self dual solutions to pseudo yang mills equations abstract we study pseudo yang mills fields on a compact 5 dimensional strictly pseudoconvex cr manifold m i e critical points to the functional y m b d 1 2 m h r d 2 d 2 on the space c e h of all connections d on a hermitian vector bundle e h over m such that d h 0 if a d c e h r d 0 g tr r d d 0 and d a is an absolute minimum to y m b a r then i b tr r d 0 and ii d is self dual or anti self dual according to the sign of c 2 d m p 2 d m 1 2 m p 1 d p 1 d where p k d is the k th chern form of e d and provided c 2 d is constant on a

stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

parallel solutions of very large sparse lyapunov equations by balanced bbd decompositions in this paper a method for the parallel solution of large sparse matrix equations is presented the method is based on the balanced bordered block diagonal bbd decomposition which is applied in conjunction with the successive over relaxation sor iterative method to solve large lyapunov equations in the kronecker sum representation a variety of experimental results will be presented including solutions of lyapunov equations with matrices as large as 1993 spl times 1993

pseudo einstein structures on cr manifolds etant donne une variete cr compacte non degeneree on cherche une structure pseudohermitienne associee pour laquelle le tenseur de ricci pseudohermitien est un multiple scalaire de la forme de levi

yang mills fields on cr manifolds we study pseudo yang mills fields on a compact strictly pseudoconvex cr manifold m i e the critical points of the functional pym d 12 m hrd 2 d n where d is a connection in a hermitian cr holomorphic vector bundle e h m let 0 cn be a smoothly bounded strictly pseuodoconvex domain and g the bergman metric on we show that boundary values db of yang mills fields d on g are pseudo yang mills fields on provided that itrdb 0 and inrd 0 on h if s1 c m m is the canonical circle bundle and d is a yang mills field with respect to the fefferman metric f of m then d is a pseudo yang mills field on m the yang mills equations dr d 0 project on the euler lagrange equations bdrd 0 of the variational principle pym d 0 provided that itrd 0 when m has vanishing pseudohermitian ricci curvature the pullback d of the cr invariant tanaka connection d of e h is a yang mills field on c m we derive the second variation formula d2pym dt dt2 t 0 m sbd d n dt d

on the dirichlet problem for the harmonic vector fields equation abstract we show that any harmonic with respect to the bergman metric vector field tangent to the levi distribution of the foliation by level sets of the defining function z k z z 1 n 1 of a strictly pseudoconvex bounded domain c n which is smooth up to the boundary must vanish on if n 5 and u t is a harmonic vector field with u c 2 then u 0

extending bicg and bicr methods to solve the stein tensor equation abstract the biconjugate gradients bicg and biconjugate residual bicr methods are attractive ways for solving nonsymmetric linear system a x b in this paper the bicg and bicr methods are extended to solve the high order stein tensor equation the convergent properties of the developed iterative algorithms are studied numerical examples are provided to confirm the theoretical results which demonstrate that the proposed algorithms are effective and feasible for solving the stein tensor equation

smooth solutions of degenerate laplacians on strictly pseudoconvex domains on considere la caracterisation des solutions lisses globales de u 0 sur un domaine general strictement pseudoconvexe ou est le laplacien pour la metrique avec forme de kakler i 2 log 1

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

some null solutions of the yang mills equations and cauchy riemann structures some new null solutions of the yang mills equations with congruences of twisting shear free and null geodesics are obtained these are test fields each of them is defined on a lorentzian manifold with a metric tensor adapted to the cauchy riemann geometry associated with the congruence some examples of solutions on minkowski space time are also given among them is a solution with a congruence of twisting shear free and null geodesics different from the robinson congruence

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

optimal alternating direction implicit parameters for nonsymmetric systems of linear equations the determination of optimal parameters for the well known alternating direction implicit adi method leads to a minimization problem for rational functions of given degree l on compact sets e and f that contain the eigenvalues of the matrices involved if e and f are real intervals the solution of this problem was given by zolotarev more than 100 years ago and by wachspress in the 1960s however in the complex case e g if the linear system arises from the discretization of non self adjoint elliptic differential equations little is known about this problem in this paper two approaches to the zolotarev problem in the complex plane are considered first generalized leja points are considered as an example for rational functions which are asymptotically minimal in addition for rectangular eigenvalue domains e and f the exact minimal solutions for l 2 are constructed and it is shown by numerical examples that the results obtained using the optimal parameters of degree 2 are comparable to tho

