estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

on the superlinear and quadratic convergence of primal dual interior point linear programming algorithms this paper presents a convergence rate analysis for interior point primal dual linear programming algorithms conditions that guarantee q superlinear convergence are identified in two distinct theories both state that under appropriate assumptions q superlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero the first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation q superlinear convergence was always observed the second theory requires the restrictive assumption of primal nondegeneracy however it gives the surprising result that q superlinear convergence can still be attained even if centering is not phased out provided the iterates asymptotically approach the central path the latter theory is extended to produce a satisfactory q quadratic convergence theory it requir

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

convergence of a fully conservative volume corrected characteristic method for transport problems we consider the convergence of a volume corrected characteristics mixed method vccmm for advection diffusion systems it is known that without volume correction the method is first order convergent provided there is a nondegenerate diffusion term we consider the advective part of the system and give some properties of the weak solution with these properties we prove that the volume corrected method with no diffusion term gives a lower order l 1 convergence rate of mathcal o h sqrt delta t h delta t r where r is related to the accuracy of the characteristic tracing this result compares favorably to godunov s method but avoids the cfl constraint so large time steps can be taken in practice in fact godunov s method converges at mathcal o h 1 2 which is our result for delta t ch where now c is not limited however the optimal choice delta t ch 2 2r 1 gives a better rate mathcal o h 2r 2r 1 than godunov s method e g mathcal o h 2 3 if r 1 with a nondegenerate diffusion term we obtain an l 2 error estimate for the problem we also prove the existence of and give an error estimate for a perturbed velocity field for which the volume is locally conserved finally some convergence tests are given to verify the optimal convergence rate for r 1

a conservative semi lagrangian multi tracer transport scheme cslam on the cubed sphere grid a conservative multi tracer transport algorithm on the cubed sphere based on the semi lagrangian approach cslam has been developed the scheme relies on backward trajectories and the resulting upstream cells polygons are approximated with great circle arcs biquadratic polynomial functions are used for approximating the density distribution in the cubed sphere grid cells the upstream surface integrals associated with the conservative semi lagrangian scheme are computed as line integrals by employing the gauss green theorem the line integrals are evaluated using a combination of exact integrals and high order gaussian quadrature the upstream cell trajectories information and computation of weights of integrals can be reused for each additional tracer the cslam scheme is extensively tested with various standard benchmark test cases of solid body rotation and deformational flow in both cartesian and spherical geometry and the results are compared with those of other published schemes the cslam scheme is accurate robust and moreover the edges and vertices of the cubed sphere discontinuities do not affect the overall accuracy of the scheme the cslam scheme exhibits excellent convergence properties and has an option for enforcing monotonicity the advantages of introducing cross terms in the fully two dimensional biquadratic density distribution functions are also examined in the context of cartesian as well as the cubed sphere grid which has six local sub domains with discontinuous edges and corners

a fully mass and volume conserving implementation of a characteristic method for transport problems the characteristics mixed method considers the transport not of a single point or fluid particle but rather the mass in an entire region of fluid this mass is transported along the characteristic curves of the hyperbolic part of the transport equation and the scheme thereby produces very little numerical dispersion conserves mass locally and can use long time steps however since the shape of a characteristic trace back region must be approximated in numerical implementation its volume may be incorrect resulting in inaccurate concentration densities and further inaccurate reaction dynamics we present a simple modification to the characteristics mixed method that conserves both mass and volume of the transported fluid regions our algorithm also handles boundary conditions through a space time change of variables in the trace back routines which allows the boundary to be treated as if it were interior to the domain nearly point sources such as wells present special difficulties since characteristic trace back curves converge in their vicinity we also present techniques that allow one to conservatively implement wells the techniques are illustrated in five numerical examples

