multicolor fourier analysis of the multigrid method for quadratic fem discretizations abstract quadratic finite element methods offer some advantages for the numerical solution of partial differential equations pdes due to their improved approximation properties in comparison to linear approaches the algebraic linear systems arising from the discretization of pdes by this kind of methods require an efficient resolution and multigrid methods provide a good way to solve this problem to design geometric multigrid methods local fourier analysis lfa is a very useful tool however lfa for quadratic finite element discretizations can not be performed in a standard way since the discrete operator is defined by different stencils depending on the location of the points in the grid in this work a multicolor local fourier analysis is presented to analyze multigrid solvers for this type of discretizations with the help of this analysis some point wise and line wise smoothers are analyzed some results showing the good correspondence between the two grid convergence factors predicted by the analysis and the experimentally computed asymptotic convergence factors are presented finally this analysis is applied to design a very efficient multigrid solver for semi structured triangular grids based on a hybrid smoother

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

an alternative least squares formulation of the navier stokes equations with improved mass conservation the focus of this paper is on incompressible flows in three dimensions modeled by least squares finite element methods lsfem and using a novel reformulation of the navier stokes equations lsfem are attractive because the resulting discrete equations yield symmetric positive definite systems of algebraic equations and the functional provides both a local and global error measure on the other hand it has been documented for existing reformulations that certain types of boundary conditions and high aspect ratio domains can yield very poor mass conservation it has also been documented that improved mass conservation with lsfem can be achieved by strengthening the coupling between the pressure and velocity the new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure velocity coupling along the inflow and outflow boundaries

smoothed aggregation multigrid solvers for high order discontinuous galerkin methods for elliptic problems abstract we develop a smoothed aggregation based algebraic multigrid solver for high order discontinuous galerkin discretizations of the poisson problem algebraic multigrid is a popular and effective method for solving the sparse linear systems that arise from discretizing partial differential equations however high order discontinuous galerkin discretizations have proved challenging for algebraic multigrid the increasing condition number of the matrix and loss of locality in the matrix stencil as p increases in addition to the effect of weakly enforced dirichlet boundary conditions all contribute to the challenging algebraic setting we propose a smoothed aggregation approach that addresses these difficulties in particular the approach effectively coarsens degrees of freedom centered at the same spatial location as well as degrees of freedom at the domain boundary moreover the character of the near null space particularly at the domain boundary is captured by interpolation one classic prolongation smoothing step of weighted jacobi is also shown to be ineffective at high order and a more robust energy minimization approach is used along with block relaxation that more directly utilizes the block diagonal structure of the discontinuous galerkin discretization finally we conclude by examining numerical results in support our proposed method

cubature rules of prescribed merit we introduce a criterion for the evaluation of multidimensional quadrature or cubature rules for the hypercube this is the merit of a rule which is closely related to its trigonometric degree and which reduces to the zaremba figure of merit in the case of a lattice rule we derive a family of rules q k s having dimension s and merit 2 k these rules seem to be competitive with lattice rules with respect to the merit that can be achieved with a given number of abscissas

lattice rules of minimal and maximal rank with good figures of merit for periodic integrands with unit period in each variable certain error bounds for lattice rules are conveniently characterised by the figure of merit which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n distinct nodes then becomes that of finding rules with large values of this paper presents efficient search methods for the discovery of rank 1 rules and of maximal rank rules of high order which possess good figures of merit

a spectral element least squares formulation for incompressible navier stokes flows using triangular nodal elements we present a least squares formulation for the numerical solution of incompressible flows using high order triangular nodal elements the fekete points of the triangle are used as nodes and numerical integration is performed using tensor product gauss legendre rules in a collapsed coordinate system for the standard triangle a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c 0 element expansions in each triangle the numerical results demonstrate spectral convergence for smooth solutions excellent conservation of mass for steady and unsteady problems of the inflow outflow type and the flexibility of using triangles to partition domains where the use of quadrangles would be cumbersome or inefficient

a generalized sampling theorem for locally compact abelian groups we present a sampling theorem for locally compact abelian groups the sampling sets are finite unions of cosets of a closed subgroup this generalizes the well known case of non equidistant but periodic sampling on the real line for non bandlimited functions an l1 type estimate for the aliasing error is given we discuss the application of the theorem to a class of sampling sets in r s give a general algorithm for computer implementation present a detailed description of the implementation for the s dimensional torus group and point out connections to lattice rules for numerical integration

