numerical method for solving matrix coefficient elliptic equation with sharp edged interfaces solving elliptic equations with sharp edged interfaces is a challenging problem for most existing methods especially when the solution is highly oscillatory nonetheless it has wide applications in engineering and science an accurate and efficient method is desired we propose an efficient non traditional finite element method with non body fitting grids to solve the matrix coefficient elliptic equations with sharp edged interfaces extensive numerical experiments show that this method is second order accurate in the l norm and that it can handle both sharp edged interface and oscillatory solutions

a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces we present a finite volume method for the solution of the two dimensional elliptic equation s x u x f x with variable discontinuous coefficients and solution discontinuities on irregular domains the method uses bilinear ansatz functions on cartesian grids for the solution u x resulting in a compact nine point stencil the resulting linear problem has been solved with a standard multigrid solver singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two step asymptotic approach the method achieves second order of accuracy in the l and l2 norm

mib galerkin method for elliptic interface problems material interfaces are omnipresent in the real world structures and devices mathematical modeling of material interfaces often leads to elliptic partial differential equations pdes with discontinuous coefficients and singular sources which are commonly called elliptic interface problems the development of high order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades despite of significant advances challenges remain in the construction of high order schemes for nonsmooth interfaces i e interfaces with geometric singularities such as tips cusps and sharp edges the challenge of geometric singularities is amplified when they are associated with low solution regularities e g tip geometry effects in many fields the present work introduces a matched interface and boundary mib galerkin method for solving two dimensional 2d elliptic pdes with complex interfaces geometric singularities and low solution regularities the cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure consequently the interface cuts through elements to ensure the continuity of classic basis functions across the interface two sets of overlapping elements called mib elements are defined near the interface as a result differentiation can be computed near the interface as if there is no interface interpolation functions are constructed on mib element spaces to smoothly extend function values across the interface a set of lowest order interface jump conditions is enforced on the interface which in turn determines the interpolation functions the performance of the proposed mib galerkin finite element method is validated by numerical experiments with a wide range of interface geometries geometric singularities low regularity solutions and grid resolutions extensive numerical studies confirm the designed second order convergence of the mib galerkin method in the l and l 2 errors some of the best results are obtained in the present work when the interface is c 1 or lipschitz continuous and the solution is c 2 continuous

solving elliptic problems with discontinuities on irregular domains the voronoi interface method we introduce a simple method dubbed the voronoi interface method to solve elliptic problems with discontinuities across the interface of irregular domains this method produces a linear system that is symmetric positive definite with only its right hand side affected by the jump conditions the solution and the solution s gradients are second order accurate and first order accurate respectively in the l norm even in the case of large ratios in the diffusion coefficient this approach is also applicable to arbitrary meshes additional degrees of freedom are placed close to the interface and a voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach both the locations of the additional degrees of freedom and their voronoi discretizations are straightforward in two and three spatial dimensions

on preconditioning the treecode accelerated boundary integral tabi poisson boltzmann solver abstract we recently developed a treecode accelerated boundary integral tabi solver for solving poisson boltzmann pb equation 1 the solver has combined advantages in accuracy efficiency memory and parallelization as it applies a well posed boundary integral formulation to circumvent many numerical difficulties associated with the pb equation and uses an o n log n treecode to accelerate the gmres iterative solver however as observed in our previous work 2 occasionally when the mesh generator produces low quality triangles the number of gmres iterations required to solve the discretized boundary integral equations a x b could be large to address this issue we design a preconditioning scheme using preconditioner matrix m such that m 1 a has much improved condition while m 1 z can be rapidly computed for any vector z in this scheme the matrix m carries the interactions between boundary elements on the same leaf only in the tree structure thus is block diagonal with many computational advantages the sizes of the blocks in m are conveniently controlled by the treecode parameter n 0 the maximum number of particles per leaf the numerical results show that this new preconditioning scheme improves the tabi solver with significantly reduced iteration numbers and better accuracy particularly for protein sets on which tabi solver previously converges slowly in addition this preconditioning scheme potentially can improve the condition number of various multipole method accelerated boundary elements solvers in scattering fluids elasticity etc

