stochastic eulerian lagrangian methods for fluid structure interactions with thermal fluctuations we present approaches for the study of fluid structure interactions subject to thermal fluctuations a mixed mechanical description is utilized combining eulerian and lagrangian reference frames we establish general conditions for operators coupling these descriptions stochastic driving fields for the formalism are derived using principles from statistical mechanics the stochastic differential equations of the formalism are found to exhibit significant stiffness in some physical regimes to cope with this issue we derive reduced stochastic differential equations for several physical regimes we also present stochastic numerical methods for each regime to approximate the fluid structure dynamics and to generate efficiently the required stochastic driving fields to validate the methodology in each regime we perform analysis of the invariant probability distribution of the stochastic dynamics of the fluid structure formalism we compare this analysis with results from statistical mechanics to further demonstrate the applicability of the methodology we perform computational studies for spherical particles having translational and rotational degrees of freedom we compare these studies with results from fluid mechanics the presented approach provides for fluid structure systems a set of rather general computational methods for treating consistently structure mechanics hydrodynamic coupling and thermal fluctuations

the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

perturbation bounds for least squares problems with equality constraints we present some new results on the perturbation analysis for least squares problems with equality constraints based on an equivalent formation of the problem and the optimal perturbation result for general least squares problems

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

an analysis of real fourier domain based adaptive algorithms implemented with the hartley transform using cosine sine symmetries the least mean squared lms algorithm and its variants have been the most often used algorithms in adaptive signal processing however the lms algorithm suffers from a high computational complexity especially with large filter lengths the fourier transform based block normalized lms fbnlms reduces the computation count by using the discrete fourier transform dft and exploiting the fast algorithms for implementing the dft even though the savings achieved with the fbnlms over the direct lms implementation are significant the computational requirements of fbnlms are still very high rendering many real time applications like audio and video estimation infeasible the hartley transform based bnlms hbnlms is found to have a computational complexity much less than and a memory requirement almost of the same order as that of the fbnlms this paper is based on the cosine and sine symmetric implementation of the discrete hartley transform dht which is the key in reducing the computational complexity of the fbnlms by 33 asymptotically with respect to multiplications the parallel implementation of the discrete cosine transform dct in turn can lead to more efficient implementations of the hbnlms

a simple variant of the mizuno todd ye predictor corrector algorithm and its objective function free complexity in this paper we propose a simple variant of the mizuno todd ye predictor corrector algorithm for linear programming our variant employs a simple and a natural scaling invariant search directio

upper bound and stability of scaled pseudoinverses for given matrices x and d where d is positive definite diagonal a weighed pseudoinverse of x is defined by x d x rm h d 2 x x rm h d 2 and an oblique projection of x is defined by p d xx d when x is of full column rank stewart 3 and o leary 2 found sharp upper bound of oblique projections p d which is independent of d and an upper bound of weighed pseudoinverse x d by using the bound of p d in this paper we discuss the sharp upper bound of x d over a set d of positive diagonal matrices which does not depend on the upper bound of p d and the stability of x d over d

immersed boundary method for variable viscosity and variable density problems using fast constant coefficient linear solvers i numerical method and results we present a general variable viscosity and variable density immersed boundary method that is first order accurate in the variable density case and for problems possessing sufficient regularity second order accurate in the constant density case the viscosity and density are considered material properties and are defined by a dynamically updated tesselation empirical convergence rates are reported for a test problem of a two dimensional viscoelastic shell with spatially varying material properties the reduction to first order accuracy in the variable density case can be avoided by using an iterative scheme although this approach may not be efficient enough for practical use in our time stepping scheme both the inertial and viscous terms are split into two parts a constant coefficient part that is treated implicitly and a variable coefficient part that is treated explicitly this splitting allows the resulting equations to be solved efficiently using fast constant coefficient linear solvers and i