a matrix crs iterative method for solving a class of coupled sylvester transpose matrix equations abstract in this paper we apply kronecker product and vectorization operator to extend the conjugate residual squared crs method for solving a class of coupled sylvester transpose matrix equations some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature numerical results validate that the proposed method can be much more efficient than some existing methods

twistor cr manifolds and three dimensional conformal geometry a cr i e partially complex 5 manifold is contructed as a sphere bundle over an arbitrary 3 manifold with conformal metric this so called twistor cr manifold is show to capture completely the original geometry and necessary and sufficient conditions are given for an abstract cr manifold to arise via the construction the above correspondence is then used to prove that a twistor cr manifold is locally imbeddable as a real hypersurface in c3 only if it is real analytic with respect to a suitable atlas introduction there is a very beautiful interplay between the conformal geometry of riemannian manifolds and complex analysis examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self dual riemannian 4 manifolds penrose 1976 atiyah et al 1978 intermediating between these two theories is a relationship between 3dimensional conformal geometry and cr partially complex manifolds of dimension 5 in this paper we will explain this relationship focusing on the imbedding problem when can these partially complex manifolds be realized as real hypersurfaces in complex 3 manifolds the idea of associating a cr manifold with a conformal riemannian 3 manifold originated with penrose 1975 1983 who starting with a real analytic space like hypersurface in a real analytic lorentzian 4 manifold constructed a real hypersurface in a complex 3 manifold if the space like hypersurface is totally geodesic or merely all umbilic penroses s cr manifold coincides in our terminology with the twistor cr manifold of the conformal structure induced on the space like hypersurface after establishing notation and conventions in 0 principal results on cr manifolds are reviewed and the notions of cr vector bundle and cr contact form are introduced in 1 a general machine for constructing involutive distributions is produced and in 2 this construction is used to associate to every conformal riemannian 3 manifold a cr 5 manifold the cr manifolds arising in this way are characterized abstractly in 4 in terms of properties presented in 3 then in 5 it is shown that these cr manifolds are isomorphic to real hypersurfaces of complex received by the editors march 15 1983 and in revised form july 8 1983 1980 mathematics subject classificcation primary 32f25 secondary 53a30 83c99 work supported in part by nsf grant mcs 812 0790 001984 american mathematical society 0002 9947 84 1 00 25 per page

preconditioned hss iteration method and its non alternating variant for continuous sylvester equations abstract bai 2010 proposed an efficient hermitian and skew hermitian splitting hss iteration method for solving a broad class of large sparse continuous sylvester equations to further improve the efficiency of the hss method in this paper we present a preconditioned hss phss iteration method and its non alternating variant nphss for this matrix equation the convergence properties of the phss and nphss methods are studied in depth and the quasi optimal values of the iteration parameters for the two methods are also derived moreover to reduce the computational cost we establish the inexact variants of the two iteration methods numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants

deterministic continuation of stochastic metastable equilibria via lyapunov equations and ellipsoids numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary partial and delay differential equations here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations we stress that we do not develop a new technical tool but that we combine results and methods from probability theory dynamical systems numerical analysis optimization and control theory into an algorithm that augments classical equilibrium continuation methods in particular we use ellipsoids defining regions of high concentration of sample paths it is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numer

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

numerical solution of the lyapunov equation by approximate power iteration abstract we present the approximate power iteration api algorithm for the computation of the dominant invariant subspace of the solution x of large order lyapunov equations ax xa t q 0 without first computing the matrix x itself the api algorithm is an iterative procedure that uses krylov subspace bases in computing estimates of matrix vector products x v in a power iteration sequence application of the api algorithm requires that a a t x admits a good low rank solution then api provides an orthogonal basis of a subspace that closely approximates the dominant x invariant subspace of corresponding dimension analytical convergence results are also presented