new constructions of quadratic bent functions in polynomial form new quadratic bent functions in polynomial form are constructed in this paper the constructions give new boolean bent generalized boolean bent and p ary bent functions based on z 4 valued quadratic forms a simple method provides several new constructions of generalized boolean bent functions from these generalized boolean bent functions a method is presented to transform them into boolean bent and semi bent functions moreover many new p ary bent functions can also be obtained by applying similar methods

testing linearity against smooth transition autoregressive models summary we study a general univariate smooth transition autoregressive star model it contains as a special case the self exciting threshold autoregressive setar model we present three tests for testing linearity against star models and discuss their properties the power of the tests in small samples is investigated by simulation when the alternative is the logistic star model one of the tests is identical to tsay s 1986 test statistic and is recommended only in a special case of the two remaining tests with wider applicability one seems superior to the other in small samples it is also more powerful than the cusum test recently proposed for testing linearity against setar models

a galerkin formulation of the mib method for three dimensional elliptic interface problems we develop a three dimensional 3d galerkin formulation of the matched interface and boundary mib method for solving elliptic partial differential equations pdes with discontinuous coefficients i e the elliptic interface problem the present approach builds up two sets of elements respectively on two extended subdomains which both include the interface as a result two sets of elements overlap each other near the interface fictitious solutions are defined on the overlapping part of the elements so that the differentiation operations of the original pdes can be discretized as if there was no interface the extra coefficients of polynomial basis functions which furnish the overlapping elements and solve the fictitious solutions are determined by interface jump conditions consequently the interface jump conditions are rigorously enforced on the interface the present method utilizes cartesian meshes to avoid the mesh generation in conventional finite element methods fems we implement the proposed mib galerkin method with three different elements namely rectangular prism element five tetrahedron element and six tetrahedron element which tile the cartesian mesh without introducing any new node the accuracy stability and robustness of the proposed 3d mib galerkin are extensively validated over three types of elliptic interface problems in the first type interfaces are analytically defined by level set functions these interfaces are relatively simple but admit geometric singularities in the second type interfaces are defined by protein surfaces which are truly arbitrarily complex the last type of interfaces originates from multiprotein complexes such as molecular motors near second order accuracy has been confirmed for all of these problems to our knowledge it is the first time for an fem to show a near second order convergence in solving the poisson equation with realistic protein surfaces additionally the present work offers the first known near second order accurate method for c 1 continuous or h 2 continuous solutions associated with a lipschitz continuous interface in a 3d setting

bbz _4 valued quadratic forms and quaternary sequence families in this paper zopf4 valued quadratic forms defined on a vector space over gf 2 are studied a classification of such forms is established distinguishing zopf4 valued quadratic forms only by their rank and whether the associated bilinear form is alternating this result is used to compute the distribution of certain exponential sums which occur frequently in the analysis of quaternary codes and quaternary sequence sets the concept is applied as follows when t 0 or m is odd the correlation distribution of family s t consisting of quaternary sequences of length 2 m 1 is established then motivated by practical considerations a subset s t of family s t is defined and the correlation distribution of family s t is given for odd and even m

a kernel free boundary integral method for elliptic boundary value problems this paper presents a class of kernel free boundary integral kfbi methods for general elliptic boundary value problems bvps the boundary integral equations reformulated from the bvps are solved iteratively with the gmres method during the iteration the boundary and volume integrals involving green s functions are approximated by structured grid based numerical solutions which avoids the need to know the analytical expressions of green s functions the kfbi method assumes that the larger regular domain which embeds the original complex domain can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast fourier transform fft based poisson helmholtz solvers or those based on geometric multigrid iterations are applicable the structured grid based solutions are obtained with standard finite difference method fdm or finite element method fem where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme numerical results demonstrating the efficiency and accuracy of the kfbi methods are presented it is observed that the number of gmres iterations used by the method for solving isotropic and moderately anisotropic bvps is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals with the standard second order fems and fdms the kfbi method shows a second order convergence rate in accuracy for all of the tested dirichlet neumann bvps when the anisotropy of the diffusion tensor is not too strong