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

the cholesky factorization in interior point methods the paper concerns the cholesky factorization of symmetric positive definite matrices arising in interior point methods our investigation is based on a property of the cholesky factorization which interprets small diagonal values during factorization as degeneracy in the scaled optimization problem a practical scaling independent technique based on the above property is developed for the modified cholesky factorization of interior point methods this technique increases the robustness of cholesky factorizations performed during interior point iterations when the optimization problem is degenerate our investigations show also the limitations of interior point methods with the recent implementation technology and floating point arithmetic standard we present numerical results on degenerate linear programming problems of netlib

efficient computation of the dft with only a subset of input or output points ways of efficiently computing the discrete fourier transform dft when the number of input and output data points differ are discussed the two problems of determining whether the length of the input sequence or the length of the output sequence is reduced can be found to be duals of each other and the same methods can to a large extent be used to solve both the algorithms utilize the redundancy in the input or output to reduce the number of operations below those of the fast fourier transform fft algorithms the usual pruning method is discussed and an efficient algorithm called transform decomposition is introduced it is based on a mixture of a standard fft algorithm and the horner polynomial evaluation scheme equivalent to the one in goertzel s algorithms it requires fewer operations and is more flexible than pruning the algorithm works for power of two and prime factor algorithms as well as for real input data

stabilization of mehrotra s primal dual algorithm and its implementation abstract in this paper we apply a stabilization procedure proposed by kovacevic vujcic and asic to the mehrotra s primal dual interior point algorithm for linear programming transformations of the dual problem corresponding to the stabilization procedure are considered the stabilization procedure and mehrotra s algorithm are implemented in the package mathematica a number of highly degenerate test examples are used to compare the modified mehrotra s method with respect to the original one and the codes pcx lipsol and mosek we also provide numerical results on some examples from the netlib test set

on bounds for scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a positive definite diagonal matrix in a recent paper stewart considered the weighted pseudoinverse x d x t dx 1 x t d and the associated oblique projection p d xx d and gave bounds independent of d for the norms of these matrices in this note we answer a question he raised by showing that the bounds are computable

uniform boundedness of a preconditioned normal matrix used in interior point methods solving systems of linear equations with normal matrices of the form a d2 at is a key ingredient in the computation of search directions for interior point algorithms in this article we establish that a well known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as d varies over the set of all positive diagonal matrices in particular we show that when a is the node arc incidence matrix of a connected directed graph with one of its rows deleted then the condition number of the corresponding preconditioned normal matrix is bounded above by m n m 1 where m and n are the number of nodes and arcs of the network

perturbation analysis for the projection of a point to an affine set abstract we present a perturbation analysis for the projection of a point to an affine set in r n the matrix involved is not assumed to be full rank and the perturbation is arbitrary error bounds for the solution under the rank preserving and rank nondecreasing perturbations are obtained the results given here appear to be the most general up to now

a fast recursive algorithm for computing the discrete cosine transform the discrete cosine transform dct is widely applied in various fields including image data compression because it operates like the karhunen loeve transform for stationary random data this paper presents a recursive algorithm for dct with a structure that allows the generation of the next higher order dct from two identical lower order dct s as a result the method for implementing this recursive dct requires fewer multipliers and adders than other dct algorithms

fast hartley transform pruning the discrete hartley transform dht is discussed as a tool for the processing of real signals fast hartley transform fht algorithms which compute the dht in a time proportional to n log sub 2 n exist in many applications such as interpolation and convolution of signals a significant number of zeros are padded to the nonzero valued samples before the transform is computed it is shown that for such situations significant savings in the number of additions and multiplications can be obtained by pruning the fht algorithm the modifications in the fht algorithm as a result of pruning are developed and implemented in an fht subroutine the amount of savings in the operation is determined

product form cholesky factorization in interior point methods for second order cone programming second order cone programming socp problems are typically solved by interior point methods as in linear programming lp interior point methods can in theory solve socps in polynomial time and can in practice exploit sparsity in the problem data specifically when cones of large dimension are present the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an lp here we propose a product form cholesky factorization pfcf approach and show that it is more numerically stable than the alternative sherman morrison woodbury approach we derive several pfcf variants and compare their theoretical perfomance finally we prove that the elements of l in the cholesky factorizations ldlt that arise in interior point methods for socp are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path

