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

superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

the symmetry group of a finite frame abstract we define the symmetry group of a finite frame as a group of permutations on its index set this group is closely related to the symmetry group of vale and waldron 2005 12 for tight frames they are isomorphic when the frame is tight and has distinct vectors the symmetry group is the same for all similar frames in particular for a frame its dual and canonical tight frames it can easily be calculated from the gramian matrix of the canonical tight frame further a frame and its complementary frame have the same symmetry group we exploit this last property to construct and classify some classes of highly symmetric tight frames

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

the noncommutative akns system projection to matrix systems countable superposition and soliton like solutions starting from the recent work on noncommutative akns systems for functions with values in the bounded operators on a banach space it is shown how their formal 1 soliton solution depending on operator parameters can be mapped to solutions of matrix akns systems the main result is rather general solution formulas for matrix akns systems the most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton like solutions of the classical akns system

correction for the dislocation of curved surfaces caused by the psf in 2d and 3d ct images conventional edge detection methods suffer from the dislocation of curved surfaces due to the psf we propose a new method that uses the isophote curvature to circumvent this it is accurate for objects with locally constant curvature even for small objects like blood vessels and in the presence of noise

new types of soliton solutions we announce a detailed investigation of limits of n soliton solutions of the korteweg devries kdv equation as n tends to infinity our main results provide new classes of kdv solutions including in particular new types of soliton like reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional schrodinger operators and explicitly construct smooth and real valued potentials that yield a purely absolutely continuous spectrum on the nonnegative real axis and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of the negative real axis

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

new classes of toda soliton solutions we provide a detailed investigation of limits of n soliton solutions of the toda lattice as n tends to infinity our principal results yield new classes of toda solutions including in particular new kinds of soliton like i e reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional jacobi operators and explicitly construct tri diagonal matrices that yield a purely absolutely continuous spectrum in 1 1 and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of mr bs 1 1

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 nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

an operator theoretic approach to the toda lattice equation abstract we treat the toda lattice equation with operator methods and derive an explicit solution formula in terms of determinants as an application we investigate solutions which are given by special settings in the finite dimensional case matrices in jordan canonical form give rise to a new class of solutions within this class the well known n soliton solutions can be recovered by the special choice of diagonal matrices moreover using diagonal operators we get solutions depending on an infinite number of parameters we comprehensively discuss the case involving diagonal operators and show that it can be reduced to a very particular situation

on the location error of curved edges in low pass filtered 2 d and 3 d images the authors study the location error of curved edges in two and three dimensional images after analog and digital low pass filtering the zero crossing of a second derivative filter is a well known edge localization criterion the second derivative in gradient direction sdgd produces a predictable bias in edge location towards the centers of curvature while the linear laplace filter produces a shift in the opposite direction their sum called plus plus laplace sdgd leads to an edge detector that finds curved edges one order more accurately than its constituents this argument holds irrespective of the dimension the influence of commonly used low pass filters such as the psf originating from diffraction limited optics using incoherent light 2 d the gaussian filter with variable cutoff point d d and the isotropic uniform filter d d is studied

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

sampling and reconstruction of operators we study the recovery of operators with a bandlimited kohn nirenberg symbol from the action of such operators on a weighted impulse train a procedure we refer to as sampling of operators kailath bello and later kozek and the authors have shown that the sampling of operators is possible if the symbol of the operator is bandlimited to a set with area less than one in this paper we develop the theory of the sampling of operators in analogy with the classical theory of sampling of bandlimited functions we define the notions of sampling set and sampling rate for operators and give necessary and sufficient conditions on the sampling rate that depend on the size and geometry of the bandlimiting set we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions we give necessary and sufficient conditions on the bandlimiting set under which sampling of operators is possible by their action on a given periodically weighted delta train we show that under mild geometric conditions on the bandlimiting set classes of operators are bandlimited to an unknown set of area less than one half permit sampling and reconstruction we generalize two results of the heckel and bolcskei concerning sampling of operators with area not greater than one half and less than one respectively by finding a larger class of operators to which they apply operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly time varying impulse response the results in this paper are rooted in work by bello and kailath in the 1960s

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

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

linear and nonlinear image deblurring a documented study nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods this paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging where reconstruction of high frequency detail is of prime interest and where avoidance of a priori smoothness constraints is a major concern the theoretical basis behind each of the following nonlinear procedures is examined the lucy richardson method the maximum likelihood e m algorithm the poisson maximum a posteriori method and the nunez llacer version of the maximum entropy method a linear iterative method vancittert s iteration is also studied it is shown that each of the first three methods as well as vancittert s method lack a necessary ingredient for successful solution of the ill posed deblurring problem while in the maximum entropy method the enforced smoothness may have adverse consequences in medical imaging a direct linear method the slow evolution from the continuation boundary secb method designed specifically for nonsmooth images is also considered that method is stabilized by constraining the blurring operator as well as the solution and does not require smoothness constraints it is shown that useful error estimates can be obtained in the secb method while this is impossible in tikhonov s method without a priori bounds on derivatives of the unknown solution reconstruction experiments on low noise synthetic mri data show that thousands of iterations are necessary to achieve sufficient resolution in the iterative procedures however the secb method provides higher resolution at considerable savings in computer time at high noise levels the iterative algorithms are shown to diverge at these same noise levels the secb method produces reconstructions comparable in quality to those that would be obtained in the iterative methods were one able to terminate the divergent algorithm at that iteration which best approximates the true solution in the l1 norm