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

an improved index of interactivity for fuzzy numbers in this paper we will introduce a new index of interactivity between marginal possibility distributions a and b of a joint possibility distribution c the starting point of our approach is to equip each c level set of c with a uniform probability distribution then the probabilistic correlation coefficient between its marginal probability distributions is interpreted as an index of interactivity between the c level sets of a and b then we define the index of interactivity between a and b as the weighted average of these indexes over the set of all membership grades this new index of interactivity is meaningful for the whole family of joint possibility distributions

the determination of canonical forms for lattice quadrature rules abstract lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s dimensional unit hypercube us 0 1 s for a given lattice rule say ql a set of points l the integration lattice regularly spaced in all of rs is generated by a finite number of rational vectors the abscissa set for ql is then p ql l us it is known that p ql is a finite abelian group under addition modulo the integer lattice zs and that ql f may be written in the form of a nonrepetitive multiple sum q l f 1 n 1 n m j 1 1 n 1 j m 1 n m f j 1 n 1 z 1 j m n m z m known as a canonical form in which denotes addition modulo zs in this form zi zs m is called the rank and n1 n2 nm are called the invariants of ql and n i 1 n i for i 1 2 m 1 the rank and invariants are uniquely determined for a given lattice rule in this paper we provide a construction of a canonical form for a lattice rule ql given a generator set for the lattice l we then show how the rank and invariants of ql may be determined directly from the generators of the dual lattice l

algebraic multigrid for higher order finite elements two related approaches for solving linear systems that arise from a higher order finite element discretization of elliptic partial differential equations are described the first approach explores direct application of an algebraic based multigrid method amg to iteratively solve the linear systems that result from higher order discretizations while the choice of basis used on the discretization has a significant impact on the performance of the solver results indicate that amg is capable of solving operators from both poisson s equation and a first order system least squares fosls formulation of stoke s equation in a scalable manner nearly independent of basis order p for 3

weighted least squares finite elements based on particle imaging velocimetry data the solution of the navier stokes equations requires that data about the solution is available along the boundary in some situations such as particle imaging velocimetry there is additional data available along a single plane within the domain and there is a desire to also incorporate this data into the approximate solution of the navier stokes equation the question that we seek to answer in this paper is whether two dimensional velocity data containing noise can be incorporated into a full three dimensional solution of the navier stokes equations in an appropriate and meaningful way for addressing this problem we examine the potential of least squares finite element methods lsfem because of their flexibility in the enforcement of various boundary conditions further by weighting the boundary conditions in a manner that properly reflects the accuracy with which the boundary values are known we develop the weighted lsfem the potential of weighted lsfem is explored for three different test problems the first uses randomly generated gaussian noise to create artificial experimental data in a controlled manner and the second and third use particle imaging velocimetry data in all test problems weighted lsfem produces accurate results even for cases where there is significant noise in the experimental data

lattice rules by component scaling the authors introduce a theory of rectangular scaling of integer lattices this may be used to construct families of lattices they determine the relation between the zaremba index rho lambda of various members of the same family it appears that if one member of a family has a high index some of the other family members of higher order may have extraordinarily high indices they have applied a technique based on this theory to lists of good lattices available this has enabled the authors to construct lists of excellent previously unknown lattices of high order in three and four dimensions and of moderate order in five dimensions 10 refs 12 tabs

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

fractional programming approach to fuzzy weighted average this paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average based on the cut representation of fuzzy sets and the extension principle a pair of fractional programs is formulated to find the cut of fuzzy weighted average owing to the special structure of the fractional programs in most cases the optimal solution can be found analytically consequently the exact form of the membership function can be derived by taking the inverse function of the cut for other cases a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method examples are given for illustration

correlation of intuitionistic fuzzy sets in probability spaces in this paper we study the concepts of correlation and correlation coefficient of intuitionistic fuzzy sets in probability spaces in the case of finite spaces these results give the results of gerstenkorn and manko 1991 and in the case of infinite spaces also give the correct forms of yu s 1993 results