new perturbation results for equality constrained least squares problems abstract we present some perturbation results for least squares problems with equality constraints relative errors are obtained on perturbed solutions and lagrange multipliers of the problem based on the equivalence of the problem to a consistent system of linear equations

error analysis and implementation aspects of deferred correction for equality constrained least squares problems the deferred correction procedure recently proposed by van loan on the method of weighting for equality constrained least squares problems siam j numer anal 22 1985 pp 851 864 solves the problem of minimizing f ex 2 subject to cx g by repeatedly solving the related least squares problem of minimizing left left begin array 20 c tau delta g delta f end array right left begin array 20 c tau c e end array right delta x right 2 for some large weight tau the convergence behavior of the procedure is shown to be greatly dependent upon error analysis considerations an analysis of its convergence with an approximation factorization of left begin array 20 c tau c e end array right is given an implementation improvement to the procedure that makes it more robust is also given van loan s conjecture that the method converges for all but ill conditioned problems is shown to be true

modified cholesky factorizations in interior point algorithms for linear programming we investigate a modified cholesky algorithm typical of those used in most interior point codes for linear programming cholesky based interior point codes are popular for three reasons their implementation requires only minimal changes to standard sparse cholesky algorithms allowing us to take full advantage of software written by specialists in that area they tend to be more efficient than competing approaches that use alternative factorizations and they perform robustly on most practical problems yielding good interior point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned we investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis illustrating our results with computational experiments finally we point out the potential limitations of this approach

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

fast computation of the discrete cosine transform and the discrete hartley transform a new factorization of the discrete hartley transform dht is presented it is used to derive new algorithms for the dht and the discrete cosine transform dct with reduced number of multiplications

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

equivalent formulae for the supremum and stability of weighted pseudoinverses during recent decades there have been a great number of research articles studying interior point methods for solving problems in mathematical programming and constrained optimization stewart and o leary obtained an upper bound for scaled pseudoinverses sup w1 2x w1 2 2 of a matrix x w p where p is a set of diagonal positive definite matrices we improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses forsgren further generalized these results to derive the supremum of weighted pseudoinverses sup w1 2x w1 2 2 where w p p is a set of diagonally dominant positive semidefinite matrices by using a signature decomposition of weighting matrices w and by applying the binet cauchy formula and cramer s rule for determinants the results are also extended to equality constrained linear least squares problems in this paper we extend forsgren s results to a general complex matrix x to establish several equivalent formulae for sup w1 2x w1 2 2 where p is a w p set of diagonally dominant positive semidefinite matrices or a set of weighting matrices arising from solving equality constrained least squares problems we also discuss the stability property of these weighted pseudoinverses

an immersed boundary method for simulating vesicle dynamics in three dimensions we extend our previous immersed boundary ib method for 3d axisymmetric inextensible vesicle in navier stokes flows hu et al 2014 17 to general three dimensions despite a similar spirit in numerical algorithms to the axisymmetric case the fully 3d numerical implementation is much more complicated and is far from straightforward a vesicle membrane surface is known to be incompressible and exhibits bending resistance as in 3d axisymmetric case instead of keeping the vesicle locally incompressible we adopt a modified elastic tension energy to make the vesicle surface patch nearly incompressible so that solving the unknown tension lagrange multiplier for the incompressible constraint can be avoided nevertheless the new elastic force derived from the modified tension energy has exactly the same mathematical form as the original one except the different definitions of tension the vesicle surface is discretized on a triangular mesh where the elastic tension and bending force are calculated on each vertex lagrangian marker in the ib method of the triangulation a series of numerical tests on the present scheme are conducted to illustrate the robustness and applicability of the method we perform the convergence study for the immersed boundary forces and the fluid velocity field we then study the vesicle dynamics in various flows such as quiescent simple shear and gravitational flows our numerical results show good agreements with those obtained in previous theoretical experimental and numerical studies