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

solving some large scale semidefinite programs via the conjugate residual method most current implementations of interior point methods for semidefinite programming use a direct method to solve the schur complement equation sce m delta y h in computing the search direction when the number of constraints is large the problem of having insufficient memory to store m can be avoided if an iterative method is used instead numerical experiments have shown that the conjugate residual cr method typically takes a huge number of steps to generate a high accuracy solution on the other hand it is difficult to incorporate traditional preconditioners into the sce except for block diagonal preconditioners we decompose the sce into a 2 2 block system by decomposing delta y similarly for h into two orthogonal components with one lying in a certain subspace that is determined from the structure of m numerical experiments on semidefinite programming problems arising from the lovasz theta function of graphs and maxcut problems show that high accuracy solutions can be obtained with a moderate number of cr steps using the proposed equation

an interior point method with a primal dual quadratic barrier penalty function for nonlinear semidefinite programming in this paper we consider an interior point method for nonlinear semidefinite programming yamashita yabe and harada presented a primal dual interior point method in which a nondifferentiable merit function was used by using shifted barrier kkt conditions we propose a differentiable primal dual merit function within the framework of the line search strategy and prove its global convergence property under weaker assumptions than the method of yamashita yabe and harada

a primal dual variant of the iri imai algorithm for linear programming a local acceleration method for primal dual potential reduction algorithms is introduced the method developed here uses modified newton search directions to minimize the tanabe todd ye tty potential function and can be regarded as a primal dual variant of the iri imai algorithm based on the multiplicative analogue of karmarkar s potential function when iterates are close to an optimal solution the tty potential function has negative curvature along the generated search directions therefore large reductions in the potential function can be obtained guaranteeing polynomial and quadratic convergence to nondegenerate solutions

measurement of time variant linear channels the goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals in this paper it is shown that the spreading support of the operator that is the support of the symplectic fourier transform of the kohn nirenberg symbol of the operator has area less than one then the operator is identifiable if the spreading support is larger than one then the operator is not identifiable the shape of the support region is essentially arbitrary thereby proving a conjecture of bello the input signal considered is a weighted delta train where the weights are the window function of a finite gabor system whose elements satisfy a certain robust completeness property

classifying ct image data into material fractions by a scale and rotation invariant edge model a fully automated method is presented to classify 3 d ct data into material fractions an analytical scale invariant description relating the data value to derivatives around gaussian blurred step edges arch model is applied to uniquely combine robustness to noise global signal fluctuations anisotropic scale noncubic voxels and ease of use via a straightforward segmentation of 3 d ct images through material fractions projection of noisy data value and derivatives onto the arch yields a robust alternative to the standard computed gaussian derivatives this results in a superior precision of the method the arch model parameters are derived from a small but over determined set of measurements data values and derivatives along a path following the gradient uphill and downhill starting at an edge voxel the model is first used to identify the expected values of the two pure materials named and and thereby classify the boundary second the model is used to approximate the underlying noise free material fractions for each noisy measurement an iso surface of constant material fraction accurately delineates the material boundary in the presence of noise and global signal fluctuations this approach enables straightforward segmentation of 3 d ct images into objects of interest for computer aided diagnosis and offers an easy tool for the design of otherwise complicated transfer functions in high quality visualizations the method is applied to segment a tooth volume for visualization and digital cleansing for virtual colonoscopy

toda flows with infinitely many variables abstract the toda flow and related flows extend naturally to operators in hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

estimation of overspread scattering functions in many radar scenarios the radar target or the medium is assumed to possess randomly varying parts the properties of the target are described by a random process known as the spreading function its second order statistics under the wssus assumption are given by the scattering function recent developments in operator sampling theory suggest novel channel sounding procedures that allow for the determination of the spreading function given complete statistical knowledge of the operator echo from a single sounding by a weighted pulse train we construct and analyze a novel estimator for the scattering function based on these findings our results apply whenever the scattering function is supported on a compact subset of the time frequency plane we do not make any restrictions either on the geometry of this support set or on its area our estimator can be seen as a generalization of the averaged periodogram estimator for the case of a non rectangular geometry of the support set of the scattering function

toda soliton limits on general backgrounds abstract starting from an arbitrary background solution of the toda lattice we study limits of n soliton solutions on this given background as n tends to infinity this yields a new class of solutions of the toda lattice simultaneously we solve an inverse spectral problem for one dimensional jacobi operators we explicitly construct jacobi operators whose spectrum contains a given countable bounded set of eigenvalues and whose absolutely continuous spectrum coincides with that of a given background operator

commutation methods for jacobi operators abstract we offer two methods of inserting eigenvalues into spectral gaps of a given background jacobi operator the single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi bounded background jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps moreover we prove unitary equivalence of the commuted operators restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues with the original background operator in addition we compute the matrix valued weyl m functions of the commuted operator in terms of the background weyl m functions finally we show how to iterate the above methods and give explicit formulas for various quantities such as eigenfunctions and spectra of the iterated operators in terms of the corresponding background quantities and scattering matrix concrete applications include an explicit realization of the isospectral torus for algebro geometric finite gap jacobi operators and the n soliton solutions of the toda and kac van moerbeke lattice equations with respect to arbitrary background solutions

a feasible primaldual interior point method for linear semidefinite programming in this paper we consider a feasible primaldual interior point method for linear semidefinite programming problem sdp based on alizadehhaeberlyoverton aho direction monteiro 1997 firstly and by a new and simple technique we establish the existence and uniqueness of optimal solution of the perturbed problem sdp and its convergence to optimal solution of sdp next we present new different alternatives to calculate the displacement step after we establish the convergence of the obtained algorithm and we show that its complexity is o nln 1 x0 s0 finally we present some numerical simulations which show the effectiveness of the algorithm developed in this work

sampling of stochastic operators we develop sampling methodology aimed at identifying stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function the data that we use to reconstruct the operator or in some cases only the autocorrelation of the spreading function is based on the response of the unknown operator to a known deterministic test signal