mib method for elliptic equations with multi material interfaces elliptic partial differential equations pdes are widely used to model real world problems due to the heterogeneous characteristics of many naturally occurring materials and man made structures devices and equipments one frequently needs to solve elliptic pdes with discontinuous coefficients and singular sources the development of high order elliptic interface schemes has been an active research field for decades however challenges remain in the construction of high order schemes and particularly for nonsmooth interfaces i e interfaces with geometric singularities the challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other high order methods for elliptic equations with multi material interfaces have not been reported in the literature to our knowledge the present work develops matched interface and boundary mib method based schemes for solving two dimensional 2d elliptic pdes with geometric singularities of multi material interfaces a number of new mib schemes are constructed to account for all possible topological variations due to two material interfaces the geometric singularities of three material interfaces are significantly more difficult to handle three new mib schemes are designed to handle a variety of geometric situations and topological variations although not all of them the performance of the proposed new mib schemes is validated by numerical experiments with a wide range of coefficient contrasts geometric singularities and solution types extensive numerical studies confirm the designed second order accuracy of the mib method for multi material interfaces including a case where the derivative of the solution diverges

a numerical method for solving elasticity equations with interface involving multi domains and triple junction points solving elasticity equations with interfaces on multiple domains has wide applications in engineering and science corner singularities make it difficult for most existing solvers to deal with a triple junction in the case of nonelastic problems therefore constructing an efficient and accurate solver for an elasticity problem with multiple domains is a challenge in this paper an efficient non traditional finite element method with non body fitted grids is proposed to solve elliptical elasticity equations with multi domains and triple junction points numerical experiments show that this method is approximately second order accurate in the l norm for piecewise smooth solutions

a matched interface and boundary method for solving multi flow navier stokes equations with applications to geodynamics we have developed a second order numerical method based on the matched interface and boundary mib approach to solve the navier stokes equations with discontinuous viscosity and density on non staggered cartesian grids we have derived for the first time the interface conditions for the intermediate velocity field and the pressure potential function that are introduced in the projection method differentiation of the velocity components on stencils across the interface is aided by the coupled fictitious velocity values whose representations are solved by using the coupled velocity interface conditions these fictitious values and the non staggered grid allow a convenient and accurate approximation of the pressure and potential jump conditions a compact finite difference method was adopted to explicitly compute the pressure derivatives at regular nodes to avoid the pressure velocity decoupling numerical experiments verified the desired accuracy of the numerical method applications to geophysical problems demonstrated that the sharp pressure jumps on the clast newtonian matrix are accurately captured for various shear conditions moderate viscosity contrasts and a wide range of density contrasts we showed that large transfer errors will be introduced to the jumps of the pressure and the potential function in case of a large absolute difference of the viscosity across the interface these errors will cause simulations to become unstable

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

linearity testing for fuzzy rule based models in this paper we introduce a linearity test for fuzzy rule based models in the framework of time series modeling to do so we explore a family of statistical models the regime switching autoregressive models and the relations that link them to the fuzzy rule based models from these relations we derive a lagrange multiplier linearity test and some properties of the maximum likelihood estimator needed for it finally an empirical study of the goodness of the test is presented

several classes of codes and sequences derived from a bbz_ 4 valued quadratic form let m and k be positive integers with m gcd m k being odd for a r and b l the exponential sum x litr ax 2bx2k 1 is studied systematically in this paper where i 1 r gr 4 m is a galois ring l is the teichmuller set of r and tr is the trace function from the galois ring r to z4 through the discussions on the solutions of certain equations and the newly developed theory of z4 valued quadratic forms the distribution of the exponential sum is completely determined as its applications we can determine the lee weight and hamming weight distributions of a class of codes ck over z4 and the correlation distribution of a quaternary sequence family uk respectively furthermore the hamming weight distributions of the binary codes obtained from ck under the most significant bit msb and gray maps are also determined for the msb map sequences of uk the nontrivial maximal correlation value is given and the correlation distribution is determined for the gray map sequences of uk it should be noted that the distribution of the exponential sum for the case gcd m k 1 is obtained for the first time and then the corresponding codes and sequences are novel

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

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

an improved non traditional finite element formulation for solving three dimensional elliptic interface problems solving elliptic equations with interfaces has wide applications in engineering and science the real world problems are mostly in three dimensions while an efficient and accurate solver is a challenge some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions although traditional finite element method using body fitted grid is well established the expensive cost of mesh generation is an issue in this paper an efficient non traditional finite element method with non body fitted grids is proposed to solve elliptic interface problems the special cases when the interface cuts though grid points are handled carefully rather than perturbing the cutting point away to apply the method for general case both dirichlet and neumann boundary conditions are considered numerical experiments show that this method is approximately second order accurate in the l norm and l2 norm for piecewise smooth solutions the large sparse matrix for our linear system also has nice structure and properties