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

numerical methods for the stochastic landau lifshitz navier stokes equations the landau lifshitz navier stokes llns equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes this paper examines explicit eulerian discretizations of the full llns equations several computational fluid dynamics approaches are considered including maccormack s two step lax wendroff scheme and the piecewise parabolic method and are found to give good results for the variance of momentum fluctuations however neither of these schemes accurately reproduces the fluctuations in energy or density we introduce a conservative centered scheme with a third order runge kutta temporal integrator that does accurately produce fluctuations in density energy and momentum a variety of numerical tests including the random walk of a standing shock wave are considered and results from the stochastic llns solver are compared with theory when available and with molecular simulations using a direct simulation monte carlo algorithm

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

a modified layered step interior point algorithm for linear programming the layered step interior point algorithm was introduced by vavasis and ye the algorithm accelerates the path following interior point algorithm and its arithmetic complexity depends only on the coefficient matrixa the main drawback of the algorithm is the use of an unknown big constant bar chi a in computing the search direction and to initiate the algorithm we propose a modified layered step interior point algorithm which does not use the big constant in computing the search direction the constant is required only for initialization when a well centered feasible solution is not available and it is not required if an upper bound on the norm of a primal dual optimal solution is known in advance the complexity of the simplified algorithm is the same as that of vavasis and ye 1998 the mathematical programming society inc published by elsevier science b v

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

an embedding approach to frequency domain and subband adaptive filtering frequency domain and subband implementations improve the computational efficiency and the convergence rate of adaptive schemes the well known multidelay adaptive filter mdf belongs to this class of block adaptive structures and is a dft based algorithm we develop adaptive structures that are based on the trigonometric transforms discrete cosine transform dct and discrete sine transform dst and on the discrete hartley transform dht as a result these structures involve only real arithmetic and are attractive alternatives in cases where the traditional dft based scheme exhibits poor performance the filters are derived by first presenting a derivation for the classical dft based filter that allows us to pursue these extensions immediately the approach used in this paper also provides further insights into subband adaptive filtering

robust maximum likelihood estimation of multivariable dynamic systems this paper examines the problem of estimating linear time invariant state space system models in particular it addresses the parametrization and numerical robustness concerns that arise in the multivariable case these difficulties are well recognised in the literature resulting for example in extensive study of subspace based techniques as well as recent interest in data driven local co ordinate approaches to gradient search solutions the paper here proposes a different strategy that employs the expectation maximisation em technique the consequence is an algorithm that is iterative with associated likelihood values that are locally convergent to stationary points of the gaussian likelihood function furthermore theoretical and empirical evidence presented here establishes additional attractive properties such as numerical robustness avoidance of difficult parametrization choices the ability to naturally and easily estimate non zero initial conditions and moderate computational cost moreover since the methods here are maximum likelihood based they have associated known and asymptotically optimal statistical properties

adaptive filtering using hartley transform and overlap save method a novel method of implementing fir finite impulse response adaptive filters using the discrete hartley transform dht and the overlap save method is proposed it not only has the advantage of fast convergence for highly correlated input signals but also provides a significant reduction in computation over the conventional time domain and frequency domain least mean square lms adaptive filters compared with the previously proposed unconstrained frequency domain flms algorithm the convergence performance is similar but an approximate saving of 30 in computation can be obtained for the usual filter length

effects of finite precision arithmetic on interior point methods for nonlinear programming we show that the effects of finite precision arithmetic in forming and solving the linear system that arises at each iteration of primal dual interior point algorithms for nonlinear programming are benign provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path following methods when we replace the standard assumption that the active constraint gradients are independent by the weaker mangasarian fromovitz constraint qualification rapid convergence usually is attainable even when cancellation and roundoff errors occur during the calculations in deriving our main results we prove a key technical result about the size of the exact primal dual step this result can be used to modify existing analysis of primal dual interior point methods for convex programming making it possible to extend the superlinear local convergence results to the nonconvex case

