fast fourier transforms of piecewise constant functions we present an algorithm for the evaluation of the fourier transform of piecewise constant functions of two variables the algorithm overcomes the accuracy problems associated with computing the fourier transform of discontinuous functions in fact its time complexity is o n2 logn np log2 1 v log3 1 where is the accuracy n is the size of the problem p is the perimeter of the set of discontinuities and v is its number of vertices the algorithm is based on the lagrange interpolation formula and the green s theorem which are used to preprocess the data before applying the fast fourier transform it readily generalizes to higher dimensions and to piecewise smooth functions

decomposition of hessenberg dae systems to state space form abstract an algorithm is given for symbolically decoupling the solutions to a linear time dependent differential algebraic equation ez a t z t z t r s in hessenberg form into state and algebraic components the state variables are the solutions to an ordinary differential equation with initial conditions restricted to a subspace of rs while the algebraic components are linear functions of the state variables involving derivatives of the coefficients and input functions up to order r 1 where r is the index of the system this decomposition provides closed form solutions to linear hessenberg daes in terms of the fundamental solutions of the state variable system the implications of the algorithm for computing consistent initial conditions for certain singular optimal control problems and for numerical solutions are briefly discussed

uniqueness of completions for linear time varying differential algebraic equations abstract the extent to which a completion x g t x r i 0 r i t f i t of a linear ti me varying differential algebraic equation e t x t f t x t f t is unique is carefully examined the implications for numerical methods for solving daes based on differentiated equations are discussed

numerical solution of nonlinear differential equations with algebraic constraints i convergence results for backward differentiation formulas in this paper we investigate the behavior of numerical ode methods for the solution of systems of differential equations coupled with algebraic constraints systems of this form arise frequently in the modelling of problems from physics and engineering we study some particular examples from electrical networks fluid dynamics and constrained mechanical systems we show that backward differentiation formulas converge with the expected order of accuracy for these systems

large eddy simulation of turbulent channel flows with conservative ido scheme the resolution of a numerical scheme in both physical and fourier spaces is one of the most important requirements to calculate turbulent flows a conservative form of the interpolated differential operator ido cf scheme is a multi moment eulerian scheme in which point values and integrated average values are separately defined in one cell since the ido cf scheme using high order interpolation functions is constructed with compact stencils the boundary conditions are able to be treated as easy as the 2nd order finite difference method fdm it is unique that the first order spatial derivative of the point value is derived from the interpolation function with 4th order accuracy and the volume averaged value is based on the exact finite volume formulation so that the ido cf scheme has higher spectral resolution than conventional fdms with 4th order accuracy the computational cost to calculate the first order spatial derivative with non uniform grid spacing is one third of the 4th order fdm for a large eddy simulation les we use the coherent structure model csm in which the model coefficient is locally obtained from a turbulent structure extracted from a second invariant of the velocity gradient tensor and the model coefficient correctly satisfies asymptotic behaviors to walls the results of the ido cf scheme with the csm for turbulent channel flows are compared to the fdm with the csm and dynamic smagorinsky model as well as the direct numerical simulation dns by moser et al adding the sub grid scale stress tensor of les to the ido cf scheme improves the profile of the mean velocity in comparison with an implicit eddy viscosity of the ido cf upwind scheme the ido cf scheme with the csm gives better turbulent intensities than conventional fdms with the same number of grid points the turbulent statistics calculated by ido cf scheme are in good agreement with the dns at the various values of reynolds number re t 180 395 and 590 it is found that the ido cf scheme is suitable for the turbulent flow computation and improves the turbulent statistics with compact stencils

exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids this paper is concerned with two important elements in the high order accurate spatial discretization of finite volume equations over arbitrary grids one element is the integration of basis functions over arbitrary domains which is used in expressing various spatial integrals in terms of discrete unknowns the other consists of quadrature approximations to those integrals only polynomial basis functions applied to polyhedral and polygonal grids are treated here nontriangular polygonal faces are subdivided into a union of planar triangular facets and the resulting triangulated polyhedron is subdivided into a union of tetrahedra the straight line segment triangle and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices results are given both in terms of tensor products and products of cartesian coordinates the exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains using a coordinate free formulation simple and rational procedures are developed to derive virtually all quadrature formulas including some previously unpublished four symmetry groups of quadrature points are introduced to derive gauss formulas while their limiting forms are used to derive lobatto formulas representative gauss and lobatto formulas are tabulated the relative efficiency of their application to polyhedral and polygonal grids is detailed the extension to higher degrees of precision is discussed