fuzzy measures for a correlation coefficient of fuzzy numbers under tw the weakest t norm based fuzzy arithmetic operations recently the sup min convolution based on zadeh s extension principle has been used by liu and kao fuzzy measures for correlation coefficient of fuzzy numbers fuzzy sets and systems 128 2002 267 275 to calculate a fuzzy correlation coefficient they used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient it is well known that t w the weakest t norm based addition and multiplication preserve the shape of l r fuzzy numbers in this paper we consider the computational aspect of the t w based extension principle when the principle is applied to a correlation coefficient of l r fuzzy numbers we give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

newton multigrid least squares fem for the v v p formulation of the navier stokes equations we solve the v v p vorticity velocity pressure formulation of the stationary incompressible navier stokes equations based on the least squares finite element method for the discrete systems we use a conjugate gradient cg solver accelerated with a geometric multigrid preconditioner for the complete system in addition we employ a krylov space smoother inside of the multigrid which allows a parameter free smoothing combining this linear solver with the newton linearization we construct a very robust and efficient solver we use biquadratic finite elements to enhance the mass conservation of the least squares method for the inflow outflow problems and to obtain highly accurate results we demonstrate the advantages of using the higher order finite elements and the grid independent solver behavior through the solution of three stationary laminar flow problems of benchmarking character the comparisons show excellent agreement between our results and those of the galerkin mixed finite element method as well as available reference solutions

an application of diophantine approximation to the construction of rank 1 lattice quadrature rules lattice quadrature rules were introduced by frolov 1977 sloan 1985 and sloan and kachoyan 1987 they are quasi monte carlo rules for the approximation of integrals over the unit cube in r s and are generalizations of number theoretic rules introduced by korobov 1959 and hlawka 1962 themselves generalizations in a sense of rectangle rules for approximating one dimensional integrals and trapezoidal rules for periodic integrands error bounds for rank 1 rules are known for a variety of classes of integrands for periodic integrands with unit period in each variable these bounds are conveniently characterized by the figure of merit p which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n nodes then becomes that of finding rules with large values of p this paper presents a new approach based on the theory of simultaneous diophantine approximation which uses a generalized continued fraction algorithm to construct rank 1 rules of high order

intermediate rank lattice rules for multidimensional integration every lattice rule in s dimensions may be characterised by an integer m lying between 1 and s inclusive called the rank or index of the rule and a set of m positive integers called the invariants earlier work has shown that in a certain precise sense lattice rules of rank s are better than the commonly used rank 1 rules here this earlier work is extended by showing that a similar result holds for certain lattice rules of intermediate rank m s

some properties of rank 2 lattice rules a rank 2 lattice rule is a quadrature rule for the unit ital s dimensional hypercube of the form ital qf 1 ital n sub 1 ital n sub 2 summation sub ital j sub 1 1 ital j sub 2 1 sup ital n sub 1 summation sub ital j sub 2 1 sup ital n sub 2 ital bar f ital j sub 1 ital z1 ital n sub 1 ital j sub 2 ital z2 ital n sub 2 which cannot be re expressed in an analogous form with a single sum here ital bar f is a periodic extension of ital f and ital z sub 1 ital z sub 2 are integer vectors in this paper we discuss these rules in detail in particular we categorize a special subclass whose leading one and two dimensional projections contain the maximum feasible number of abscissas we show that rules of this subclass can be expressed uniquely in a simple tricycle form

first order system least squares fosls for coupled fluid elastic problems mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the eulerian fluid domain is not known a priori it is at least partially determined by the displacement of the elastic solid in this paper a first order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled the discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration the numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom the results also show that the choice of iteration method from fully coupled to fully decoupled does not significantly effect computational cost but it does influence the error in the solution

the existence of efficient lattice rules for multidimensional numerical integration in this contribution to the theory of lattice rules for multidimensional numerical integration we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants which extends an earlier result of the author for lattice rules of rank 1

on mass conserving least squares methods least squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations including symmetric positive definite discrete operators and a sharp error measure one of the potential drawbacks especially in three dimensions is that mass conservation is achieved only in a least squares sense and underresolved solutions are especially vulnerable to poor conservation for the stationary navier stokes equations which are typically rewritten as a larger system of first order equations the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used a new first order system reformulation of the navier stokes equations is presented that admits a wider range of mass conserving boundary conditions this new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems a square channel backward facing step and branching tubes with two generations of bifurcations