a strong bound on the integral of the central path curvature and its relationship with the iteration complexity of primal dual path following lp algorithms the main goals of this paper are to i relate two iteration complexity bounds derived for the mizuno todd ye predictor corrector mty p c algorithm for linear programming lp and ii study the geometrical structure of the lp central path the first iteration complexity bound for the mty p c algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path the second iteration complexity bound derived recently by the authors using the notion of crossover events introduced by vavasis and ye is expressed in terms of a scale invariant condition number associated with m n constraint matrix of the lp in this paper we establish a relationship between these bounds by showing that the first one can be majorized by the second one we also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by vavasis and ye in view of the behavior of their layered least squares path following lp method that the central path consists of mathcal o n 2 long but straight continuous parts while the remaining curved part is relatively short

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

on computing the discrete hartley transform the discrete hartley transform dht is a real valued transform closely related to the dft of a real valued sequence bracewell has recently demonstrated a radix 2 decimation in time fast hartley transform fht algorithm in this paper a complete set of fast algorithms for computing the dht is developed including decimation in frequency radix 4 split radix prime factor and winograd transform algorithms the philosophies of all common fft algorithms are shown to be equally applicable to the computation of the dht and the fht algorithms closely resemble their fft counterparts the operation counts for the fht algorithms are determined and compared to the counts for corresponding real valued fft algorithms the fht algorithms are shown to always require the same number of multiplications the same storage and a few more additions than the real valued fft algorithms even though computation of the fht takes more operations in some situations the inherently real valued nature of the discrete hartley transform may justify this extra cost

weighted projections and riesz frames abstract let h be a separable hilbert space and e k k 1 a fixed orthonormal basis of h motivated by many papers on scaled projections angles of subspaces and oblique projections we define and study the notion of compatibility between a subspace and the abelian algebra of diagonal operators in the given basis this is used to refine previous work on scaled projections and to obtain a new characterization of riesz frames

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

simulation of osmotic swelling by the stochastic immersed boundary method we develop computational methods for the simulation of osmotic swelling phenomena relevant to microscopic vesicles containing transformable solute molecules we introduce stochastic immersed boundary methods sibms that can capture osmotically driven fluid transport through semipermeable elastic membranes subject to thermal fluctuations we also develop numerical methods to handle within sibms an elastic shell model for a neo hookean material our extended sibm allows for capturing osmotic swelling phenomena driven by concentration changes and interactions between a discrete collection of confined particles while accounting for the thermal fluctuations of the semipermeable membrane and the hydrodynamic transport of solvent we use our computational methods to investigate osmotic phenomena in regimes that go beyond the classical van t hoff theory we develop statistical mechanics theories for osmotic swelling of vesicles when there are significant interactions between particles that can transform over tim

a product form cholesky factorization method for handling dense columns in interior point methods for linear programming cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods ipms for linear programming lp its main advantages being numerical stability and efficiency for sparse systems however in the presence of dense columns there is dramatic loss of efficiency a typical approach to remedy this problem is to apply the sherman morrison woodbury smw update formula to the dense part this approach while being very efficient is not numerically stable here we propose using product form cholesky factorization to handle dense columns the proposed approach is essentially as stable as the original cholesky factorization and nearly as efficient as the smw approach we demonstrate these properties both theoretically and computationally a key part of our theoretical analysis is the proof that the elements of the cholesky factors of the matrices that arise in ipms for lp are uniformly bounded as the duality gap converges to zero