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

unified analysis of finite volume methods for second order elliptic problems we establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions under the framework optimal order convergence in the h 1 and l 2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finite volume methods applied to second order elliptic problems

a note on the iterative solutions of general coupled matrix equation abstract recently ding and chen f ding t chen on iterative solutions of general coupled matrix equations siam j control optim 44 2006 2269 2284 developed a gradient based iterative method for solving a class of coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified so that the iterative solutions are obtained by applying hierarchical identification principle in this note by considering the coupled sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm we also propose some faster algorithms and present some numerical results

on modified hss iteration methods for continuous sylvester equations we introduce and analyze a modification of the hermitian and skew hermitian splitting iteration method for solving large sparse continuous sylvester equations with non hermitian and complex symmetric positive definite semidefinite matrices it is found that the modified hermitian and skew hermitian splitting mhss iteration method is unconditionally convergent each iteration in this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices these two systems can be solved inexactly numerical results show that the mhss iteration method and its inexact variant are efficient and robust solvers for this class of continuous sylvester equations

an analysis of block successive overrelaxation for a class of matrices with complex spectra optimal block successive overrelaxation bsor parameters are derived for a special class of matrices having complex spectra these matrices arise from the central difference approximation to singularly perturbed convection diffusion equations using a coarse grid and thus have large cell reynolds number a simple problem is examined in detail and it is shown that bsor is an excellent method for cell reynolds numbers near 1 and that the rate of convergence is invariant to the cell reynolds number for large cell reynolds number

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

new wavelet based approach for time domain simulations a new numerical method for the time domain solution of maxwell s equations in linear media is proposed in this paper the field quantities derived from the spatial discretization of maxwell s equations are expanded in the time domain by wavelets on the interval this choice yields a new arrangement of the unknowns into a matrix instead of the usual vector and transforms the differential equations in time in a sylvester 1983 matrix equation the memory requirements are proportional to the number of spatial unknowns and the time evolution of the space quantities is obtained with better accuracy than in conventional marching on time techniques

a numerical algorithm for lyapunov equations in this paper we give a new numerical iterative method for lyapunov matrix equations which is called kronecker products iterative method kpim by means of the kronecker products we get an iterative convergent sequence from the lyapunov matrix equations at the same time we give its algorithms and corresponding improved algorithm in order to reduce the computation cost finally we discuss how to choose the optimal parameter of kpim for the continuous lyapunov matrix equations in the case of symmetry

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

solution of underdetermined sylvester equations in sensor array signal processing abstract we propose efficient solution of an underdetermined matrix equation the matrix equations studied here appear in the study of subspace based high resolution multiparameter direction of arrival estimation the proposed gradient based technique is at least an order of magnitude computational improvement over the existing approaches simulation results are provided to illustrated the proposed technique

fast algorithms for the sylvester equation ax xb t c for given matrices a fm m b fn n and c fm n over an arbitrary field f the matrix equation ax xbt c has a unique solution x fm n if and only if a and b have disjoint spectra we describe an algorithm that computes the solution x for m n n with o n logn arithmetic operations in f where 2 is such that m m matrices can be multiplied with o m arithmetic operations e g 2 376 it seems that before no better bound than o m3 n3 arithmetic operations was known the state of the art in numerical analysis is o n3 m3 flops but these algorithms due to bartels stewart and golub nash van loan involve schur decompositions i e they compute the eigenvalues of at least one of a and b and can hence not be transferred for general f

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

the fefferman metric and pseudo hermitian invariants c fefferman has shown that a real strictly pseudoconvex hypersurface in complex n space carries a natural conformai lorentz metric on a circle bundle over the manifold this paper presents two intrinsic constructions of the metric valid on an abstract cr manifold one is in terms of tautologous differential forms on a natural circle bundle the other is in terms of webster s pseudohermitian invariants these results are applied to compute the connection and curvature forms of the fefferman metric explicitly

an adaptive discontinuous finite volume method for elliptic problems an adaptive discontinuous finite volume method is developed and analyzed in this paper we prove that the adaptive procedure achieves guaranteed error reduction in a mesh dependent energy norm and has a linear convergence rate numerical results are also presented to illustrate the theoretical analysis

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

minimal residual methods augmented with eigenvectors for solving sylvester equations and generalized sylvester equations in this paper we propose minimal residual methods augmented with eigenvectors for solving large sylvester matrix equations ax xb c and generalized sylvester matrix equations axb x c the subspace from which the approximate solution is extracted is the kronecker product subspace of two augmented krylov subspaces numerical experiments report the effectiveness of these methods

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

a discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

a new discontinuous finite volume method for elliptic problems we develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients an optimal order error estimate is obtained in a mesh dependent norm an l2 error estimate is also obtained

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