further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

a fast poisson solver for complex geometries robust fast solvers for the poisson equation have generally been limited to regular geometries where direct methods based on fourier analysis or cyclic reduction and multigrid methods can be used while multigrid methods can be applied in irregular domains and to a broader class of partial differential equations they are difficult to implement in a robust fashion since they require an appropriate hierarchy of coarse grids which are not provided in many practical situations in this paper we present a new fast poisson solver based on potential theory rather than on direct discretization of the partial differential equation our method combines fast algorithms for computing volume integrals and evaluating layer potentials on a grid with a fast multipole accelerated integral equation solver the amount of work required is o m log m n where m is the number of interior grid points and n is the number of points on the boundary asymptotically the cost of our method is just twice that of a standard poisson solver on a rectangular domain in which the problem domain can be embedded independent of the complexity of the geometry

superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming this paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming sdp under the assumptions that the semidefinite program has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero the interior point algorithm considered here closely resembles the mizuno todd ye predictor corrector method for linear programming which is known to be quadratically convergent it is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 1 2 r the proof relies on a careful analysis of the central path for sdp it is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap

pseudospectral solution of the two dimensional navier stokes equations in a disk an efficient and accurate algorithm for solving the two dimensional 2d incompressible navier stokes equations on a disk with no slip boundary conditions is described the vorticity stream function formulation of these equations is used and spatially the vorticity and stream functions are expressed as fourier chebyshev expansions the poisson and helmholtz equations which arise from the implicit explicit time marching scheme are solved as banded systems using a postconditioned spectral tau method the polar coordinate singularity is handled by expanding fields radially over the entire diameter using a parity modified chebyshev series and building partial regularity into the vorticity the no slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint significant gains in run times were realized by parallelizing the code in message passage interface mpi

numerical solution for the steady motion of a viscous fluid inside a circular boundary using integral conditions abstract the problem of determining the two dimensional steady motion of a viscous incompressible fluid which is injected radially over one small arc of a circle and ejected radially over another arc is considered and examples are given of both symmetrical and asymmetrical flows the motion is governed by the navier stokes equations and the method of solution is based on the use of truncated fourier series representations for the stream function and vorticity in the angular polar coordinate the navier stokes equations are reduced to ordinary differential equations in the radial variable and these sets of equations are solved using finite difference methods but with the boundary vorticity calculated using global integral conditions rather than local finite difference approximations one of the objects of the investigation is to relate this method to a previous study which did not use integral conditions and also to a recent study which uses an integro differential method which is different in concept but which also uses integral conditions a brief review of previous work on the problem is given comparisons of present and previous results are excellent

the numerical solution of the biharmonic equation by conformal mapping the solution to the biharmonic equation in a simply connected region omega in the plane is computed in terms of the goursat functions the boundary conditions are conformally transplanted to the disk with a numerical conformal map a linear system is obtained for the taylor coefficients of the goursat functions the coefficient matrix of the linear system can be put in the form i k where k is the discretization of a compact operator k can be thought of as the composition of a block hankel matrix with a diagonal matrix the compactness leads to clustering of eigenvalues and the hankel structure yields a matrix vector multiplication cost of o n log n thus if the conjugate gradient method is applied to the system then superlinear convergence will be obtained numerical results are given to illustrate the spectrum clustering and superlinear convergence

a note on the nonexistence of sum of squares certificates for the bessis moussa villani conjecture the algebraic reformulation of the bessis moussa villani conjecture is equivalent to a family of dimension free tracial inequalities involving positive semidefinite matrices sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by klep and schweighofer sums of hermitian squares and the bmv conjecture j stat phys 133 739 2008 later the existence of these certificates has been settled for all but one case which is resolved in this note

avoiding numerical cancellation in the interior point method for solving semidefinite programs the matrix variables in a primal dual pair of semidefinite programs are getting increasingly ill conditioned as they approach a complementary solution multiplying the primal matrix variable with a vector from the eigenspace of the non basic part will therefore result in heavy numerical cancellation this effect is amplified by the scaling operation in interior point methods a complete example illustrates these numerical issues in order to avoid numerical problems in interior point methods we propose to maintain the matrix variables in a cholesky form we discuss how the factors of the v space cholesky form can be updated after a main iteration of the interior point method with nesterov todd scaling an analogue for second order cone programming is also developed numerical results demonstrate the success of this approach