global error bounds for convex conic problems this paper aims at deriving and proving some lipschitzian type error bounds for convex conic problems in a simple way first it is shown that if the recession directions satisfy slater s condition then a global lipschitzian type error bound holds alternatively if the feasible region is bounded then the ordinary slater condition guarantees a global lipschitzian type error bound these can be considered as generalizations of previously known results for inequality systems which also follow from general results by bauschke and borwein in siam review 38 1996 pp 367 426 and bauschke borwein and li in a 1997 report however the proofs in the current paper are considerably simpler some of the results are generalized to the intersection of multiple shifted cones with different shifts under slater s condition alone a global lipschitzian type error bound does not hold it is shown however that such an error bound holds for a specific conic region for linear systems we establish that the sharp constant involved in hoffman s error bound is nothing but the condition number for linear programming as used by vavasis and ye in math programming 74 1996 pp 79 120

a norm bound for projections with complex weights abstract we consider projections weighted by a complex diagonal matrix these projections arise in the solution of alternating current ac electrical networks a norm bound for weighted projections in the case of an ill conditioned real positive definite weight matrix was established by dikin stewart todd and others we show that this bound generalizes to the complex case

on the method of weighting for equality constrained least squares problems the generalized singular value decomposition is used to analyze the problem of minimizing ax b 2 subject to the constraint bx d a byproduct of the analysis is a new iterative procedure that can be used to improve an approximate solution obtained via the method of weights all that is required to implement the procedure is a single qr factorization these developments turn out to be of interest when a and b are sparse and for the case when systolic architectures are used to carry out the computations

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

the geometry of the set of scaled projections abstract the set of scaled projections of a vector onto the column space of a matrix has recently been of interest to several authors the aim of the present investigation is to obtain a detailed description of the geometry of this set in our main result we show by construction that the set of scaled projections is the union of finitely many polytopes the proof makes use of a theorem on the alternative

algebraic properties of the rank deficient equality constrained and weighted least squares problems abstract we study the algebraic properties of the solutions of the equality constrained least squares problem min f s v kf g v 2 with s f v h lf v 2 min z c n v v h lz v 2 lse and corresponding weighted least squares problem wls in which l and l k may not be of full rank general formulas for the solutions are given and the algebraic relations between the lse and the wls solutions are obtained

a polynomial predictor corrector trust region algorithm for linear programming in this paper we present a scaling invariant interior point predictor corrector type algorithm for linear programming lp whose iteration complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the lp constraint matrix at the predictor stage the algorithm either takes the step along the standard affine scaling as direction or a new trust region type direction whose construction depends on a scaling invariant bipartition of the variables determined by the as direction this contrasts with the layered least squares direction introduced in s vavasis and y ye math program 74 1996 pp 79 120 whose construction depends on multiple layered partitions of the variables that are not scaling invariant moreover it is shown that the overall arithmetic complexity of the algorithm weakly depends on the right hand side and the cost of the lp in view of the work involved in the computation of the trust region steps

the interior point revolution in optimization history recent developments and lasting consequences interior methods are a pervasive feature of the optimization landscape today but it was not always so although interior point techniques primarily in the form of barrier methods were widely used during the 1960s for problems with nonlinear constraints their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method during the 1970s barrier methods were superseded nearly to the point of oblivion by newly emerging and seemingly more efficient alternatives such as augmented lagrangian and sequential quadratic programming methods by the early 1980s barrier methods were almost universally regarded as a closed chapter in the history of optimization this picture changed dramatically in 1984 when narendra karmarkar announced a fast polynomial time interior method for linear programming in 1985 a formal connection was established between his method and classical barrier methods since then interior methods have continued to transform both the theory and practice of constrained optimization we present a condensed unavoidably incomplete look at classical material and recent research about interior methods

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

a note on properties of condition numbers abstract the condition numbers described by stewart todd and others for a matrix a are given by the suprema of the norms of all the oblique projection operators respectively on the image space of a t and the null space of a this paper shows that these two numbers coincide