fast fourier transform for discontinuous functions in computational electromagnetics and other areas of computational science and engineering fourier transforms of discontinuous functions are often required we present a fast algorithm for the evaluation of the fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast fourier transform fft algorithm we call this the discontinuous fft algorithm for n sample points the complexity of the algorithm is o spl nu np spl nu nlog n where p is the interpolation order and spl nu is the oversampling factor the method also provides a new nonuniform fft algorithm for continuous functions numerical experiments demonstrate the high efficiency and accuracy of this discontinuous fft algorithm

the numerical solution of higher index differential algebraic equations by implicit methods this paper studies the order stability and convergence properties of implicit runge kutta irk methods applied to differential algebraic systems with index greater than one these methods do not in general attain the same order of accuracy for higher index differential algebraic systems as they do for index 1 systems or for purely differential systems necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems irk methods applied to nonlinear semi explicit index 2 systems are studied and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems finally some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff errors on the solution

a structural form for higher index semistate equations i theory and applications to circuit and control theory abstract we investigate the theoretical properties of a structural form for higher index linear time varying semistate systems x a t x b t u 1 t 0 c t x d t u 2 t where d t is singular but has constant rank for all t 0 t many systems of interest in circuit and control theory exhibit the form discussed in particular circuits with differentiators and linear quadratic regulator problems with higher order singular arcs in addition to the semiexplicit structure the systems we describe are characterized by a block upper hessenberg coefficient a t a zero pattern in c t and b t and an invertibility condition on the components of c b and the subdiagonal blocks of a we show that there exists a coordinate system in which the hessenberg form is equivalent to a semiexplicit index 1 system embedded between two linear chains of differentiators in a companion paper we establish the convergence and stability of backward differentiation formulas bdf applied to the index 4 hessenberg form extending the theory for the index 2 and index 3 special cases

the regular fourier matrices and nonuniform fast fourier transforms for any triple of positive integers m n q the matrix f m n q called the m n q regular fourier matrix is defined the regular fourier matrices f m n q are then applied to set up new algorithms for nonuniform fast fourier transforms numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity the algorithms require o n cdot log 2n arithmetic operations where n is the number of data points

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

order results for implicit runge kutta methods applied to differential algebraic systems in this paper we study the order stability and convergence properties of implicit runge kutta methods applied to a relatively simple class of nonlinear differential algebraic systems these methods often do not attain the same order of accuracy for differential algebraic systems as they do for purely differential systems we derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy and we present results on the stability and convergence properties of these methods

robust rank procedures for the behrens fisher problem abstract the problem of testing for equality of the medians of two populations is considered the standard distribution free tests for this problem require that the two populations have the same shape to maintain their nominal significance level under the null hypothesis a method is given to modify many of these tests so they can be used to test for equality of the medians with fewer assumptions on the shapes of the two populations the method is demonstrated using the mann whitney wilcoxon statistic

an accurate conformal fourier transform method for 3d discontinuous functions fourier transform of discontinuous functions are often encountered in computational electromagnetics and other areas in this work a highly accurate fast conformal fourier transform cft algorithm is proposed to evaluate the finite fourier transform of 3d discontinuous functions a curved tetrahedron mesh combined with curvilinear coordinate transform instead of the cartesian grid is adopted to flexibly model an arbitrary shape of the discontinuity boundary this enables us to take full advantages of high order interpolation and gaussian quadrature methods to achieve highly accurate fourier integration results with a low sampling density the 3d nonuniform fast fourier transform nufft helps to keep the complexity of the proposed algorithm to that similar to the traditional 3d fft algorithm therefore the proposed cft algorithm can achieve order of magnitude higher accuracy than 3d fft with lower sampling density and similar computation time the convergence is proved and verified

a spectral fc solver for the compressible navier stokes equations in general domains i explicit time stepping we present a fourier continuation fc algorithm for the solution of the fully nonlinear compressible navier stokes equations in general spatial domains the new scheme is based on the recently introduced accelerated fc method which enables use of highly accurate fourier expansions as the main building block of general domain pde solvers previous fc based pde solvers are restricted to linear scalar equations with constant coefficients the fc methodology presented in this text thus constitutes a significant generalization of the previous fc schemes as it yields general domain fc solvers for nonlinear systems of pdes while not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent fourier expansions including high order convergence essentially spectrally accurate dispersion relations and much milder cfl constraints than those imposed by polynomial based spectral methods since for example the spectral radius of the fc first derivative grows linearly with the number of spatial discretization points we demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid dynamics and nonlinear acoustics