component by component construction of good intermediate rank lattice rules it is known that the generating vector of a rank 1 lattice rule can be constructed component by component to achieve strong tractability error bounds in both weighted korobov spaces and weighted sobolev spaces since the weights for these spaces are nonincreasing the first few variables are in a sense more important than the rest we thus propose to copy the points of a rank 1 lattice rule a number of times in the first few dimensions to yield an intermediate rank lattice rule we show that the generating vector and in weighted sobolev spaces the shift also of an intermediate rank lattice rule can also be constructed component by component to achieve strong tractability error bounds in certain circumstances these bounds are better than the corresponding bounds for rank 1 lattice rules

the ultratriangular form for prime power lattice rules a recent development in the theory of lattice rules has been the introduction of the unique ultratriangular d z form for prime power rules it is known that any lattice rule may be decomposed into its sylow p components these components are prime power rules each of which has a unique ultratriangular form by reassembling these ultratriangular forms in a defined way it is possible to obtain a canonical form for any lattice rule a special case occurs when the ultratriangular forms of each of the sylow p components have a consistent set of column indices in this case it is possible to obtain a unique canonical d z form given the column indices and the invariants for an ultratriangular form we may obtain a formula for the number of ultratriangular forms and hence the number of prime power lattice rules having these column indices and invariants

principal component analysis of fuzzy data using autoassociative neural networks this paper describes an extension of principal component analysis pca allowing the extraction of a limited number of relevant features from high dimensional fuzzy data our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as pca without explicit matrix diagonalization fuzzy input values are propagated through the network using fuzzy arithmetics and the weights are adjusted to minimize a suitable error criterion the inputs being taken as target outputs the concept of correlation coefficient is extended to fuzzy numbers allowing the interpretation of the new features in terms of the original variables experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data

a characterization of the extension principle abstract we provide an alternative characterization of the extension principle of fuzzy sets this characterization is based upon using relations to represent mappings we apply this new characterization to develop an extension for non deterministic mappings

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

the index of merit of kth copy integration lattices r2d2lri is an automatic two dimensional cubature algorithm that demonstrates the practical value of using an augmentation sequence consisting of 2 k 2 copy lattices as a basis for numerical integration this paper investigates use of similar embedded augmentation sequences in higher dimensions by developing theoretical results relating to the index of merit of s dimensional 2 k s copy lattices generated from rank 1 simple lattices the theoretical results can be used to guide the search for good augmentation sequences in s dimensions in the sense of high index of merit

fuzzy portfolio allocation models through a new risk measure and fuzzy sharpe ratio a new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper beyond the well known sharpe ratio i e the reward to variability ratio in modern portfolio theory we initiate the so called fuzzy sharpe ratio in the fuzzy modeling context in addition to the introduction of the new risk measure we also put forward the reward to uncertainty ratio to assess the portfolio performance in fuzzy modeling corresponding to two approaches based on rm t rm m and rm t rm w fuzzy arithmetic two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized while the fuzzy sharpe ratio is maximized these models are solved by the fuzzy approach or by the genetic algorithm ga solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean variance optimization mvo model used prevalently in the financial literature in terms of portfolio performance evaluated by the fuzzy sharpe ratio and the reward to uncertainty ratio the model using rm t rm w fuzzy arithmetic results in higher performance portfolios than those obtained by both the mvo and the fuzzy model which employs rm t rm m fuzzy arithmetic we also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using ga although ga can offer a series of well diversified portfolio solutions diagrammed in a pareto frontier

robustness and scalability of algebraic multigrid algebraic multigrid amg is currently undergoing a resurgence in popularity due in part to the dramatic increase in the need to solve physical problems posed on very large unstructured grids while amg has proved its usefulness on various problem types it is not commonly understood how wide a range of applicability the method has in this study we demonstrate that range of applicability while describing some of the recent advances in amg technology moreover in light of the imperatives of modern computer environments we also examine amg in terms of algorithmic scalability finally we show some of the situations in which standard amg does not work well and indicate the current directions taken by amg researchers to alleviate these difficulties

correlation and information energy of interval valued fuzzy numbers in this paper we introduce the correlation coefficient of interval valued fuzzy numbers and study some of their properties furthermore we put forward the concept of information energy and how to compare interval valued fuzzy numbers by applying them