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

some properties of the hessian of the logarithmic barrier function more than twenty years ago murray and lootsma showed that hessian matrices of the logarithmic barrier function become increasingly ill conditioned at points on the barrier trajectory as the solution is approached this paper explores some further characteristics of the barrier hessian we first show that except in two special cases the barrier hessian is ill conditioned in an entire region near the solution at points in a more restricted region including the barrier trajectory itself this ill conditioning displays a special structure connected with subspaces defined by the jacobian of the active constraints we then indicate how a cholesky factorization with diagonal pivoting can be used to detect numerical rank deficiency in the barrier hessian and to provide information about the underlying subspaces without making an explicit prediction of the active constraints using this subspace information a close approximation to the newton direction can be calculated by solving linear systems whose condition reflects that of the original problem

characterizations bounds and probabilistic analysis of two complexity measures for linear programming problems this note studies a a condition number used in the linear programming algorithm of vavasis and ye 14 whose running time depends only on the constraint matrix a m n and a a variant of another condition number due to ye 17 that also arises in complexity analyses of linear programming problems we provide a new characterization of a and relate a and a furthermore we show that if a is a standard gaussian matrix then e ln a o min mlnn n thus the expected running time of the vavasis ye algorithm for linear programming problems is bounded by a polynomial in m and n for any right hand side and objective coefficient vectors when a is randomly generated in this way as a corollary of the close relation between a and a we show that the same bound holds for e ln a

a new algorithm to compute the discrete cosine transform a new algorithm is introduced for the 2m point discrete cosine transform this algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms and it makes the system simpler

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

unconstrained frequency domain adaptive filter an unconstrained frequency domain least mean squares uflms algorithm is presented the algorithm is based on the overlap save technique used in frequency domain filtering the proposed algorithm converges to the wiener solution without the time domain constraint on the impulse response as proposed by ferrara for a large number of taps geq32 the uflms algorithm offers a significant reduction in computation another advantage is its fast convergence for highly correlated input signals the performance of the algorithm is illustrated by a simulation

barrier function based model predictive control a new formulation of nonlinear model predictive control mpc is developed by including a weighted barrier function in the control objective while the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set in addition the resulting optimisation problem to be solved at each control step is effectively unconstrained and thus amenable to elegant optimisation techniques the barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier conventional mpc may be seen as a limiting case of the new class as the barrier weighting itself approaches zero we pay particular attention to the novel approach of fixing the weighting parameter to some positive value possibly large and observe that this provides a degree of controller caution near constraint boundaries we construct an ellipsoidal invariant set by exploiting the geometry of self concordant functions and show nominal closed loop stability for this class of controllers under full state feedback

on scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a diagonal matrix with positive diagonal elements the weighted pseudoinverse defined by x d x t dx 1 x t d and the associated oblique projection p d xx d arise in many applications in this paper we show that the norms of both matrices are bounded by numbers that are independent of d

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

a new iteration complexity bound for the mty predictor corrector algorithm in this paper we present a new iteration complexity bound for the mizuno todd ye predictor corrector mty p c primal dual interior point algorithm for linear programming the analysis of the paper is based on the important notion of crossover events introduced by vavasis and ye for a standard form linear program min ctx ax b x ge 0 with decision variable x in re n e show that the mty p c algorithm started from a well centered interior feasible solution with duality gap n mu 0 finds an interior feasible solution with duality gap less than n eta in cal o t mu 0 eta n 3 5 log bar chi a iterations where t t equiv min n 2 log log t log t for all t 0 and bar chi a is a scaling invariant condition number associated with the matrix a more specifically bar chi a is the infimum of all the conditions numbers bar chi ad where d varies over the set of positive diagonal matrices under the setting of the turing machine model our analysis yields an cal o n 3 5 l a min n 2 log l l iteration complexity bound for the mty p c algorithm to find a primal dual optimal solution where la and l are the input sizes of the matrix a and the data a b c respectively this contrasts well with the classical iteration complexity bound for the mty p c algorithm which depends linearly on l instead of log l