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

local global neural networks a new approach for nonlinear time series modeling we propose the local global neural networks model within the context of time series models this formulation encompasses some already existing nonlinear models and also admits the mixture of experts approach we emphasize the linear expert case and extensively discuss the theoretical aspects of the model stationarity conditions existence consistency and asymptotic normality of the parameter estimates and model identifiability the proposed model consists of a mixture of stationary and nonstationary linear models and is able to describe intermittent dynamics the system spends a large fraction of time in a bounded region but sporadically develops an instability that grows exponentially for some time and then suddenly collapses intermittency is a commonly observed behavior in ecology and epidemiology fluid dynamics and other natural systems a model building strategy is also considered and the parameters are estimated by concentrated maximum likelihood the procedure is illustrated with two real

interface solutions of partial differential equations with point singularity abstract partial differential equations pdes such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips existing numerical methods for such pdes suffer from the constraint of this low regularity and fail to deliver an optimal high order rate of convergence we approach this problem by i choosing an artificial interface to enclose the center of low regularity and ii representing the solution in the interior of artificial interface as unknown linear combination of known low regularity modes of the solution this results in an interface formulation of the original pde and the linear combination is represented in the interface conditions by enforcing the smooth component of the numerical solution in the interior domain to be approximately zero a least squares problem is obtained for the unknown coefficients the solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the pde in the exterior domain the potential of our interface formulation is favorably demonstrated by numerical experiments on one and two dimensional poisson equations with low regularity solutions high order numerical solutions of unknown coefficients and pdes are obtained this proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems we address the relations between our interface formulation and domain decomposition methods as well as a regularization strategy for the poisson boltzmann equation with singular charge density

on quadratic bent functions in polynomial forms in this correspondence we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields the results improve some previous work moreover we solve a problem left by yu and gong in 2006

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

new binomial bent functions over the finite fields of odd characteristic the p ary function f x mapping gf p4k to gf p and given by f x tr4k xp3k p2k pk 1 x2 is proven to be a weakly regular bent function and the exact value of its walsh transform coefficients is found this is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic the proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

on bent and semi bent quadratic boolean functions the maximum length sequences also called m sequences have received a lot of attention since the late 1960s in terms of linear feedback shift register lfsr synthesis they are usually generated by certain power polynomials over a finite field and in addition are characterized by a low cross correlation and high nonlinearity we say that such a sequence is generated by a semi bent function some new families of such function represented by f x spl sigma sub i 1 sup n 1 2 c sub i tr x 2 sup i 1 n odd and c sub i spl isin f sub 2 have recently 2002 been introduced by khoo et al we first generalize their results to even n we further investigate the conditions on the choice of c sub i for explicit definitions of new infinite families having three and four trace terms also a class of nonpermutation polynomials whose composition with a quadratic function yields again a quadratic semi bent function is specified the treatment of semi bent functions is then presented in a much wider framework we show how bent and semi bent functions are interlinked that is the concatenation of two suitably chosen semi bent functions will yield a bent function and vice versa finally this approach is generalized so that the construction of both bent and semi bent functions of any degree in certain range for any n spl ges 7 is presented n being the number of input variables

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

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

constructions of quadratic bent functions in polynomial forms in this correspondence the constructions and enumerations of all bent functions represented by a polynomial form of f x spl sigma sub i 1 sup n 2 1 c sub i tr x sup 1 2 i c sub n 2 tr sub 1 sup n sup 2 x sup 1 2 n 2 c sub i spl isin sub 2 f sub 2 are presented for special cases of n using an iterative approach the construction of bent functions of n variables with degree n 2 is also provided using the constructed quadratic bent functions

a simple weak formulation for solving two dimensional diffusion equation with local reaction on the interface abstract in this paper we propose a numerical method for solving two dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface we use finite element method coupled with newton s method to deal with the jump conditions and to linearize the system it is easy to implement the grid used here is body fitting grids based on the idea of semi cartesian grid numerical experiments show that this method is nearly second order accurate in the l norm

finite difference ghost point multigrid methods on cartesian grids for elliptic problems in arbitrary domains in this paper we present a numerical method for solving elliptic equations in an arbitrary domain described by a level set function with general boundary conditions dirichlet neumann robin etc on cartesian grids using finite difference discretization and non eliminated ghost values a system of n i n g equations in n i n g unknowns is obtained by finite difference discretization on the n i internal grid points and second order interpolation to define the conditions for the n g ghost values the resulting large sparse linear system is then solved by a multigrid technique the novelty of the papers can be summarized as follows general strategy to discretize the boundary condition to second order both in the solution and its gradient a relaxation of inner equations and boundary conditions by a fictitious time method inspired by the stability conditions related to the associated time dependent problem with a convergence proof for the first order scheme an effective geometric multigrid which maintains the structure of the discrete system at all grid levels it is shown that by increasing the relaxation step of the equations associated to the boundary conditions a convergence factor close to the optimal one is obtained several numerical tests including variable coefficients anisotropic elliptic equations and domains with kinks show the robustness efficiency and accuracy of the approach

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

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

a sharp interface finite volume method for elliptic equations on cartesian grids we present a second order sharp interface finite volume method for the solution of the three dimensional elliptic equation x u x f x with variable coefficients on cartesian grids in particular we focus on interface problems with discontinuities in the coefficient the source term the solution and the fluxes across the interface the method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27 point stencil singularities associated with vanishing partial volumes of intersected grid cells are removed by a two term asymptotic approach in contrast to the 2d method presented by two of the authors in m oevermann r klein a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal of computational physics 219 2006 749 769 we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz this simplifies the treatment of the different cut cell types and avoids additional special operations for degenerated interface topologies the resulting set of linear equations has been solved with a bicgstab solver preconditioned with an algebraic multigrid in various testcases including large ratios and non smooth interfaces the method achieves second order of accuracy in the l and l 2 norm

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

proofs of two conjectures on ternary weakly regular bent functions in this paper we study ternary monomial functions of the form f x trn axd where x isin bbf 3 n and trn bbf 3 nrarr bbf 3 is the absolute trace function using a lemma of hou stickelberger s theorem on gauss sums and certain ternary weight inequalities we show that certain ternary monomial functions arising in the 2006 ieee transactions on information theory paper vol 52 pp 2018 2032 2006 are weakly regular bent thus settling a conjecture of helleseth and kholosha we also prove that the coulter matthews bent functions are weakly regular

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

an augmented immersed finite element method for variable coefficient elliptic interface problems in two and three dimensions abstract an augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients by introducing the jump of normal derivative of the solution along the interface as an augmented variable the interface problem is solved iteratively on cartesian meshes using the generalized minimal residual gmres iterative method within each iteration we need to solve a system of linear equations with the same symmetric and positive definite matrix but with different right hand sides the sequence of linear equations is solved efficiently by the multi grid solvers since the set up of coarse grid matrices and decompositions is only computed once extensive numerical examples show that the number of gmres iterations is nearly independent of the mesh size when a diagonal like preconditioning is applied thus the computational work of the proposed method is linearly proportional to the number of grid nodes implementation details are given and extensive numerical examples in two and three dimensions are provided to show the accuracy and efficiency of the proposed method

quaternary constant amplitude codes for multicode cdma a constant amplitude code is a code that reduces the peak to average power ratio papr in multicode code division multiple access mc cdma systems to the favorable value 1 in this paper quaternary constant amplitude codes codes over z 4 of length 2 m with error correction capabilities are studied these codes exist for every positive integer m while binary constant amplitude codes cannot exist if m is odd every word of such a code corresponds to a function from the binary m tuples to z4 having the bent property i e its fourier transform has magnitudes 2 m 2 several constructions of such functions are presented which are exploited in connection with algebraic codes over z4 in particular quaternary reed muller kerdock and delsarte goethals codes to construct families of quaternary constant amplitude codes mappings from binary to quaternary constant amplitude codes are presented as well

a flexible coefficient smooth transition time series model we consider a flexible smooth transition autoregressive star model with multiple regimes and multiple transition variables this formulation can be interpreted as a time varying linear model where the coefficients are the outputs of a single hidden layer feedforward neural network this proposal has the major advantage of nesting several nonlinear models such as the self exciting threshold autoregressive setar the autoregressive neural network ar nn and the logistic star models furthermore if the neural network is interpreted as a nonparametric universal approximation to any borel measurable function our formulation is directly comparable to the functional coefficient autoregressive far and the single index coefficient regression models a model building procedure is developed based on statistical inference arguments a monte carlo experiment showed that the procedure works in small samples and its performance improves as it should in medium size samples several real examples are also addressed

gradient recovery for elliptic interface problem iii nitsche s method abstract this is the third paper on the study of gradient recovery for elliptic interface problems in our previous works guo and yang 2017 23 we developed gradient recovery methods for elliptic interface problems based on body fitted meshes and immersed finite element methods despite the efficiency and accuracy that these methods bring to recover the gradient there are still some cases in unfitted meshes where skinny triangles appear in the generated local body fitted triangulations that destroy the accuracy of recovered gradient near the interface in this paper we propose a gradient recovery technique based on nitsche s method for elliptic interface problems which avoids the loss of accuracy of gradient near the interface caused by skinny triangles we analyze the supercloseness between the gradient of the numerical solution by the nitsche s method and the gradient of the interpolation of the exact solution which leads to the superconvergence of the proposed gradient recovery method we also present several numerical examples to validate the theoretical results

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

a simple method for matrix valued coefficient elliptic equations with sharp edged interfaces abstract the traditional finite element method has a number of nice properties and thus it is highly desired for matrix valued coefficient elliptic equations with sharp edged interfaces however its efficient implementation with body fitting grids for such problems is highly nontrivial in this paper we propose a simple finite element method with body fitting grids based on semi cartesian grid which makes its implementation fairly straightforward even for complicated geometry all possible situations that the interface cuts the grid are considered further the symmetry and positive definiteness of the resulting matrix are ensured for positive definite coefficients numerical experiments show that it is second order accurate in the l norm and numerically very stable

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

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

second order finite difference ghost point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface abstract in this paper we propose a second order accurate numerical method to solve elliptic problems with discontinuous coefficients with general non homogeneous jumps in the solution and its gradient in 2d and 3d the method consists of a finite difference method on a cartesian grid in which complex geometries boundaries and interfaces are embedded and is second order accurate in the solution and the gradient itself in order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface two numerical values are assigned on grid points that are close to the interface a real value that represents the numerical solution on that grid point and a ghost value that represents the numerical solution extrapolated from the other side of the interface obtained by enforcing the assigned non homogeneous jump conditions on the solution and its flux the method is also extended to the case of matrix coefficient the linear system arising from the discretization is solved by an efficient multigrid approach unlike the 1d case grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient a proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor comparable with the one obtained by local fourier analysis for rectangular domains the method is robust enough to handle large jump in the coefficients order of accuracy monotonicity of the errors and good convergence factor are maintained by the scheme

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

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

smooth transition autoregressive models and fuzzy rule based systems functional equivalence and consequences in this work we will explore the theoretical connections existing between fuzzy rule based systems frbs applied on univariate time series and two statistical reference tools the autoregressive ar models and the smooth transition autoregressive star model we will show that a tsk fuzzy rule happens to be a localised ar model and that a star model can hence be interpreted as a restricted frbs several consequences derive from this fact and we will explore some of them including a statistical inference based procedure to incrementally build frbs or the linguistic interpretation of star models finally an econometric example is given to illustrate the core idea

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

a class of partially adaptive one step m estimators for the non linear regression model with dependent observations in this paper we consider a class of partially adaptive one step m estimators for the non linear regression model with dependent observations those estimators adapt themselves with respect to a measure of the tailthickness of the disturbance distribution as well as to a measure of the scale the large sample behavior of those estimators is examined theoretically for general disturbance distributions and numerically for various specific ones the estimators considered are motivated by the student r maximum likelihood estimator given appropriate specifications of the adaptation parameter the estimators are asymptotically efficient on the family of student t distributions including the normal distribution

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

a simplified reproducing kernel method for 1 d elliptic type interface problems abstract in this paper an efficient numerical method is proposed for the 1d elliptic type interface problems we first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems the convergence of the proposed method is discussed finally some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method