identifying structure of nonsmooth convex functions by the bundle technique we consider the problem of minimizing nonsmooth convex functions defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite hessian on the set of interest this is a particular case of functions with primal dual gradient structure a notion closely related to the so called cal vu space decomposition at a given point nonsmoothness is locally restricted to the directions of the subspace cal v while along the subspace cal u the behavior of the function is twice differentiable constructive identification of the two subspaces is important because it opens the way to devising fast algorithms for nonsmooth optimization by following iteratively the manifold of smoothness on which superlinear cal u newton steps can be computed in this work we show that for the class of functions in consideration the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points provided a minimizer satisfies the nondegeneracy and strong transversality conditions

new active set identification for general constrained optimization and minimax problems abstract the purpose of this paper is to discuss the problem of identifying the active constraints for general constrained nonlinear programming and constrained minimax problems at an isolated local solution facchinei fischer and kanzow 4 proposed an effective technique which can identify the active set in a neighborhood of an isolated local solution for nonlinear programming and han jian and li 5 improved and extended this to minimax problems in this work a new active constraint identification set is constructed not only is it tighter than the previous two identification sets but also it can be used effectively in penalty algorithms without strict complementarity and linear independence it is shown that the new identification technique can accurately identify the active constraints of nonlinear programming and constrained minimax problems some numerical examples illustrate the identification technique

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

a comparative study on methods for convergence acceleration of iterative vector sequences we discuss several methods for accelerating the convergence of the iterative solution of nonlinear equation systems commonly in use and point to interrelations between them in particular we investigate two of the most sophisticated schemes namely the anderson mixing and the broyden update both generalized to the consideration of arbitrarily many previous iterations for the broyden method we give a new derivation which is much simpler than that recently proposed by vanderbilt and louie we show that if the additional parameters invented by these authors in order to increase flexibility are used to optimize the convergence of the iteration process they in fact cancel out in addition we prove that in this optimal case the anderson mixing and the broyden update as applied to the inverse jacobian are fully identical thus we come to the conclusion that neither of these schemes is superior moreover we show that broyden update of the inverse jacobian is superior to updating the jacobian itself finally we propose an extension of the anderson mixing which avoids the numerical difficulties all these methods are faced with

finite element simulation of three dimensional particulate flows using mixture models in this paper we discuss the numerical treatment of three dimensional mixture models for semi dilute and concentrated suspensions of particles in incompressible uids the generalized navier stokes system and the continuity equation for the volume fraction of the disperse phase are discretized using an implicit high resolution nite element scheme and maximum principles are enforced using algebraic ux correction to prevent the volume fractions from exceeding the maximum packing limit a conservative overshoot limiter is applied to the converged convective

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

accelerated block coordinate relaxation for regularized optimization we discuss minimization of a smooth function to which is added a separable regularization function that induces structure in the solution a block coordinate relaxation approach with proximal linearized subproblems yields convergence to critical points while identification of the optimal manifold under a nondegeneracy condition allows acceleration techniques to be applied on a reduced space the work is motivated by experience with an algorithm for regularized logistic regression and computational results for the algorithm on problems of this type are presented

nonlinear krylov and moving nodes in the method of lines we report on some successes and problem areas in the method of lines from our work with moving node finite element methods first we report on our nonlinear krylov accelerator for the modified newton s method on the nonlinear equations of our stiff ode solver since 1990 it has been robust simple cheap and automatic on all our moving node computations we publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations second we discuss the need for reliable automatic choice of spatially variable time steps third we discuss the need for robust and efficient iterative solvers for the difficult linearized equations jx b of our stiff ode solver here the 1997 thesis of zulu xaba has made significant progress