ill conditioning and computational error in interior methods for nonlinear programming ill conditioning has long been regarded as a plague on interior methods but its damaging effects have rarely been documented in fact implementors of interior methods who ignore warnings about the dire consequences of ill conditioning usually manage to compute accurate solutions we offer some insight into this seeming contradiction by analyzing ill conditioning within a primal dual method in which the full usually well conditioned primal dual matrix is transformed to a condensed inherently ill conditioned matrix mpd we show that ill conditioning in the exact condensed matrix closely resembles that known for the primal barrier hessian and then examine the influence of cancellation in the computed constraints using the structure of mpd various bounds are obtained on the absolute accuracy of the computed primal dual steps without cancellation the portion of the computed x step in the small space of mpd a subspace close to the null space of the jacobian of the active constraints has an absolute error bound comparable to machine precision and its large space component has a much smaller error bound with cancellation the usual case the absolute error bounds for both the small and large space components of the computed x step are comparable to machine precision in either case the absolute error bound for the computed multiplier steps associated with active constraints is comparable to machine precision the computed multiplier steps for inactive constraints although converging to zero retain approximately full relative precision because of errors in forming the right hand side the absolute error in the computed solution of the full well conditioned primal dual system is shown to be comparable to machine precision thus under quite general conditions ill conditioning in mpd does not noticeably impair the accuracy of the computed primal dual steps a similar analysis applies to search directions obtained by direct solution of the primal newton equations

hopdm version 2 12 a fast lp solver based on a primal dual interior point method hopdm is an implementation of the primal dual interior point method for solving large scale linear programming lp problems hopdm stands for higher order primal dual method its newest version 2 12 of may 1995 is a robust and efficient code that compares favourably with today s commercial lp solvers

a variant of the vavasis ye layered step interior point algorithm for linear programming in this paper we present a variant of vavasis and ye s layered step path following primal dual interior point algorithm for linear programming our algorithm is a predictor corrector type algorithm which uses from time to time the layered least squares lls direction in place of the affine scaling as direction it has the same iteration complexity bound of vavasis and ye s algorithm namely cal o n 3 5 log bar chi a n where n is the number of nonnegative variables and bar chi a is a certain condition number associated with the constraint matrix a vavasis and ye s algorithm requires explicit knowledge of bar chi a which is very hard to compute or even estimate in order to compute the layers for the lls direction in contrast our algorithm uses the as direction at the current iterate to determine the layers for the lls direction and hence does not require the knowledge of bar chi a a variant with similar properties and with the same complexity has been developed by megiddo mizuno and tsuchiya math programming 82 1998 pp 339 355 however their algorithm needs to compute n lls directions on every iteration while ours computes at most one lls direction on any given iteration

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

consistent independent component analysis and prewhitening we study the statistical merits of two techniques used in the literature of independent component analysis ica first we analyze the characteristic function based ica method chfica and study its statistical properties such as consistency spl radic n consistency and robustness against small additive noise second we study the validity of prewhitening a preprocessing technique used by many ica algorithms as applied to the chfica method in particular we establish the surprising effectiveness of this technique even when some components have heavy tails and others do not a fast new algorithm implementing the prewhitened chfica method is also provided

regularization techniques in interior point methods regularization techniques i e modifications on the diagonal elements of the scaling matrix are considered to be important methods in interior point implementations so far regularization in interior point methods has been described for linear programming problems in which case the scaling matrix is diagonal it was shown that by regularization free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill posed linear programming problems in this paper we study the factorization of the augmented system arising in interior point methods in our investigation we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite but not diagonal we show that regularization techniques may be applied beyond the linear programming case

the quick fourier transform an fft based on symmetries this paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete fourier transform dft we develop an algorithm called the quick fourier transform qft that reduces the number of floating point operations necessary to compute the dft by a factor of two or four over direct methods or goertzel s method for prime lengths by further application of the idea to the calculation of a dft of length 2 sup m we construct a new o nlogn algorithm with computational complexities comparable to the cooley tukey algorithm we show that the power of two qft can be implemented in terms of discrete sine and cosine transforms the algorithm can be easily modified to compute the dft with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real