first and second order methods for semidefinite programming in this paper we survey the most recent methods that have been developed for the solution of semidefinite programs we first concentrate on the methods that have been primarily motivated by the interior point ip algorithms for linear programming putting special emphasis in the class of primal dual path following algorithms we also survey methods that have been developed for solving large scale sdp problems these include first order nonlinear programming nlp methods and more specialized path following ip methods which use the preconditioned conjugate gradient or residual scheme to compute the newton direction and the notion of matrix completion to exploit data sparsity

on two interior point mappings for nonlinear semidefinite complementarity problems extending our previous work monteiro and pang 1996 this paper studies properties of two fundamental mappings associated with the family of interior point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices the first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case these trajectories completely fill up the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program unlike the approach based on the theory of maximal monotone maps taken by shida and shindoh 1996 and shida shindoh and kojima 1995 our approach is based on the theory of local homeomorphic maps in nonlinear analysis

fast direct solver for the biharmonic equation on a disk and its application to incompressible flows we develop a simple and efficient fft based fast direct solver for the biharmonic equation on a disk the biharmonic equation is split into a coupled system of harmonic problems we first use the truncated fourier series expansion to derive a set of coupled singular odes then we solve those singular equations by second order finite difference discretizations using a radial grid with shifting a half mesh away from the origin we can handle the coordinate singularity easily without pole conditions the sherman morrison formula is then applied to solve the resultant linear system in a cost efficient way the computational complexity of the method consists of o mnlog 2n arithmetic operations for mxn grid points the numerical accuracy check and some applications to the incompressible navier stokes flows inside a disk are conducted

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

a new notion of weighted centers for semidefinite programming the notion of weighted centers is essential in v space interior point algorithms for linear programming although there were some successes in generalizing this notion to semidefinite programming via weighted center equations we still do not have a generalization that preserves two important properties 1 each choice of weights uniquely determines a pair of primal dual weighted centers and 2 the set of all primal dual weighted centers completely fills up the relative interior of the primal dual feasible region this paper presents a new notion of weighted centers for semidefinite programming that possesses both uniqueness and completeness furthermore it is shown that under strict complementarity these weighted centers converge to weighted centers of optimal faces finally this convergence result is applied to homogeneous cone programming where the central paths defined by a certain class of optimal barriers for homogeneous cones are shown to converge to analytic centers of optimal faces in the presence of strictly complementary solutions

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

on the identification of the optimal partition of second order cone optimization problems this paper discusses the identification of the optimal partition of second order cone optimization soco by giving some condition numbers which only depend on the soco problem itself we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition mathcal b mathcal n mathcal r and mathcal t for soco problems can be identified along the central path when the barrier parameter mu is small enough then we generalize the results to a specific neighborhood of the central path

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

integral equation methods for stokes flow and isotropic elasticity in the plane we present a class of integral equation methods for the solution of biharmonic boundary value problems with applications to two dimensional stokes flow and isotropic elasticity the domains may be multiply connected and finite infinite or semi infinite in extent our analytic formulation is based on complex variables and our fast multipole based iterative solution procedure requireso n operations wherenis the number of nodes in the discretization of the boundary the performance of the methods is illustrated with several large scale numerical examples

affine scaling algorithm fails for semidefinite programming in this paper we introduce an affine scaling algorithm for semidefinite programming sdp and give an example of a semidefinite program such that the affine scaling algorithm converges to a non optimal point both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and they are non degenerate everywhere

slanted matrices banach frames and sampling in this paper we present a rare combination of ab stract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sam pling problems we show that for a large class of slanted matrices boundedness below of the corresponding operator in p for some p implies boundedness below in p for all p we use the established result to enrich our understanding of banach frames and obtain new results for irregular sampling problems we also present a version of a non commutative wiener s lemma for slanted matri ces

vorticity boundary conditions and boundary vorticity generation for two dimensional viscous incompressible flows abstract in this paper we present boundary conditions appropriate for the vorticity formulation of the two dimensional incompressible viscous navier stokes equations these boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type i e vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied the results of computations with this finite difference method are presented for flow past a circular cylinder a difference scheme and computational results for a model problem the prandtl boundary layer equations describing flow over a semi infinite flat plate are also presented

limiting behavior of trajectories generated by a continuation method for monotone complementarity problems defining the mapping f from the 2n dimensional euclidean space r2n into itself by fx y x1y1 xn yn y1 f1x yn fnx for every x y r2n we write the cp the complementarity problem with a mapping f rn rn as the system of equations fx y 0 and x y r 2n where r 2n denotes the nonnegative orthant of r2n under the assumption that f is a monotone function on r n we show that f maps the positive orthant r 2n of r2n homeomorphically this result then ensures the existence of a trajectory consisting of the solutions x y t of the system of equations fx y wt and x y t r 2n 0 1 where w is a continuous mapping from the unit interval 0 1 into fr 2n such that w0 0 if t 0 the system coincides with the cp we study limiting behavior of the trajectory as t 0 and give some sufficient conditions for the trajectory to lead to a solution of the cp this approach is an extension of the one used in the polynomially bounded algorithm recently given by kojima mizuno and yoshise for solving positive semidefinite linear complementarity problems