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

an ale pfem method for the numerical simulation of two phase mixture flow abstract a new finite element method is proposed to simulate two phase mixture problems with free surfaces and or large moving boundaries the two phases are treated as interpenetrating continua within a two fluid model in contrast to the conventional two fluid model formulations in which the governing equations are written in the eulerian eulerian description the modified two fluid model has the governing equations of one fluid phase written in the lagrangian description while the respective equations of the other phase is formulated in the arbitrary lagrangian eulerian ale description with relative velocities defined with respect to the moving mesh associated with the first fluid phase the finite element method fem together with stabilization techniques the characteristic based split cbs and the algebraic flux correction afc method is employed to solve the equations numerically in space and time the nodes of the computational mesh are updated together with the flow of the phase described in the lagrangian description the computational mesh is updated by applying techniques used in the so called particle finite element method pfem characterized by regenerating the mesh in each computational step according to the updated nodal positions while the current boundary of the domain is identified using shape method the performance of the method is demonstrated by means of three 2d and 3d examples including a mixing process of two fluids in a stirred tank a free surface flow a 3d two phase channel flow a phase separation problem and the flow of two immiscible fluids

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

the group finite element formulation abstract the introduction of trial solutions for groups of variables within a finite element formulation is shown to provide a significant economy with no loss of accuracy for steady solutions of the two dimensional burgers equations with linear rectangular elements the application of the group finite element to the convective term produces a threefold reduction in execution time with a small improvement in accuracy operation count estimates indicate even larger economies in three dimensions or if higher order nonlinearities occur or if higher order shape functions are used applications of the group formulation are described for inviscid and viscous laminar and turbulent separated and unseparated and incompressible and compressible flows

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

an approximate decomposition algorithm for convex minimization for nonsmooth convex optimization robert mifflin and claudia sagastizabal introduce a vu space decomposition algorithm in mifflin and sagastizabal 2005 11 an attractive property of this algorithm is that if a primal dual track exists this algorithm uses a bundle subroutine with the inclusion of a simple line search it is proved to be globally and superlinearly convergent however a drawback is that it needs the exact subgradients of the objective function which is expensive to compute in this paper an approximate decomposition algorithm based on proximal bundle type method is introduced that is capable to deal with approximate subgradients it is shown that the sequence of iterates generated by the resulting algorithm converges to the optimal solutions of the problem numerical tests emphasize the theoretical findings

some results about vector extrapolation methods and related fixed point iterations abstract in this work we consider polynomial extrapolation methods for vector sequences we give a unified presentation of these methods and a new recursive algorithm for the implementation of the modified minimal polynomial extrapolation mmpe method properties operation counts and storage requirements of the algorithm will be given the second part of the work is devoted to the application of these methods for solving systems of nonlinear equations a sufficient condition for quadratic convergence is given

svd based initialization a head start for nonnegative matrix factorization we describe nonnegative double singular value decomposition nndsvd a new method designed to enhance the initialization stage of nonnegative matrix factorization nmf nndsvd can readily be combined with existing nmf algorithms the basic algorithm contains no randomization and is based on two svd processes one approximating the data matrix the other approximating positive sections of the resulting partial svd factors utilizing an algebraic property of unit rank matrices simple practical variants for nmf with dense factors are described nndsvd is also well suited to initialize nmf algorithms with sparse factors many numerical examples suggest that nndsvd leads to rapid reduction of the approximation error of many nmf algorithms

explicit and implicit fem fct algorithms with flux linearization a new approach to the design of flux corrected transport fct algorithms for continuous linear multilinear finite element approximations of convection dominated transport problems is pursued the algebraic flux correction paradigm is revisited and a family of nonlinear high resolution schemes based on zalesak s fully multidimensional flux limiter is considered in order to reduce the cost of flux correction the raw antidiffusive fluxes are linearized about an auxiliary solution computed by a high or low order scheme by virtue of this linearization the costly computation of solution dependent correction factors is to be performed just once per time step and there is no need for iterative defect correction if the governing equation is linear a predictor corrector algorithm is proposed as an alternative to the hybridization of high and low order fluxes three fem fct schemes based on the runge kutta crank nicolson and backward euler time stepping are introduced a detailed comparative study is performed for linear convection diffusion equations

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

active set newton methods and partial smoothness diverse optimization algorithms correctly identify in finite time intrinsic constraints that must be active at optimality analogous behavior extends beyond optimization to systems involving part

towards efficient runge kutta methods for stiff systems a special class of implicit runge kutta methods is developed for the numerical solution of stiff initial value problems these formulae are derived from known singly implicit methods by adding one or more extra diagonally implicit stages it is hoped that this modification of the original method will lead to an overall gain in efficiency and an analysis of the advantages of making this enhancement is presented

the norsett time integration methodology for finite element transient analysis this paper presents a set of methods for time integration of problems arising from finite element semidiscretizations the purpose is to obtain computationally efficient methods which possess higher order accuracy and controllable dissipation in the spurious high modes the methods are developed and analysed by a general collocation methodology which leads to the class of norsett approximants an algorithmic parameter is used to achieve an effective control over numerical dissipation moreover a simple and efficient implementation scheme is presented at each time step algorithms based on p order collocation polynomials require the solution of p sets of linear algebraic equations with the same coefficient matrix in this way a single factorization is needed and no transformations are required to recover the approximate solution at the end or within the time interval to demonstrate the performance of the proposed algorithms a wide experimental evaluation is carried out on typical test problems in finite element transient analysis

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

an inequality constrained nonlinear kalman bucy smoother by interior point likelihood maximization kalman bucy smoothers are often used to estimate the state variables as a function of time in a system with stochastic dynamics and measurement noise this is accomplished using an algorithm for which the number of numerical operations grows linearly with the number of time points all of the randomness in the model is assumed to be gaussian including other available information for example a bound on one of the state variables is non trivial because it does not fit into the standard kalman bucy smoother algorithm in this paper we present an interior point method that maximizes the likelihood with respect to the sequence of state vectors satisfying inequality constraints the method obtains the same decomposition that is normally obtained for the unconstrained kalman bucy smoother hence the resulting number of operations grows linearly with the number of time points we present two algorithms the first is for the affine case and the second is for the nonlinear case neither algorithm requires the optimization to start at a feasible sequence of state vector values both the unconstrained affine and unconstrained nonlinear kalman bucy smoother are special cases of the class of problems that can be handled by these algorithms

optimality identifiability and sensitivity around a solution of an optimization problem an identifiable subset of the feasible region is one containing all nearby solutions after small perturbations to the problem a quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are minimal this new notion lays a broad and intuitive variational analytic foundation for optimality conditions sensitivity and active set methods

discontinuous finite volume element discretization for coupled flow transport problems arising in models of sedimentation the sedimentation consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the stokes problem written in terms of the mixture flow velocity and the pressure field here both the viscosity and the forcing term depend on the local solids concentration a semi discrete discontinuous finite volume element dfve scheme is proposed for this model the numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field whereas the pressure is approximated by piecewise constant elements the unique solvability of both the nonlinear continuous problem and the semi discrete dfve scheme is discussed and optimal convergence estimates in several spatial norms are derived properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples including flows under gravity with changing direction a secondary settling tank in an axisymmetric setting and batch sedimentation in a tilted cylindrical vessel

generic minimizing behavior in semialgebraic optimization we present a theorem of sard type for semialgebraic set valued mappings whose graphs have dimension no larger than that of their range space the inverse of such a mapping admits a single valued analytic localization around any pair in the graph for a generic value parameter this simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems typical semialgebraic problems have finitely many critical points around each of which they admit a unique active manifold analogue of an active set in nonlinear optimization moreover such critical points satisfy strict complementarity and second order sufficient conditions for optimality are indeed necessary

a flux corrected transport algorithm for handling the close packing limit in dense suspensions convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level in this paper we are concerned with solving continuity equations that govern the evolution of volume fractions in eulerian models of disperse two phase flows an implicit galerkin finite element approximation is equipped with a flux limiter for the convective terms the fully multidimensional limiting strategy is based on a flux corrected transport fct algorithm this nonlinear high resolution scheme satisfies a discrete maximum principle for divergence free velocities and ensures positivity preservation for arbitrary velocity fields to enforce an upper bound that corresponds to the maximum packing limit an fct like overshoot limiter is applied to the converged convective fluxes at the end of each time step this postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model numerical results for 2d implosion problems illustrate the performance of the proposed limiting procedure

