method allows us to formulate a monotone discretization for which the existence of a dmp can be theoretically proved moreover on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the multi point flux approximation mpfa methods numerical experiments confirm the effectiveness of the considered monotonicity conditions

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

uniform coverage designs for molecule selection in screening for drug discovery chemists often select a large subset of molecules from a very large database e g select 1 000 molecules from 100 000 to generate diverse leads for drug optimization highly active compounds in several structurally different chemical classes are sought molecules can be characterized by numerical descriptors and the chosen subset should cover the descriptor space or subspaces formed by several descriptors we propose a method that concentrates on low dimensional subspaces a criterion for uniformity of coverage and a fast exchange algorithm to optimize the criterion these methods are illustrated by using a national cancer institute database

orthogonal array based latin hypercubes abstract in this article we use orthogonal arrays oa s to construct latin hypercubes besides preserving the univariate stratification properties of latin hypercubes these strength r oa based latin hypercubes also stratify each r dimensional margin therefore such oa based latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general latin hypercubes we prove that when used for integration the sampling scheme with oa based latin hypercubes offers a substantial improvement over latin hypercube sampling

jensen s functional and polynomials in several variables in this paper we obtain some lower bounds of the integral 0 2 p 0 2 plog p e i q 1 e i q n d q 12 p d q n2 p if the polynomial p z 1 z n in several variables has a concentration at low degrees measured by 1 l p norm under l p norm 1 p 2 2 l p norm under l p norm p 2 3 l p quasi norm under l p quasi norm 0

flux reconstruction and solution post processing in mimetic finite difference methods abstract we present a post processing technique for the mimetic finite difference solution of diffusion problems in mixed form our post processing method yields a piecewise linear approximation of the scalar variable that is second order accurate in the l2 norm on quite general polyhedral meshes including non convex and non matching elements the post processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables this technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied the post processing method is computationally inexpensive optimal performance is confirmed by numerical experiments

orthogonal column latin hypercubes and their application in computer experiments abstract latin hypercubes have been frequently used in conducting computer experiments in this paper a class of orthogonal latin hypercubes that preserves orthogonality among columns is proposed applying an orthogonal latin hypercube design to a computer experiment benefits the data analysis in two ways first it retains the orthogonality of traditional experimental designs the estimates of linear effects of all factors are uncorrelated not only with each other but also with the estimates of all quadratic effects and bilinear interactions second it can facilitate nonparametric fitting procedures because one can select good space filling designs within the class of orthogonal latin hypercubes according to selection criteria

centered l 2 discrepancy of random sampling and latin hypercube design and construction of uniform designs in this paper properties and construction of designs under a centered version of the l2 discrepancy are analyzed the theoretic expectation and variance of this discrepancy are derived for random designs and latin hypercube designs the expectation and variance of latin hypercube designs are significantly lower than that of random designs while in dimension one the unique uniform design is also a set of equidistant points low discrepancy designs in higher dimension have to be generated by explicit optimization optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance

block regularized repeated learning testing for estimating generalization error abstract repeated learning testing rlt is a popular cross validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning however the variance of the estimator of the generalization error in the rlt method is easily affected by random partitions poor data partitioning may cause a large overlap between any two training sets of rlt and enlarge the variance in the rlt estimator of the generalization error thus in this study we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which rlt is called the block regularized rlt brlt we theoretically prove that the brlt estimator has a smaller variance than does the rlt estimator specifically the variance of the rlt estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal furthermore we provide easy to use construction algorithms of brlt s partition set for several training set size and partition count settings by adopting two level orthogonal arrays we also illustrate brlt s optimal properties with several simulated and real world examples

balanced repeated replications based on orthogonal multi arrays summary to obtain variance estimates in stratified sampling with possibly unequal numbers of primary sampling units per stratum one can either randomly divide the units in each stratum into two groups and apply the balanced half samples method to these groups or use mixed orthogonal arrays to form a set of balanced subsamples with the first technique resulting variance estimates can be very inefficient and with the second technique it can be difficult to construct orthogonal arrays with a small number of replicates for a given stratified design the proposed extension of the orthogonal array method to orthogonal multi arrays retains all of the orthogonal array method s desirable properties but allows greater flexibility in construction and thus can handle more complex stratified designs using fewer replicates

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity