numerical wave propagation for the triangular p1dg p2 finite element pair the f plane and b plane wave propagation properties are examined for discretisations of the linearised rotating shallow water equations using the p1 d g p2 finite element pair on arbitrary triangulations in planar geometry a discrete helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations in the f plane planar geometry coriolis force independent of space case this decomposition is used to obtain decoupled equations for the geostrophic modes the inertia gravity modes and the inertial oscillations as has been noticed previously the geostrophic modes are steady the helmholtz decomposition is used to show that the resulting inertia gravity wave equation is third order accurate in space in general the p1 d g p2 finite element pair is second order accurate so this leads to very accurate wave propagation it is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f and which do not propagate a restriction of the p1 d g velocity space is proposed in which these modes are not present leading to a finite element discretisation which is completely free of spurious modes the helmholtz decomposition also allows a simple derivation of the quasi geostrophic limit of the discretised p1 d g p2 equations in the b plane planar geometry coriolis force linear in space case resulting in a rossby wave equation which is also third order accurate this means that the dispersion relation for the wave propagation is very accurate an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles

energy conserving explicit local time stepping for second order wave equations locally refined meshes impose severe stability constraints on explicit time stepping methods for the numerical simulation of time dependent wave phenomena to overcome that stability restriction local time stepping methods are developed which allow arbitrarily small time steps precisely where small elements in the mesh are located when combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix the resulting discrete numerical scheme is explicit is inherently parallel and exactly conserves a discrete energy starting from the standard second order leap frog scheme time stepping methods of arbitrary order of accuracy are derived numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory

an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

rapid development and adjoining of transient finite element models abstract recent advances in high level finite element systems have allowed for the symbolic representation of discretisations and their efficient automated implementation as model source code this allows for the extremely compact implementation of complex non linear models in a handful of lines of high level code in this work we extend the high level finite element fenics system to introduce an abstract representation of the temporal discretisation this enables the similarly rapid development of transient finite element models efficiency is achieved via aggressive optimisations that exploit the temporal structure such as automated pre assembly and caching of forms and the robust re use of matrix factorisations and preconditioner data the resulting models are as fast or faster than hand optimised finite element codes the high level representation of the system remains extremely compact and easily manipulated this structure is exploited to derive the associated discrete adjoint model automatically with the adjoint model inheriting the performance advantages of the forward model combined this provides a system for the rapid development of efficient transient models together with their discrete adjoints

a structure preserving split finite element discretization of the split wave equations abstract we introduce a new finite element fe discretization framework applicable for covariant split equations the introduction of additional differential forms df that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric dependent closure equations that apply the hodge star operator our discretization framework conserves this geometrical structure and provides for all dfs proper fe spaces such that the differential operators here gradient and divergence hold in strong form we introduce lowest possible order discretizations of the split 1d wave equations in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant fe spaces omitting partial integrations approximating the hodge star by nontrivial galerkin projections gp the two discrete metric equations follow by projections onto either the piecewise constant gp0 or piecewise linear gp1 space out of the four possible realizations our framework gives us three schemes with significantly different behavior the split scheme using twice gp1 is unstable and shares the dispersion relation with the p1 p1 fe scheme that approximates both variables by piecewise linear spaces p1 the split schemes that apply a mixture of gp1 and gp0 share the dispersion relation with the stable p1 p0 fe scheme that applies piecewise linear and piecewise constant p0 spaces however the split schemes exhibit second order convergence for both quantities of interest for the split scheme applying twice gp0 we are not aware of a corresponding standard formulation to compare with though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find fe schemes by new combinations of fe spaces

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

numerical study of the interface errors of finite difference simulations of seismic waves numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion which is only valid for homogeneous media however there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers the error associated with media that have two layers is called an interface error and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite difference schemes fds for homogeneous media we evaluated a detailed numerical study of the interface error for three staggered grid fds that are commonly used in the simulation of seismic wave propagation we determined that a standard staggered grid scheme ssgs also known as the virieux scheme a rotated staggered grid scheme rsgs and a lebedev scheme ls preserve the second order of convergence at horizontal vertical solid solid interfaces when the medium parameters have been properly modified such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density however for a fluid solid interface aligned with the grid line a second order convergence can only be achieved by an ssgs in addition the presence of a fluid solid interface reduces the order of convergence for the ls and the rsgs to a first order of convergence the presence of inclined interfaces makes high order second and more convergence impossible

higher order triangular finite elements with mass lumping for the wave equation in this article we construct new higher order finite element spaces for the approximation of the two dimensional 2d wave equation these elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping these formulas are constructed explicitly error estimates are provided for the corresponding semidiscrete problem finally higher order finite difference time discretizations are proposed and various numerical results are shown

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

symplectic hamiltonian hdg methods for wave propagation phenomena abstract we devise the first symplectic hamiltonian hybridizable discontinuous galerkin hdg methods for the acoustic wave equation we discretize in space by using a hamiltonian hdg scheme that is an hdg method which preserves the hamiltonian structure of the wave equation and in time by using symplectic diagonally implicit and explicit partitioned runge kutta methods the fundamental feature of the resulting scheme is that the conservation of a discrete energy which is nothing but a discrete version of the original hamiltonian is guaranteed we present numerical experiments which indicate that the method achieves optimal approximations of order k 1 in the l 2 norm when polynomials of degree k 0 and runge kutta time marching methods of order k 1 are used in addition by means of post processing techniques and by increasing the order of the runge kutta method to k 2 we obtain superconvergent approximations of order k 2 in the l 2 norm for the displacement and the velocity we also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative hdg schemes of similar accuracy properties for long time simulations

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

the general coupled matrix equations over generalized bisymmetric matrices abstract in the present paper by extending the idea of conjugate gradient cg method we construct an iterative method to solve the general coupled matrix equations j 1 p a ij x j b ij m i i 1 2 p including the generalized coupled lyapunov and sylvester matrix equations as special cases over generalized bisymmetric matrix group x 1 x 2 x p by using the iterative method the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors when the general coupled matrix equations are consistent over generalized bisymmetric matrices a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors the least frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen in addition the optimal approximation generalized bisymmetric solution group to a given matrix group x 1 x 2 x p in frobenius norm can be obtained by finding the least frobenius norm generalized bisymmetric solution group of new general coupled matrix equations the numerical results indicate that the iterative method works quite well in practice

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

design of strictly positive real systems using constant output feedback the authors present a linear matrix inequality lmi approach to the strictly positive real spr synthesis problem find an output feedback k such that the closed loop system t s is spr the authors establish that if no such constant output feedback k exists then no dynamic output feedback with a proper transfer matrix exists to make the closed loop system spr the existence of k to guarantee the spr property of the closed loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

common hermitian and positive solutions to the adjointable operator equations ax c xb d let a be a c algebra for any hilbert a modules h and k let l h k be the set of adjointable operators from h to k let h k l be hilbert a modules a c l h k and b d l l h in this paper we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions x l h to the equations ax c xb d and obtain the formulae for the general forms of these solutions some results known for finite matrices and hilbert space operators are extended to the adjointable operators acting on hilbert c modules

interior penalty discontinuous galerkin time domain method based on wave equation for 3 d electromagnetic modeling built on the vector wave equation an interior penalty discontinuous galerkin time domain ipdg we method is presented for 3 d real life electromagnetic modeling in this paper we develop the interior penalty fluxes corresponding to different types of boundary conditions which are commonly used in electromagnetic modeling to construct the ipdg we system in local elements in addition to model the incident wave excitation the total field scattered field approach is incorporated into the proposed method the stability of the ipdg we method is studied analytically and demonstrated numerically we also compare the performance between wave equation based finite element time domain method the maxwell s equation based discontinuous galerkin time domain method and the proposed method both in cpu time and memory usage which demonstrates that the ipdg we method is very competitive among time domain solvers besides the presented method shows good characteristics including the energy conservation and the optimal convergence rate of o h p 1 numerical examples are presented to illustrate the validity versatility and potential of the proposed method in 3 d electromagnetic problems modeling

a procedure for the galerkin method for a vibrating system in this paper we apply the galerkin method to the problem of vibration of a one dimensional system with free end conditions giving a maple r procedure we test the results on two numerical examples

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

symmetric positive semidefinite and positive definite real solutions of ax xat and ax yb abstract given the equations ax xa t and ax yb with arbitrary nonzero real matrices a and b of the same size we seek all real solutions x and y which are 1 symmetric 2 symmetric and positive semidefinite and 3 symmetric and positive definite necessary and sufficient conditions for the existence of such solutions and their general forms are derived

an iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation axb c an iteration method is constructed to solve the linear matrix equation axb c over symmetric x by this iteration method the solvability of the equation axb c over symmetric x can be determined automatically when the equation axb c is consistent over symmetric x its solution can be obtained within finite iteration steps and its least norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new matrix equation axb c finally numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation axb c

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

optimal geostatistical model selection in many fields of science predicting variables of interest over a study region based on noisy data observed at some locations is an important problem two popular methods for the problem are kriging and smoothing splines the former assumes that the underlying process is stochastic whereas the latter assumes it is purely deterministic kriging performs better than smoothing splines in some situations but is outperformed by smoothing splines in others however little is known regarding selecting between kriging and smoothing splines in addition how to perform variable selection in a geostatistical model has not been well studied in this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on approximately unbiased estimation of mean squared prediction errors using a data perturbation technique the proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors and is shown to be optimal in terms of two mean sq

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

the use of generalized inverses in restricted maximum likelihood the calculus of generalized inverses and related concepts in matrix algebra is applied to the general restricted maximum likelihood problem some new results on g inverses kronecker products and matrix differentials are presented for the restricted maximum likelihood problem we obtain generalizations of the well known results of aitchison and silvey 1 we use the approach recently developed by heijmans and magnus 13 14 to allow for non i i d observations a nonlinear seemingly unrelated regressions model with possibly singular covariance matrix and linear restrictions nlsursr is analyzed and the linear expenditure system les is discussed as a special case

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

on minimal solutions of the matrix equation ax yb 0 abstract in the case when one of the ranks of matrices a or b is full all self adjoint pairs of solutions of the equation ax yb 0 are given necessary and sufficient conditions for the existence of nonnegative and positive definite solutions are proved without any condition on ranks and with a given solution x it is shown that maximal and minimal solutions for y do not exist in nontrivial cases it is also proved that the minimal nonnegative solution y exists an explicit formula for this solution is given

the symmetric solution of the matrix equations ax ya c axat bybt c and atxa btxb c d abstract the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations i ax ya c ii axa t byb t c and iii a t xa b t xb c d are derived in addition the minimum 2 norm least squares symmetric solution of equation i the minimum 2 norm symmetric solution of equation ii and the least squares solution of equation iii are obtained

a ray based ipdg method for high frequency time domain acoustic wave propagation in inhomogeneous media abstract the numerical approximation of high frequency wave propagation in inhomogeneous media is a challenging problem in particular computing high frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy in this paper we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns the method is based on a discontinuous galerkin discretization together with carefully chosen basis functions to obtain the basis functions we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion also we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell we show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the ipdg method using polynomials moreover the relative errors of our method grow only moderately as the frequency increases

symmetric and skew antisymmetric solutions to systems of real quaternion matrix equations in this paper we consider symmetric and skew antisymmetric solutions to certain matrix equations over the real quaternion algebra h first a criterion for a quaternion matrix to be symmetric and skew antisymmetric is given then necessary and sufficient conditions are obtained for the matrix equation ax c and the following system a 1x c 1xb 3 c 3 to have symmetric and skew antisymmetric solutions the expressions of such solutions of the matrix equation and the system mentioned above are also given

the common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations abstract we in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations ax c xb d we derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system the general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented in addition we give a numerical example to illustrate the results of this paper

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

analogs of cramer s rule for the minimum norm least squares solutions of some matrix equations abstract the least squares solutions with the minimum norm of the matrix equations ax b xa b and axb d are considered in this paper we use the determinantal representations of the moore penrose inverse obtained earlier by the author and get analogs of the cramer rule for the minimum norm least squares solutions of these matrix equations

non linear petrov galerkin methods for reduced order modelling of the navier stokes equations using a mixed finite element pair abstract a new nonlinear petrov galerkin approach has been developed for proper orthogonal decomposition pod reduced order modelling rom of the navier stokes equations the new method is based on the use of the cosine rule between the advection direction in cartesian space time and the direction of the gradient of the solution a finite element pair p 1 dg p 2 which has good balance preserving properties is used here consisting of a mix of discontinuous for velocity components and continuous for pressure basis functions the contribution of the present paper lies in applying this new non linear petrov galerkin method to the reduced order navier stokes equations and thus improving the stability of rom results without tuning parameters the results of numerical tests are presented for a wind driven 2d gyre and the flow past a cylinder which are simulated using the unstructured mesh finite element cfd model in order to illustrate the numerical performance of the method the numerical results obtained show that the newly proposed pod petrov galerkin method can provide more accurate and stable results than the pod bubnov galerkin method

a discontinuous continuous low order finite element shallow water model on the sphere we study the applicability of a new finite element in atmosphere and ocean modeling the finite element under investigation combines a second order continuous representation for the scalar field with a first order discontinuous representation for the velocity field and is therefore different from continuous and discontinuous galerkin finite element approaches the specific choice of low order approximation spaces is attractive because it satisfies the ladyzhenskaya babuska brezzi condition and is at the same time able to represent the crucially important geostrophic balance the finite element is used to solve the viscous and inviscid shallow water equations on a rotating sphere we introduce the spherical geometry via a stereographic projection the projection leads to a manageable number of additional terms the associated scaling factors can be exactly represented by second order polynomials we perform numerical experiments considering steady and unsteady zonal flow flow over topography and an unstable zonal jet stream for ocean applications the wind driven stommel gyre is simulated the experiments are performed on icosahedral geodesic grids and analyzed with respect to convergence rates conservation properties and energy and enstrophy spectra the results match quite well with results published in the literature and encourage further investigation of this type of element for three dimensional atmosphere ocean modeling

runge kutta based explicit local time stepping methods for wave propagation locally refined meshes severely impede the efficiency of explicit runge kutta rk methods for the simulation of time dependent wave phenomena by taking smaller time steps precisely where the smallest elements are located local time stepping lts methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh starting from classical or low storage explicit rk methods explicit lts methods of arbitrarily high accuracy are derived when combined with an essentially diagonal finite element mass matrix the resulting time marching schemes retain the high accuracy stability and efficiency of the original rk methods while circumventing the geometry induced stiffness numerical experiments with continuous and discontinuous galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these lts rk methods

eigenvector free solutions to ax b with px xp and xh sx constraints abstract the matrix equation ax b with px xp and xh sx constraints is considered where p is a given hermitian involutory matrix and s 1 by an eigenvalue decomposition of p we equivalently transform the constrained problem to two well known constrained problems and represent the solutions in terms of the eigenvectors of p using moore penrose generalized inverses of the products generated by matrices a b and p the involved eigenvectors can be released and eigenvector free formulas of the general solutions are presented similar strategy is applied to the equations ax b xc d with the same constraints

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

a new iteration method for the matrix equation ax b an iteration method for the matrix equation ax b is constructed by this iteration method the least norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least norm least squares solutions can be obtained when the matrix equation is not consistent the related optimal approximation solution is also obtained by this iteration method a preconditioned method for improving the iteration rate is put forward finally some numerical examples are given

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

the hermitian reflexive solutions to the matrix inverse problem ax b in this paper the necessary and sufficient conditions for the solvability of the matrix inverse problem ax b in hermitian reflexive matrix set hj n x n are given and the solution expression of this matrix inverse problem is also presented moreover an optimal approximation to any matrix in the above solution set in frobenius norm is provided

linear restriction problem of hermitian reflexive matrices and its approximation in this paper we consider a linear restriction problem of hermitian reflexive matrices and its approximation by using the properties and structure of hermitian reflexive matrices and the special properties of reflexive vectors and anti reflexive vectors we convert the linear restriction problem to an equivalence problem trickily which is a special feature of this paper and is a different method from other articles then we solve this problem completely and also obtain its optimal approximate solution moreover an algorithm provided for it and the numerical examples show that the algorithm is feasible

the reflexive and anti reflexive solutions of the matrix equation ahxb c in this paper the necessary and sufficient conditions for the solvability of matrix equation a hxb c over the reflexive or anti reflexive matrices are given and the general expression of the solution for a solvable case is obtained moreover the relative optimal approximation problem is considered the explicit expressions of the optimal approximation solution and the minimum norm solution are both provided

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

on semiparametric inference of geostatistical models via local karhunen loeve expansion summary we develop a semiparametric approach to geostatistical modelling and inference in particular we consider a geostatistical model with additive components where the form of the covariance function of the spatial random error is not prespecified and thus is flexible a novel local karhunen loeve expansion is developed and a likelihood based method is devised for estimating the model parameters and statistical inference a simulation study demonstrates sound finite sample properties and a real data example is given for illustration finally the theoretical properties of the estimates are explored and in particular consistency results are established

local time stepping with the discontinuous galerkin method for wave propagation in 3d heterogeneous media modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings finite elements enable accurate simulation of time dependent wave propagation in heterogeneous media they are more costly than finite difference methods but this is compensated by their superior accuracy if the finite element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength hence with the local wave velocity however 3d complex geologic settings often contain details on a very small scale compared to the dominant wavelength requiring the mesh to contain elements that are smaller than dictated by the wavelength also limitations of the mesh generation software may produce regions where the elements are much smaller than desired in both cases this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost local time stepping lts can improve the computational efficiency and speed up the simulation we evaluated a local formulation of an lts scheme with second order accuracy for the discontinuous galerkin finite element discretization of the wave equation we tested the benefits of the scheme by considering a geologic model for a north sea type example

variable selection for binary spatial regression penalized quasi likelihood approach we consider the problem of selecting covariates in a spatial regression model when the response is binary penalized likelihood based approach is proved to be effective for both variable selection and estimation simultaneously in the context of a spatially dependent binary variable an uniquely interpretable likelihood is not available rather a quasi likelihood might be more suitable we develop a penalized quasi likelihood with spatial dependence for simultaneous variable selection and parameter estimation along with an efficient computational algorithm the theoretical properties including asymptotic normality and consistency are studied under increasing domain asymptotics framework an extensive simulation study is conducted to validate the methodology real data examples are provided for illustration and applicability although theoretical justification has not been made we also investigate empirical performance of the proposed penalized quasi likelihood approach for spatial count data to explore suitability of this method to a general exponential family of distributions

penalty factor threshold and time step bound estimations for discontinuous galerkin time domain method based on helmholtz equation in this article penalty factor threshold and time step bound in discontinuous galerkin time method based on vector wave equation dgtd we method are well estimated based on the semidiscrete form of the dgtd we method properties of the system matrices are studied and the stability condition related to the penalty factor is derived by decomposing the global system matrices of the dgtd we method into the local ones and developing an efficient iteration procedure the lower bound threshold of the penalty factor is well estimated to guarantee the positive semidefinite property of the global system matrices with the calculated penalty factor the maximum time step is analytically determined by approximating spectral radius of the local system matrix both the penalty factor bound and the maximal time step are computed element wise instead of a global system matrix operation and thus the proposed method can be efficiently applied into the large scale meshes with different types of the basis functions and boundary conditions numerical examples are presented to demonstrate the validity and good performance of the proposed methods

a high order discontinuous galerkin method for nonlinear sound waves abstract we propose a high order discontinuous galerkin scheme for nonlinear acoustic waves on polytopic meshes to model sound propagation with losses through homogeneous media we use westervelt s nonlinear wave equation with strong damping challenges in the numerical analysis lie in handling the nonlinearity in the model which involves the derivatives in time of the acoustic velocity potential and in preventing the equation from degenerating we rely in our approach on the banach fixed point theorem combined with a stability and convergence analysis of a linear wave equation with a variable coefficient in front of the second time derivative by doing so we derive an a priori error estimate for westervelt s equation in a suitable energy norm for the polynomial degree p 2 numerical experiments carried out in two dimensional settings illustrate the theoretical convergence results in addition we demonstrate efficiency of the method in a three dimensional domain with varying medium parameters where we use the discontinuous galerkin approach in a hybrid way

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

additive model building for spatial regression summary spatial regression is an important predictive tool in many scientific applications and an additive model provides a flexible regression relationship between predictors and a response variable we develop a regularized variable selection technique for building a spatial additive model we find that the methods developed for independent data do not work well for spatially dependent data this motivates us to propose a spatially weighted error norm with a group lasso type of penalty to select additive components in spatial additive models we establish the selection consistency of the approach proposed where the penalty parameter depends on several factors such as the order of approximation of additive components characteristics of the spatial weight and spatial dependence an extensive simulation study provides a vivid picture of the effects of dependent data structure and choice of a spatial weight on selection results as well as the asymptotic behaviour of the estimators as an illustrative example the method is applied to lung cancer mortality data over the period of 2000 2005 obtained from the surveillance epidemiology and end results programme national cancer institute usa

interior penalty discontinuous galerkin method for maxwell s equations energy norm error estimates we develop the symmetric interior penalty discontinuous galerkin dg method for the time dependent maxwell equations in second order form we derive optimal a priori error estimates in the energy norm for smooth solutions we also consider the case of low regularity solutions that have singularities in space

on the penalty stabilization mechanism for upwind discontinuous galerkin formulations of first order hyperbolic systems abstract penalty fluxes are dissipative numerical fluxes for high order discontinuous galerkin dg methods which depend on a penalization parameter warburton 2013 ye et al 2016 we investigate the dependence of the spectra of high order dg discretizations on this parameter and show that as its value increases the spectrum of the dg discretization splits into two disjoint sets of eigenvalues one set converges to the eigenvalues of a conforming discretization while the other set corresponds to spurious eigenvalues which are damped proportionally to numerical experiments also demonstrate that undamped spurious modes present in both in the limit of zero and large penalization parameters are damped for moderate values of the upwind parameter

the skew symmetric orthogonal solutions of the matrix equation ax b abstract an n n real matrix x is said to be a skew symmetric orthogonal matrix if xt x and xtx i using the special form of the c s decomposition of an orthogonal matrix with skew symmetric k k leading principal submatrix this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew symmetric orthogonal solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided furthermore the procrustes problem of skew symmetric orthogonal matrices is considered and the formula solutions are provided finally an algorithm is proposed for solving the first and third problems numerical experiments show that it is feasible

the least squares anti bisymmetric solution and the optimal approximation solution for sylvester equation in this paper a modified conjugate gradient iterative method for solving sylvester equation is presented by using this iterative method the least squares anti bisymmetric solution and the optimal approximation solution can be obtained here we present the derivation and theoretical analysis of our iterative method numerical results illustrate the feasibility and effectiveness of the proposed iterative method

short note a short note on the discontinuous galerkin discretization of the pressure projection operator in incompressible flow this note reports on the issue of spurious compressibility artifacts that can arise when the popular pressure projection pp method is used for unsteady simulations of incompressible flow using the symmetric interior penalty discontinuous galerkin sip dg method through a spectral analysis of the projection operator s sip dg discretization we demonstrate that the eigenfunctions of the operator do not form a basis that allows for the correct enforcement of the incompressibility constraint this short coming can cause numerical instabilities for inviscid advection dominated and density stratified flow simulations especially for long time integrations and or under resolved situations to remedy this problem we propose a local post processing projection that enforces incompressibility exactly to allow for stable and robust long time integrations

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

enriched galerkin finite element approximation for elastic wave propagation in fractured media abstract finite element methods fem are becoming increasingly popular in modeling seismic wave propagation these methods provide higher order accuracy geometrical flexibility and adaptive gridding capabilities that are not easy to incorporate in traditional finite difference methods employed for generation of synthetic seismograms moreover several studies have shown that discontinuous galerkin fem dgm is a promising approach for modeling wave propagation in fractured media here we propose an enriched galerkin fem egm for elastic wave propagation egm uses the same bilinear form as dgm and the continuous galerkin finite element spaces enriched by discontinuous piecewise constants or bilinear functions egm satisfies local equilibrium while reducing the degrees of freedom in dgm formulations in this paper we consider elastic wave propagation and derive optimal a priori error estimates for dgm and egm we present numerical examples in two spatial dimensions that confirm these theoretical results in addition we provide numerical comparisons with the spectral element method in previous work dgm has been shown to be effective in modeling elastic wave propagation in fractured media using the linear slip model we now extend these results to egm with reduced computational costs over dgm

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

lbb stability of a mixed galerkin finite element pair for fluid flow simulations we introduce a new mixed finite element for solving the 2 and 3 dimensional wave equations and equations of incompressible flow the element which we refer to as p1 d p2 uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure the aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the lbb stability condition and hence has no spurious zero energy modes the advantage of this particular element choice is that the mass matrix for velocity is block diagonal so it can be trivially inverted it also allows the order of the pressure to be increased to quadratic whilst maintaining lbb stability which has benefits in geophysical applications with coriolis forces we give a normal mode analysis of the semi discrete wave equation in one dimension which shows that the element pair is stable and demonstrate that the element is stable with numerical integrations of the wave equation in two dimensions an analysis of the resultant discrete laplace operator in two and three dimensions on various meshes which shows that the element pair does not have any spurious modes we provide convergence tests for the element pair which confirm that the element is stable since the convergence rate of the numerical solution is quadratic

the reflexive solutions of the matrix equation ax b c in this paper we study the existence of a reflexive with respect to the generalized reflection matrix p solution of the matrix equation ax b c for the special case when b i we get the result of peng and hu 1

space time spectral method for the two dimensional generalized sine gordon equation abstract in this paper we propose a space time spectral method for solving the generalized two dimensional sine gordon equation with nonhomogeneous dirichlet boundary conditions and initial conditions the proposed method is based on the legendre galerkin spectral method in space and the spectral collocation method or block spectral collocation method in time we derive a priori error estimates in both l 2 and h 1 norms for the semidiscrete formulation our formulation has spectral accuracy in both space and time our numerical results confirm the exponential convergence of the proposed method in both space and time

solutions with special structure to the linear matrix equation ax b in this article we solve the following three kinds of problems first some formulas for the minimal rank of the submatrices in a solution x of matrix equation ax b and the minimal and maximal rank of x itself are derived by using the matrix rank method from these formulas necessary and sufficient conditions are given for x to be nonsingular or the submatrices to be zero respectively second some formulas for the minimal rank of x x x x and the corresponding expressions of submatrices of x are investigated combined with matrix decomposition necessary and sufficient conditions are given for the existence of solutions to be hermitian local hermitian skew hermitian and local skew hermitian respectively third necessary and sufficient conditions are given for the existence of solutions to be local positive negative semidefinite and for some hermitian solution with zero submatrix the structure form of a positive negative semidefinite solution are obtained using results of inertia