flux corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements abstract the flux corrected transport fct algorithm is applied to the unsteady and steady state particle transport equation the proposed fct method employs the following 1 a low order positivity preserving scheme based on the application of m matrix properties 2 a high order scheme based on the entropy viscosity method introduced by guermond 1 and 3 local discrete solution bounds derived from the integral transport equation the resulting scheme is second order accurate in space enforces an entropy inequality mitigates the formation of spurious oscillations and guarantees the absence of negativities space discretization is achieved using continuous finite elements time discretizations for unsteady problems include theta schemes such as explicit and implicit euler and strong stability preserving runge kutta ssprk methods the developed fct scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady state and implicit time discretizations

generic optimality conditions for semialgebraic convex programs we consider linear optimization over a nonempty convex semialgebraic feasible region f semidefinite programming is an example if f is compact then for almost every linear objective there is a unique optimal solution lying on a unique active manifold around which f is partly smooth and the second order sufficient conditions hold perturbing the objective results in smooth variation of the optimal solution the active manifold consists locally of these perturbed optimal solutions it is independent of the representation of f and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes these results extend to unbounded sets f

efficient runge kutta integrators for index 2 differential algebraic equations in seeking suitable runge kutta methods for differential algebraic equations we consider singly implicit methods to which are appended diagonally implicit stages methods of this type are either similar to those of butcher and cash or else allow for the importation of a final derivative from a previous step for these two classes with up to three additional diagonally implicit stages we derive methods that satisfy appropriate order conditions for index 2 daes

vector extrapolation methods applications and numerical comparison the present paper is a survey of the most popular vector extrapolation methods such as the reduced rank extrapolation rre the minimal polynomial extrapolation mpe the modified minimal polynomial extrapolation mmpe the vector e algorithm vea and the topological e algorithm tea using projectors we derive a different interpretation of these methods and give some theoretical results the second aim of this work is to give a numerical comparison of the vector extrapolation methods above when they are used for practical large problems such as linear and nonlinear systems of equations

formulation analysis and numerical study of an optimization based conservative interpolation remap of scalar fields for arbitrary lagrangian eulerian methods we develop and study the high order conservative and monotone optimization based remap obr of a scalar conserved quantity mass between two close meshes with the same connectivity the key idea is to phrase remap as a global inequality constrained optimization problem for mass fluxes between neighboring cells the objective is to minimize the discrepancy between these fluxes and the given high order target mass fluxes subject to constraints that enforce physically motivated bounds on the associated primitive variable density in so doing we separate accuracy considerations handled by the objective functional from the enforcement of physical bounds handled by the constraints the resulting obr formulation is applicable to general unstructured heterogeneous grids under some weak requirements on grid proximity but not on the cell types we prove that the obr algorithm is linearity preserving in one two and three dimensions the paper also examines connections between the obr and the recently proposed flux corrected remap fcr liska et al 1 we show that the fcr solution coincides with the solution of a modified version of obr m obr which has the same objective but a simpler set of box constraints derived by using a worst case scenario because m obr fcr has a smaller feasible set preservation of linearity may be lost and accuracy may suffer for some grid configurations our numerical studies confirm this and show that obr delivers significant increases in robustness and accuracy preliminary efficiency studies of obr reveal that it is only a factor of 2 1 slower than fcr but admits 1 5 times larger time steps

model consistency of partly smooth regularizers this paper studies least square regression penalized with partly smooth convex regularizers this class of penalty functions is very large and versatile and allows to promote solutions conforming to some notion of low complexity indeed such penalties regularizers force the corresponding solutions to belong to a low dimensional manifold the so called model which remains stable when the argument of the penalty function undergoes small perturbations such a good sensitivity property is crucial to make the underlying low complexity manifold model robust to small noise in a deterministic setting we show that a generalized irrepresentable condition implies stable model selection under small noise perturbations in the observations and the design matrix when the regularization parameter is tuned proportionally to the noise level as an algorithmic implication we also prove that this condition is almost necessary for stable model recovery we then turn to the random setting where the design matrix and the noise are random and the number of observations grows large we show that under our generalized irrepresentable condition and a proper scaling of the regularization parameter the regularized estimator is model consistent in plain words with a probability tending to one as the number of measurements tends to infinity the regularized estimator belongs to the correct low dimensional model manifold this paper unifies and generalizes a large body of literature where model consistency was known to hold for instance for the lasso group lasso total variation fused lasso and nuclear trace norm regularizers as an algorithmic implication we show that under the deterministic model selection conditions the forward backward proximal splitting algorithm used to solve the penalized least square regression problem is guaranteed to identify the model manifold after a finite number of iterations finally we detail how our results extend from the quadratic loss to an arbitrary smooth and strictly convex loss function we illustrate the usefulness of our results on the problem of low rank matrix recovery from random measurements using nuclear norm minimization