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

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

quadratic convergence of potential reduction methods for degenerate problems global and local convergence properties of a primal dual interior point pure potential reduction algorithm for linear programming problems is analyzed this algorithm is a primal dual variant of the iri imai method and uses modified newton search directions to minimize the tanabe todd ye tty potential function a polynomial time complexity for the method is demonstrated furthermore this method is shown to have a unique accumulation point even for degenerate problems and to have q quadratic convergence to this point by an appropriate choice of the step sizes this is to the best of our knowledge the first superlinear convergence result on degenerate linear programs for primal dual interior point algorithms that do not follow the central path

superlinear primal dual affine scaling algorithms for lcp we describe an interior point algorithm for monotone linear complementarity problems in which primal dual affine scaling is used to generate the search directions the algorithm is shown to have global and superlinear convergence with q order up to but not including two the technique is shown to be consistent with a potential reduction algorithm yielding the first potential reduction algorithm that is both globally and superlinearly convergent

a fully conservative eulerian lagrangian method for a convection diffusion problem in a solenoidal field tracer transport is governed by a convection diffusion problem modeling mass conservation of both tracer and ambient fluids numerical methods should be fully conservative enforcing both conservation principles on the discrete level locally conservative characteristics methods conserve the mass of tracer but may not conserve the mass of the ambient fluid in a recent paper by the authors t arbogast c huang a fully mass and volume conserving implementation of a characteristic method for transport problems siam j sci comput 28 2006 2001 2022 a fully conservative characteristic method the volume corrected characteristics mixed method vccmm was introduced for potential flows here we extend and apply the method to problems with a solenoidal i e divergence free flow field the modification is a computationally inexpensive simplification of the original vccmm requiring a simple adjustment of trace back regions in an element by element traversal of the domain our numerical results show that the method works well in practice is less numerically diffuse than uncorrected characteristic methods and can use up to at least about eight times the cfl limited time step

incremental remapping as a transport advection algorithm there are two fundamentally different strategies for solving the standard transport or continuity equation corresponding to whether it is expressed as a partial differential equation or as an integral statement of conservation the more common approach is to discretize the partial differential equation and to march the solution forward in time the alternative method is to project cell volumes along lagrangian trajectories as far forward or backward in time as desired and then to remap the resulting density distribution onto some target mesh this latter approach is known as remapping remapping has many advantages not the least of which is that the time step is limited only by accuracy considerations but it tends to be expensive and complex in this paper we show that if the time step is made sufficiently short such that trajectories are confined to the nearest neighbor cells then the remapping may be written as a flux form transport algorithm and it becomes nearly as simple and efficient as standard transport schemes the resulting method called incremental remapping retains most of the advantages of general remapping these include a geometric basis for transport b compatibility of associated tracer transport with simple tracer advection i e retention of tracer monotonicity properties and c efficient handling of multiple tracers since each additional tracer adds only a relatively small incremental cost

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

the modified method of characteristics with adjusted advection the mmoc procedure for approximating the solutions of transport dominated diffusion problems does not automatically preserve integral conservation laws leading to mass balance errors in many kinds of flow problems the variant called the mmocaa discussed herein preserves the conservation law at a minor additional computational cost it is shown that its solution in either galerkin or finite difference form converges at the same rates as were proved earlier by dougl as and russell for the standard mmoc procedure

sparsenet coordinate descent with nonconvex penalties we address the problem of sparse selection in linear models a number of nonconvex penalties have been proposed in the literature for this purpose along with a variety of convex relaxation algorithms for finding good solutions in this article we pursue a coordinate descent approach for optimization and study its convergence properties we characterize the properties of penalties suitable for this approach study their corresponding threshold functions and describe a df standardizing reparametrization that assists our pathwise algorithm the mc penalty is ideally suited to this task and we use it to demonstrate the performance of our algorithm certain technical derivations and experiments related to this article are included in the supplementary materials section

asymptotic behavior of continuous trajectories for primal dual potential reduction methods this article considers continuous trajectories of the vector fields induced by primal dual potential reduction algorithms for solving linear programming problems it is known that these trajectories converge to the analytic center of the primal dual optimal face we establish that this convergence may be tangential to the central path tangential to the optimal face or in between depending on the value of the potential function parameter