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

density theorems for sampling and interpolation in the bargmann fock space we give a complete description of sampling and interpolation in the bargmann fock space based on a density concept of beurling roughly speaking a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von neumann lattice and similarly a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von neumann lattice

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

deformation of gabor systems abstract we introduce a new notion for the deformation of gabor systems such deformations are in general nonlinear and in particular include the standard jitter error and linear deformations of phase space with this new notion we prove a strong deformation result for gabor frames and gabor riesz sequences that covers the known perturbation and deformation results our proof of the deformation theorem requires a new characterization of gabor frames and gabor riesz sequences it is in the style of beurling s characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of gabor frames without inequalities from lattices to non uniform sets

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

sampling in weighted lp spaces of entire functions in cn and estimates of the bergman kernel the necessary density condition in c known for sampling and interpolation in the lp space of entire functions with a subharmonic weight is extended to the case of a 2 homogeneous plurisubharmonic weight function in c the method is by estimating the eigenvalues of a certain toeplitz concentration operator using asymptotic estimates for the bergman kernel of independent interest

pointwise estimates of weighted bergman kernels in several complex variables abstract we prove new pointwise bounds for weighted bergman kernels in c n whenever a coercivity condition is satisfied by the associated weighted kohn laplacian on 0 1 forms our results extend the ones obtained by christ in 5 our main idea is to develop a version of agmon theory originally introduced in 1 to deal with schrodinger operators for weighted kohn laplacians on 0 1 forms inspired by the fact that these are unitarily equivalent to certain generalized schrodinger operators

dilation of the weyl symbol and balian low theorem the key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their weyl symbols this observation has two implications in time frequency analysis first it implies the stability of general gabor frames under small dilations of the time frequency set previously known only for the case that the time frequency set is a lattice secondly it allows us to derive a new balian low theorem blt for gabor systems with window in the standard window class and with general time frequency families in contrast to the classical versions of blt the new blt does not only exclude orthonormal bases and riesz bases at critical density but indeed it even excludes irregular gabor frames at critical density

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

analyticity of the central path at the boundary point in semidefinite programming abstract in this paper we study the limiting behavior of the central path for semidefinite programming sdp we show that the central path is an analytic function of the barrier parameter even at the limit point provided that the semidefinite program has a strictly complementary solution a consequence of this property is that the derivatives of any order of the central path have finite limits as the barrier parameter goes to zero

search directions in the sdp and the monotone sdlcp generalization and inexact computation we propose a family of directions that generalizes many directions proposed so far in feasible interior point methods for the sdp semidefinite programming and for the monotone sdlcp semidefinite linear complementarity problem we derive the family from the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction by relaxing its centrality equation into a centrality inequality using this family we present a predictor corrector infeasible interior point method to provide a theoretical basis for inexact computation of directions in primal dual interior point methods for the sdp

asymptotic behavior of the central path for a special class of degenerate sdp problems this paper studies the asymptotic behavior of the central path x s y as 0 for a class of degenerate semidefinite programming sdp problems namely those that do not have strictly complementary primal dual optimal solutions and whose degenerate diagonal blocks of the central path are assumed to satisfy we establish the convergence of the central path towards a primal dual optimal solution which is characterized as being the unique optimal solution of a certain log barrier problem a characterization of the class of sdp problems which satisfy our assumptions are also provided it is shown that the re parametrization t 0 x t4 s t4 y t4 of the central path is analytic at t 0 the limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is finally we apply our results to the convex quadratically constrained convex programming cqccp problem and characterize the class of cqccp problems which can be formulated as sdps satisfying the assumptions of this paper in particular we show that cqccp problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class

a simple compact fourth order poisson solver on polar geometry we present a simple and efficient compact fourth order poisson solver in polar coordinates this solver relies on the truncated fourier series expansion where the differential equations of the fourier coefficients are solved by the compact fourth order finite difference scheme by shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of fourier coefficients we can easily handle coordinate singularities without pole conditions the numerical evidence confirms fourth order accuracy for the problem on an annulus and third order accuracy for the problem on a disk in addition a simple and comparably accurate approximation for the derivatives of the solution is also presented