multi p preconditioners a class of algebraic multi p methods for solving the p version of the finite element equations is first presented and discussed it is then shown how these multi p methods can be used as preconditioners for the conjugate gradient cg method in particular it is shown that given any preconditioner mp to cg a multi p preconditioner bp based on mp can be constructed which leads to a smaller condition number and hence faster convergence numerical experiments on representative problems indicate that the condition numbers after multi p preconditionings are in fact independent of p the numerical results also show greater efficiency for the preconditioned conjugate gradient pcg method with the multi p preconditioners in terms of number of iterations and cpu time when compared with two sophisticated linear equation solvers 1 a direct frontal solver specially designed for the p version of the finite element analysis 2 a highly tuned pcg code in itpack2c http www netlib org itpack 1994 preliminary comparisons of the number of iterations are also made with rockits solvers international inc boulder co 1994 a new commercial code used for the p version

algebraic multigrid for moderate order finite elements we investigate the use of algebraic multigrid amg methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with lagrangian finite elements of order at most 4 the resulting system matrices do not have the m matrix property that is required by standard analyses of classical amg and aggregation based amg methods a unified approach is presented that allows us to extend these analyses it uses an intermediate m matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix this constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations provided that jumps are located at elements boundaries for two dimensional problems it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound this analysis validates the application of the amg methods to the

integration of nonperiodic functions of two variables by fibonacci lattice rules abstract two dimensional lattice rules are applied to continuous functions over the unit square which do not have a continuous periodic extension it is shown that provided lattice points at vertices and edges are treated appropriately certain functions including all bilinear functions are integrated exactly whenever the lattice contains a possibly rotated square unit cell the fibonacci lattice with denominators f k for the nodes is then shown to have a square unit cell if and only if k is odd numerical experiments for fibonacci rules and copies of fibonacci rules confirm that there are significant different between the odd k and even k cases

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

multigrid solver for the inner problem in domain decomposition methods for p fem from the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p version finite element discretizations of elliptic boundary value problems the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner related to the dirichlet problems in the subdomains and an extension operator from the boundaries of the subdomains into their interior in the case of poisson s equation we propose a preconditioner for the problems in the subdomains which can be interpreted as the stiffness matrix resulting from an h version finite element discretization of a degenerate operator for solving the corresponding systems of finite element equations a multigrid algorithm with a special line smoother is used we prove that the convergence rate of the multigrid method is independent of the discretization parameter the proof is based on the strengthened cauchy inequality the theoretical result is confirmed by numerical examples

a stable and scalable hybrid solver for rate type non newtonian fluid models we present and analyze hybrid discretization schemes for rate type non newtonian fluids models the method employs higher order conforming approximations for velocity and pressure of the stokes equation and lower order approximations such as piecewise linear or piecewise constant for conformation tensor to reduce the accuracy gap the constitutive equation is discretized on a refined mesh which is obtained by subdividing the mesh for the velocity and pressure the temporal discretization is made by the standard semi lagrangian scheme the fully discrete nonlinear system is shown to be solved iteratively by applying the three steps 1 locating the characteristic feet of fluid particles 2 solving the constitutive equation and 3 solving the momentum and continuity equations stokes type equation to achieve a scalability of the solution process we employ an auxiliary space preconditioning method for the solution to the conforming finite element methods for the stokes equation this method is basically two grid method in which lower order finite element spaces are employed as auxiliary spaces it is shown to not only lead to the mesh independent convergence but also improve robustness and scalability stability analysis shows that if t o h d where d is the dimension of domain then the scheme admits a globally unique solution a number of full 3d test cases are provided to demonstrate the advantages of the proposed numerical techniques in relation to efficiency robustness and weak scalability

on computing the lattice rule criterion r lattice rules are integration rules for approximating integrals of periodic functions over the s dimensional unit cube one criterion for measuring the goodness of lattice rules is the quantity r this quantity is defined as a sum which contains about n sup s 1 terms where n is the number of quadrature points although various bounds involving r are known a procedure for calculating r itself does not appear to have been given previously here we show how an asymptotic series can be used to obtain an accurate approximation to r whereas an efficient direct calculation of r requires o nn sub 1 operations where n sub 1 is the largest invariant of the rule the use of this asymptotic expansion allows the operation count to be reduced to o n a complete error analysis for the asymptotic expansion is given the results of some calculations of r are also given 19 refs 2 tabs