does ell _ p minimization outperform ell _ 1 minimization in many application areas ranging from bioinformatics to imaging we are faced with the following question can we recover a sparse vector x o in mathbb r n from its undersampled set of noisy observations y in mathbb r n y a x o w the last decade has witnessed a surge of algorithms and theoretical results to address this question one of the most popular schemes is the ell p regularized least squares given by the following formulation hat x gamma p in arg min x 1 2 y ax 2 2 gamma x p p where p in 0 1 among these optimization problems the case p 1 also known as lasso is the best accepted in practice for the following two reasons first thanks to the extensive studies performed in the fields of high dimensional statistics and compressed sensing we have a clear picture of lasso s performance second it is convex and efficient algorithms exist for finding its global minima unfortunately neither of the above two properties hold for 0 leq p however they are still appealing because of the following folklores in the high dimensional statistics first hat x gamma p is closer to x o than hat x gamma 1 second if we employ iterative methods that aim to converge to a local minima of arg min x 1 2 y ax 2 2 gamma x p p then under good initialization these algorithms converge to a solution that is still closer to x o than hat x gamma 1 in spite of the existence of plenty of empirical results that support these folklore theorems the theoretical progress to establish them has been very limited this paper aims to study the above mentioned folklore theorems and establish their scope of validity starting with approximate message passing amp algorithm as a heuristic method for solving ell p regularized least squares we study the following questions first what is the impact of initialization on the performance of the algorithm second when does the algorithm recover the sparse signal x o under a good initialization third when does the algorithm converge to the sparse signal regardless of the initialization studying these questions will not only shed light on the second folklore theorem but also lead us to the answer the first one i e the performance of the global optima hat x gamma p for that purpose we employ the replica analysis 1 to show the connection between the solution of amp and hat x gamma p in the asymptotic settings this enables us to compare the accuracy of hat x gamma p and hat x gamma 1 in particular we will present an accurate characterization of the phase transition and noise sensitivity of ell p regularized least squares for every 0 leq p leq 1 our results in the noiseless setting confirm that ell p regularized least squares if gamma is tuned optimally exhibits the same phase transition for every 0 leq p and this phase transition is much better than that of lasso furthermore we show that in the noisy setting there is a major difference between the performance of ell p regularized least squares with different values of p for instance we will show that for very small and very large measurement noises p 0 and p 1 outperform the other values of p respectively 1 replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems

optimization based mesh correction with volume and convexity constraints we consider the problem of finding a mesh such that 1 it is the closest with respect to a suitable metric to a given source mesh having the same connectivity and 2 the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh this volume correction problem arises in important simulation contexts such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi lagrangian transport schemes in this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective while the prescribed cell volumes mesh validity and or cell convexity specify the constraints we solve this problem numerically using a sequential quadratic programming sqp method whose performance scales with the mesh size to achieve scalable performance we develop a specialized multigrid based preconditioner for optimality systems that arise in the application of the sqp method to the volume correction problem numerical examples illustrate the importance of volume correction and showcase the accuracy robustness and scalability of our approach

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

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

randomly spread cdma asymptotics via statistical physics this paper studies randomly spread code division multiple access cdma and multiuser detection in the large system limit using the replica method developed in statistical physics arbitrary input distributions and flat fading are considered a generic multiuser detector in the form of the posterior mean estimator is applied before single user decoding the generic detector can be particularized to the matched filter decorrelator linear minimum mean square error mmse detector the jointly or the individually optimal detector and others it is found that the detection output for each user although in general asymptotically non gaussian conditioned on the transmitted symbol converges as the number of users go to infinity to a deterministic function of a hidden gaussian statistic independent of the interferers thus the multiuser channel can be decoupled each user experiences an equivalent single user gaussian channel whose signal to noise ratio snr suffers a degradation due to the multiple access interference mai the uncoded error performance e g symbol error rate and the mutual information can then be fully characterized using the degradation factor also known as the multiuser efficiency which can be obtained by solving a pair of coupled fixed point equations identified in this paper based on a general linear vector channel model the results are also applicable to multiple input multiple output mimo channels such as in multiantenna systems

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

primal dual symmetry and scale invariance of interior point algorithms for convex optimization we present a definition of symmetric primal dual algorithms for convex optimization problems expressed in the conic form after describing a generalization of the space approach for such optimization problems we show that a symmetric space approach can be developed for a convex optimization problem in the conic form if and only if the underlying cone is homogeneous and self dual we provide an alternative definition of self scaled barriers and then conclude with a discussion of the scalings of the variables which keep the underlying convex cone invariant

a statistical mechanics approach to large system analysis of cdma multiuser detectors we present a theory based on statistical mechanics to evaluate analytically the performance of uncoded fully synchronous randomly spread code division multiple access cdma multiuser detectors with additive white gaussian noise awgn channel under perfect power control and in the large system limit application of the replica method a tool developed in the literature of statistical mechanics allows us to derive analytical expressions for the bit error rate as well as the multiuser efficiency of the individually optimum io and jointly optimum jo multiuser detectors over the whole range of noise levels the information theoretic capacity of the randomly spread cdma channel and the performance of decorrelating and linear minimum mean square error mmse detectors are also derived in the same replica formulation thereby demonstrating validity of the statistical mechanical approach