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

compressed sensing off the grid this paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples unlike previous work in compressed sensing the frequencies are not assumed to lie on a grid but can assume any values in the normalized frequency domain 0 1 an atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies which is then reformulated as an exact semidefinite program even with this continuous dictionary it is shown that o slog s log n random samples are sufficient to guarantee exact frequency localization with high probability provided the frequencies are well separated extensive numerical experiments are performed to illustrate the effectiveness of the proposed method

error bounds and limiting behavior of weighted paths associated with the sdp map x 1 2 sx 1 2 this paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming sdp obtained from centrality equations of the form x1 2sx1 2 nu w where w is a fixed positive definite matrix and nu 0 is a parameter under the assumption that the problem has a strictly complementary primal dual optimal solution it is shown that a weighted central path as a function of sqrt nu can be extended analytically beyond 0 and hence that the path converges as nu downarrow 0 characterization of the limit points of the path and its normalized first order derivatives are also provided as a consequence it is shown that a weighted central path can have two types of behavior it converges either as theta nu or as theta sqrt nu depending on whether the matrix w on a certain scaled space is block diagonal or not respectively we also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal dual optimal solutions finally in light of the results of this paper we give a characterization of a sufficient condition proposed by potra and sheng which guarantees the superlinear convergence of a class of primal dual interior point sdp algorithms

numerical solution of steady navier stokes problems using integral representations steady state incompressible navier stokes and continuity equations are recast into integral representations for the velocity and the vorticity vector using the concept of fundamental solutions integrals are discretized using finite fourier series approximations an efficient iterative procedure is established for the numerical solution of internal flow problems in two dimensions the iteration loop in this procedure is composed of a kinetic step and a kinematic step explicit computations are utilized in each step boundary vorticity values required for the kinetic step are determined from constraint equations obtained from the kinematic aspect of the problem expressed in the integral representation form several flow problems are studied numerically for various reynolds numbers flow structures are determined from numerical results obtained a bachelor s flow model of constant vorticity is verified numerically for high reynolds number closed streamline flows inside a circle

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

an integral equation approach to the incompressible navier stokes equations in two dimensions we present a collection of methods for solving the incompressible navier stokes equations in the plane that are based on a pure stream function formulation the advantages of this approach are twofold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided the disadvantage is that the solution of a nonlinear fourth order partial differential equation is required by recasting this partial differential equation as an integral equation we avoid the ill conditioning which hampers finite difference and finite element methods in this environment by using fast algorithms for the evaluation of volume integrals we are able to solve the equations using o m or o m log m operations where m is the number of points in the discretization of the domain

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions abstract answering a question of lindholm we prove strict density inequalities for sampling and interpolation in fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight in particular these spaces do not admit a set that is simultaneously sampling and interpolating to prove optimality of the density conditions we construct sampling sets with a density arbitrarily close to the critical density the techniques combine methods from several complex variables estimates for and the theory of localized frames in general reproducing kernel hilbert spaces with no analyticity assumed the abstract results on fekete points and deformation of frames may be of independent interest

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

steady viscous flow past a circular cylinder up to reynolds number 600 viscous flow past a circular cylinder becomes unstable around reynolds number re 40 with a numerical technique based on newton s method and made possible by the use of a supercomputer steady but unstable solutions have been calculated up to re 600 it is found that the wake bubble region with recirculating flow grows in length approximately linearly with re the width increases like re12 up to re 300 at which paint a transition to linear increase with re begins at the highest reynolds numbers we reached the wake resembles a pair of translating uniform vortices both touching the center line the cylinder moving in front with the same speed supplies the vorticity required to balance diffusion

the rapid evaluation of volume integrals of potential theory on general regions abstract we present a new method for the rapid high order accurate evaluation of certain volume integrals in potential theory on general irregular regions the kernels of the integrals are either a fundamental solution or a linear combination of the derivatives of a fundamental solution of a second order linear elliptic differential equation instead of using a standard quadrature formula or the exact evaluation of any integral the methods rely on rapid methods of solving the differential equation which the kernel is the solution of therefore the number of operations needed to evaluate the volume integral is essentially equal to the number of operations needed to solve the differential equation on a rectangular region with a regular grid and the method requires no evaluation of the kernel