fast fourier transform based iterative method for electromagnetic scattering from 1d flat surfaces an efficient iterative method is proposed for computing the electromagnetic fields scattered from a one dimensional 1d flat surface the new iterative method is based on a similar implementation to the conjugate gradient fast fourier transform cg fft where acceleration of the matrix vector multiplications is achieved using fast fourier transforms fft however the iterative method proposed is not based on krylov subspace expansions and is shown to converge faster than gmres fft and cgne fft while maintaining the computational complexity and memory usage of those methods analysis is presented deriving an explicit convergence criterion

effects attributable to treatment inference in experiments and observational studies with a discrete pivot summary in randomisation and permutation inference pivotal arguments remove the hypothesised treatment effect thereby basing inferences on the null distribution in which the treatment has no effect this is common for instance with additive treatment effects the current paper uses attributable effects to expand substantially the scope of pivotal arguments attributable effects are defined for three cases namely the 2 x 2 contingency table displacement effects and the mann whitney wilcoxon statistic and in each case removing an appropriate attributable effect restores the familiar null randomisation distribution of the associated statistic yielding exact inferences the procedure extends immediately for use in sensitivity analysis in nonrandomised observational studies

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

affine invariant multivariate one sample signed rank tests abstract brown and hettmansperger introduced affine invariant bivariate analogs of the sign rank and signed rank tests based on the oja median in this article affine invariant k variate extensions of the one sample signed rank test and the hodges lehmann estimate are considered the necessary distribution theory is developed and asymptotic pitman efficiencies with respect to hotelling s t 2 test under multivariate t distributions are tabulated an application of the signed rank tests to a repeated measurement setting is presented

backward differentiation approximations of nonlinear differential algebraic systems finite difference approximations of dynamical systems modeled by nonlinear semiexplicit differential algebraic equations are analyzed convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints the appropriate asymptotic convergence rates and the leading error terms are determined

a class of distribution free two sample tests based on placements abstract we consider a class of two sample distribution free tests that are appropriate for situations where one of the sample sizes is large relative to the other these procedures are based on the placements of the observations in the smaller sample among the ordered observations in the larger sample and this class of tests generalizes the mann whitney 1947 procedure in much the same way that the class of linear rank tests generalizes the equivalent wilcoxon 1945 rank sum form optimality criteria for choosing a test from this class are discussed and limiting distributions for the associated class of test statistics are determined for the case where only one of the sample sizes goes to infinity

the fast multipole method and fourier convolution for the solution of acoustic scattering on regular volumetric grids the fast multipole method fmm is applied to the solution of large scale three dimensional acoustic scattering problems involving inhomogeneous objects defined on a regular grid the grid arrangement is especially well suited to applications in which the scattering geometry is not known a priori and is reconstructed on a regular grid using iterative inverse scattering algorithms or other imaging techniques the regular structure of unknown scattering elements facilitates a dramatic reduction in the amount of storage and computation required for the fmm both of which scale linearly with the number of scattering elements in particular the use of fast fourier transforms to compute green s function convolutions required for neighboring interactions lowers the often significant cost of finest level fmm computations and helps mitigate the dependence of fmm cost on finest level box size numerical results demonstrate the efficiency of the composite method as the number of scattering elements in each finest level box is increased

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

optimal sample size planning for the wilcoxon mann whitney test there are many different proposed procedures for sample size planning for the wilcoxon mann whitney test at given type i and type ii error rates and respectively most methods assume very specific models or types of data to simplify calculations eg ordered categorical or metric data location shift alternatives etc we present a unified approach that covers metric data with and without ties count data ordered categorical data and even dichotomous data for that we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following synthetic data approach we evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities then well known relations for the ranks of the data are used for the calculations in addition to computing the necessary sample size n for a fixed allocation proportion t n1 n where n1 is the sample size in the first group and n n1 n2 is the total sample size we provide an interval for the optimal allocation rate t which minimizes the total sample size n it turns out that for certain distributions a balanced design is optimal we give a characterization of such distributions furthermore we show that the optimal choice of t depends on the ratio of the two variances which determine the variance of the wilcoxon mann whitney statistic under the alternative this is different from an optimal sample size allocation in case of the normal distribution model