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

a general topology godunov method we describe a numerical technique for solving 2 dimensional compressible multimaterial problems using a general topology mesh multimaterial problems are characterized by the presence of interfaces whose shapes may become arbitrarily complex in the course of dynamic evolution computational methods based on more conventional fixed connectivity quadrilateral meshes do not have adequate flexibility to follow convoluted interface shapes and frequently fail due to excessive mesh distortion the present method is based on a mesh of arbitrary polygonal cells because this mesh is dual to a triangulation its topology is unrestricted and it is able to accommodate arbitrary boundary shapes additionally this mesh is able to quickly and smoothly change local mesh resolution thus economizing on the number of mesh cells and it is able to improve mesh isotropy because in a region of uniform mesh the cells tend to become regular hexagons the underlying algorithms are based on those of the caveat code these consist of an explicit finite volume cell centered arbitrary lagrangian eulerian ale technique coupled with the godunov method which together are readily adaptable to a general topology mesh several special techniques have been developed for this extension to a more general mesh they include an interface propagation scheme based on huygens construction a near lagrangian mesh rezoning algorithm that minimizes advection while enhancing mesh regularity an efficient global remapping algorithm that is capable of conservatively transferring quantities from one general mesh to another and various mesh restructering algorithms such as mesh reconnection smoothing and point addition and deletion

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

godunov type method on non structured meshes for three dimensional moving boundary problems abstract this paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner the method uses the space time formulation by godunov while the discretization is conducted on non structured tetrahedral meshes using roe s approximate riemann solver an implicit time stepping and a muscl type interpolation the computation of the geometrical parameters required to take into account the movement of the boundaries is described examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

large step path following methods for linear programming part ii potential reduction method the algorithm proposed in this paper is the affine polynomial potential reduction method with new procedures for updating the lower bounds for an optimal solution of the linear programming problem a method is developed for updating the lower bounds by large steps with strict control over the duality gaps associated with each iterate two algorithms are obtained by this approach the first one has complexity o nl iterations and a very simple updating procedure the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of o sqrt n l iterations

near boundary behavior of primal dual potential reduction algorithms for linear programming this paper is concerned with selection of the parameter in the primal dual potential reduction algorithm for linear programming chosen from n sqrt n the level of determines the relative importance placed on the centering vs the newton directions intuitively it would seem that as the iterate drifts away from the central path towards the boundary of the positive orthant must be set close ton sqrt n this increases the relative importance of the centering direction and thus helps to ensure polynomial convergence in this paper we show that this is unnecessary we find for any iterate that can be sometimes chosen in a wide range n sqrt n while still guaranteeing the currently best convergence rate of o sqrt n l iterations this finding is encouraging since in practice large values of have resulted in fast convergence rates our finding partially complements the recent result of zhang tapia and dennis 1990 concerning the local convergence rate of the algorithm

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

restricted isometry constants where ell p sparse recovery can fail for 0 p leq 1 this paper investigates conditions under which the solution of an underdetermined linear system with minimal lscrp norm 0 p les 1 is guaranteed to be also the sparsest one matrices are constructed with restricted isometry constants ric delta2m arbitrarily close to 1 radic2 ap 0 707 where sparse recovery with p 1 fails for at least one m sparse vector as well as matrices with delta2m arbitrarily close to one where lscr1 minimization succeeds for any m sparse vector this highlights the pessimism of sparse recovery prediction based on the ric and indicates that there is limited room for improving over the best known positive results of foucart and lai which guarantee that lscr1 minimization recovers all m sparse vectors for any matrix with delta2m 2 3 radic2 7 ap 0 4531 these constructions are a by product of tight conditions for lscrp recovery 0 les p les 1 with matrices of unit spectral norm which are expressed in terms of the minimal singular values of 2m column submatrices compared to lscr1 minimization lscrp minimization recovery failure is shown to be only slightly delayed in terms of the ric values furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used it is shown that when lscrp optimization is attempted using an iterative reweighted lscr1 scheme failure can still occur for delta2m arbitrarily close to 1 radic2