controllability method for acoustic scattering with spectral elements we formulate the helmholtz equation as an exact controllability problem for the time dependent wave equation the problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements this approach leads to high accuracy in spatial discretization moreover the spectral element method results in diagonal mass matrices which makes the time integration of the wave equation highly efficient after discretization the exact controllability problem is reformulated as a least squares problem which is solved by the conjugate gradient method we illustrate the method with some numerical experiments which demonstrate the significant improvements in efficiency due to the higher order spectral elements for a given accuracy the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method in addition by using higher order polynomial basis the influence of the pollution effect is reduced

fuzzy measures for correlation coefficient of fuzzy numbers correlation coefficient of random variables has wide applications in statistical analysis this paper extends the applications to fuzzy environment with a methodology for calculating the correlation coefficient of fuzzy numbers developed different from previous studies the correlation coefficient calculated in this paper is a fuzzy number rather than a crisp value the idea is based on zadeh s extension principle a pair of nonlinear programs is formulated to find the cut of the fuzzy correlation coefficient from different values of the membership function of the fuzzy correlation coefficient is constructed to illustrate how to interpret the fuzzy correlation coefficient in real world applications the correlation between the technology level and management achievement from a sample of 15 machinery firms in taiwan is exemplified all indications show that the correlation between technology and management in taiwan machinery firms is rather low

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

minimal cubature formulae of trigonometric degree in this paper we construct minimal cubature formulae of trigonometric degree we obtain explicit formulae for low dimensions of arbitrary degree and for low degrees in all dimensions a useful tool is a closed form expression for the reproducing kernels in two dimensions

letter section randomization of lattice rules for numerical multiple integration here we investigate some procedures for the randomization of lattice rules for numerical multiple integration similar to those proposed by cranley and patterson 1976 for number theoretic rules currently there is no easily calculable error estimate available for general lattice rules and the randomization procedures looked at here allow the calculation of confidence intervals for the error

lattice methods for multiple integration theory error analysis and examples this paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension the concept of a lattice rule is introduced and error bounds are developed in terms of the dual lattice examples of lattice rules discussed in the paper are the rectangle rule the number theoretic good lattice methods of korobov and others the body centred cubic rule and a generalization of the latter to a family w nr the w nr rules appear to have interesting extrapolation properties both for fixed r and for r n

correlation of fuzzy numbers abstract in this paper we introduce the concepts of correlation and correlation coefficient of fuzzy numbers which could be used to measure the interrelation of fuzzy numbers we also give a brief discussion about the so called information energy of fuzzy numbers

a computer search of rank 2 lattice rules for multidimensional quadrature for certain lattice rules of rank 2 it has been shown that a unique representation exists in a form suitable for computer evaluation the present paper describes computer searches of such rules reports results and identifies rules that appear promising for the numerical evaluation of practical multidimensional integrals

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

time harmonic elasticity with controllability and higher order discretization methods the time harmonic solution of the linear elastic wave equation is needed for a variety of applications the typical procedure for solving the time harmonic elastic wave equation leads to difficulties solving large scale indefinite linear systems to avoid these difficulties we consider the original time dependent equation with a method based on an exact controllability formulation the main idea of this approach is to find initial conditions such that after one time period the solution and its time derivative coincide with the initial conditions the wave equation is discretized in the space domain with spectral elements the degrees of freedom associated with the basis functions are situated at the gauss lobatto quadrature points of the elements and the gauss lobatto quadrature rule is used so that the mass matrix becomes diagonal this method is combined with the second order central finite difference or the fourth order runge kutta time discretization as a consequence of these choices only matrix vector products are needed in time dependent simulation this makes the controllability method computationally efficient

controllability method for the helmholtz equation with higher order discretizations we consider a controllability technique for the numerical solution of the helmholtz equation the original time harmonic equation is represented as an exact controllability problem for the time dependent wave equation this problem is then formulated as a least squares optimization problem which is solved by the conjugate gradient method such an approach was first suggested and developed in the 1990s by french researchers and we introduce some improvements to its practical realization we use higher order spectral elements for spatial discretization which leads to high accuracy and lumped mass matrices higher order approximation reduces the pollution effect associated with finite element approximation of time harmonic wave equations and mass lumping makes explicit time stepping schemes for the wave equation very efficient we also derive a new way to compute the gradient of the least squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm numerical results demonstrate the significant improvements in efficiency due to the higher order spectral elements for a given accuracy spectral element method requires fewer computational operations than conventional finite element method in addition by using higher order polynomial basis the influence of the pollution effect is reduced