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

convergence theorems for least change secant update methods the purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means the theorems and proofs given here can be viewed as generalizations of those given by broyden dennis more j inst math appl 12 1973 pp 223 246 and by dennis more math comp 28 1974 pp 549 560 the analysis is done in the orthogonal projection setting of dennis schnabel siam rev 21 1980 pp 443 459 and many readers might feel that it is easier to understand the theorems here readily imply local and q superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of marwil and toint and the nonlinear least squares method of dennis gay welsch

limiting and divergence cleaning for continuous finite element discretizations of the mhd equations abstract this work introduces a new type of constrained algebraic stabilization for continuous piecewise linear finite element approximations to the equations of ideal magnetohydrodynamics mhd at the first step of the proposed flux corrected transport fct algorithm the galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed at the second step limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field the limiting procedure developed for this stage is designed to enforce local maximum principles as well as positivity preservation for the density and thermodynamic pressure additionally it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints numerical studies for 2d test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal mhd benchmarks

manifold identification in dual averaging for regularized stochastic online learning iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets when the objective consists of a loss function plus a nonsmooth regularization term the solution often lies on a low dimensional manifold of parameter space along which the regularizer is smooth when an l1 regularizer is used to induce sparsity in the solution for example this manifold is defined by the set of nonzero components of the parameter vector this paper shows that a regularized dual averaging algorithm can identify this manifold with high probability before reaching the solution this observation motivates an algorithmic strategy in which once an iterate is suspected of lying on an optimal or near optimal manifold we switch to a local phase that searches in this manifold thus converging rapidly to a near optimal point computational results are presented to verify the identification property and to illustrate the effectiveness of this approach

acceleration of the schwarz method for elliptic problems in this paper we present a family of domain decomposition methods based on aitken like acceleration of the schwarz method which is an iterative procedure with linear rate of convergence this paper is a generalization of our method first introduced in 2000 that was restricted to cartesian grids we consider the finite volume fv approximation on general quadrangle meshes introduced by faille in 1992 we first present the so called aitken schwarz procedure for a linear differential operator this solver is a direct solver but its computational cost in the general case might be prohibitive second we introduce the steffensen schwarz variant which is an iterative domain decomposition solver that can be applied to linear and nonlinear problems we show that this iterative solver has reasonable numerical efficiency and can be applied successfully to several classes of linear and nonlinear elliptic problems however the salient feature of our method is that our algorithm has high tolerance for slow networks in the context of distributed parallel computing and is attractive to use with computer architectures for which performance is limited by the memory bandwidth rather than the flop performance of the cpu this is currently the case for most parallel computers using risc processor architecture such as beowulf systems

accelerating with rank one updates abstract consider the iteration xk 1 xk h b axk for solving ax b a is n x n nonsingular we discuss rank one updates to improve h as an approximation to a 1 during the iteration the update kills and reduces singular values of i ah and thus speeds up the convergence the algorithm terminates after at most n sweeps and if all n sweeps are needed then a 1 has been computed

analysis of some vector extrapolation methods for solving systems of linear equations in this paper we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations when applied to sequences generated linearly these methods namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre are krylov subspaces methods and are respectively equivalent to the method of arnoldi and to the gcr and gmres by considering the geometrical aspect of these methods we derive new expressions for their residual norms and give a relationship between them this allows us to compare the two methods using this new approach we will show that for nonsingular skew symmetric matrices the gmres stagnates every two iterations and the restarted version gmres m m ge 2 is always convergent finally the incomplete forms are considered and some convergence results are given

on partial smoothness tilt stability and the mathcal vu decomposition under the assumption of prox regularity and the presence of a tilt stable local minimum we are able to show that a mathcal vu like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function