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

the representation of lattice quadrature rules as multiple sums we provide a classification of lattice rules applying elementary group theory we assign to each s dimensional lattice rule a rank m and a set of positive integer invariants n sub 1 n sub 2 n sub s the number nu q of abscissas required by the rule is the product n sub 1 n sub 2 n sub s and the rule may be expressed in a canonical form with m independent summations under this classification an n point number theoretic rule in the sense of korobov and conroy is a rank m 1 rule having invariants n 1 1 1 and the product trapezoidal rule using n sup s points in rank m s rule having invariants n n n besides providing a canonical form we give some of the properties of copy rules and of projections into lower dimensions

projection multilevel methods for quasilinear elliptic partial differential equations theoretical results the goal of this paper is to introduce a new multilevel solver for two dimensional elliptic systems of nonlinear partial differential equations pdes where the nonlinearity is of the type u partial v the incompressible navier stokes equations are an important representative of this class and are the target of this study using a first order system least squares fosls approach and introducing a new variable for partial v for this class of pdes we obtain a formulation in which the nonlinearity appears as a product of two different dependent variables the result is a system that is linear within each variable but nonlinear in the cross terms in this paper we introduce a new multilevel method that treats the nonlinearities directly this approach is based on a projection multilevel pml method s f mccormick multilevel projection methods for partial differential equations siam philadelphia 1992 applied to the fosls functional the implementation of the discretization process relaxation coarse grid correction and cycling strategies is discussed and optimal performance is established numerically a companion paper t a manteuffel s f mccormick and o rohrle siam j numer anal 44 2006 pp 139 152 establishes a two level convergence proof for this new multilevel method

triangular canonical forms for lattice rules of prime power order in this paper the authors develop a theory of t cycle d z representations for s dimensional lattice rules of prime power order of particular interest are canonical forms which by definition have a d matrix consisting of the nontrivial invariants among these is a family of triangular forms which besides being canonical have the defining property that their z matrix is a column permuted version of a unit upper triangular matrix triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra the authors main result is to define a unique canonical form for prime power rules this ultratriangular form is a triangular form is easy to recognize and may be derived in a straightforward manner 12 refs

on the fly adaptive smoothed aggregation multigrid for markov chains a new adaptive algebraic multigrid scheme is developed for the solution of markov chains where the hierarchy of operators is adapted on the fly in a setup process that is interlaced with the solution process the setup process feeds the solution process with improved operators while the solution process provides the adaptive setup process with better approximations on which to base further improved operators the approach is demonstrated using petrov galerkin smoothed aggregation where only the prolongation operator is smoothed while the restriction remains of low order results show that the on the fly adaptive scheme can improve the performance of multigrid solvers that require extensive setup computations in both serial and parallel environments

semi automatic sparse preconditioners for high order finite element methods on non uniform meshes high order finite elements often have a higher accuracy per degree of freedom than the classical low order finite elements however in the context of implicit time stepping methods high order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix there are many cases where simulations are limited by the memory required to store the matrix and or the algorithmic components of the linear solver we are particularly interested in preconditioned krylov methods for linear systems generated by discretization of elliptic partial differential equations with high order finite elements using a preconditioner like algebraic multigrid can be costly in terms of memory due to the need to store matrix information at the various levels we present a novel method for defining a preconditioner for systems generated by high order finite elements that is based on a much sparser system than the original high order finite element system we investigate the performance for non uniform meshes on a cube and a cubed sphere mesh showing that the sparser preconditioner is more efficient and uses significantly less memory finally we explore new methods to construct the sparse preconditioner and examine their effectiveness for non uniform meshes we compare results to a direct use of algebraic multigrid as a preconditioner and to a two level additive schwarz method