material state remapping in computational solid mechanics a computational procedure for remapping material state information from one finite element mesh to another is described the procedure is useful in connection with evolving meshes for inelastic problems as for example occur in the context of fracture simulation and adaptive mesh refinement the proposed method is based on weak enforcement of equality between corresponding fields on the two meshes where piecewise constant fields on both meshes are generalized from the quadrature point values the essential algorithmic problem is that of calculating the volume partition of an arbitrary convex region with respect to a covering set of disjoint convex regions instead of geometrically resolving the associated intersections the problem is herein approximated by a constrained optimization problem which may be readily and efficiently solved computationally this formulation is a main contribution of the paper computational examples are given that illustrate the effectiveness of the proposed procedure copyright 2002 john wiley sons ltd

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

building resolving large eddy simulations and comparison with wind tunnel experiments we perform large eddy simulations les of the flow past a scale model of a complex building calculations are accomplished using two different methods to represent the edifice the first method employs the standard gal chen and somerville terrain following coordinate transformation common in mesoscale atmospheric simulations the second method uses an immersed boundary approach in which fictitious body forces in the equations of motion are used to represent the building by attenuating the flow to stagnation within a time comparable to the time step of the model both methods are implemented in the same hydrodynamical code eulag using the same nonoscillatory forward in time nft incompressible flow solver based on the multidimensional positive definite advection transport algorithms mpdata the two solution methods are compared to wind tunnel data collected for neutral stratification profiles of the first and second order moments at various locations around the model building show good agreement with the wind tunnel data although both methods appear to be viable tools for les of urban flows the immersed boundary approach is computationally more efficient the results of these simulations demonstrate that contrary to popular opinion continuous mappings such as the gal chen and somerville transformation are not inherently limited to gentle slopes calculations for a strongly stratified case are also presented to point out the substantial differences from the neutral boundary layer flows

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

accelerated gcro dr method for solving sequences of systems of linear equations by recycling some error approximations generated during a previous cycle of iterations we present a technique for improving the convergence of gcro dr the scheme is able to mitigate the occurrence of small skip angles occurring in gcro dr and thus considerably accelerates the convergence the alternative phenomenon of the residual vectors is examined numerically and the effectiveness of the new method is illustrated by several examples from practical applications

a rational function preconditioner for indefinite sparse linear systems this paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices a by resorting to rational functions of a the algorithm decomposes the spectrum of a into two disjoint regions and approximates the restriction of a 1 on these regions separately we show a systematic way to construct these rational functions so that they can be applied stably and inexpensively an attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism moreover the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form a ci which can be useful in some applications the efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems including problems arising from the helmholtz equation in three dimensions

a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

growth and nonvanishing of restricted siegel modular forms arising as saito kurokawa lifts we study the analytic behavior of the restriction of a siegel modular form to bbb h times bbb h in the case that the siegel form is a saito kurokawa lift a formula of ichino links this behavior to a family of rm gl 3 times rm gl 2 l functions

generalized arnoldi tikhonov method for sparse reconstruction this paper introduces two new algorithms belonging to the class of arnoldi tikhonov regularization methods which are particularly appropriate for sparse reconstruction the main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2 norm regularization term to approximate a more general regularization term expressed in the p norm p geq 1 the regularization matrix can be updated both at each step and after some iterations have been performed leading to two different approaches the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible krylov subspaces the second one is based on restarting the arnoldi algorithm numerical examples are given in order to show the effectiveness of these new methods and comparisons with some other already existing algorithms are made

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

on conservation laws of navier stokes galerkin discretizations abstract we study conservation properties of galerkin methods for the incompressible navier stokes equations without the divergence constraint strongly enforced in typical discretizations such as the mixed finite element method the conservation of mass is enforced only weakly and this leads to discrete solutions which may not conserve energy momentum angular momentum helicity or vorticity even though the physics of the navier stokes equations dictate that they should we aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint and doing so leads us to a new formulation that conserves each of energy momentum angular momentum enstrophy in 2d helicity and vorticity for reference the usual convective formulation does not conserve most of these quantities several numerical experiments are performed which verify the theory and test the new formulation

compatible fluxes for van leer advection application of the standard second order upstream centered advection method of van leer to a coupled transport equation system of the form t a a u 0 can produce artificial extrema in the ratiot a even thoughtsatisfies a simple advection equation and is expected to preserve monotonicity hereuis velocity is mass density andais a conserved quantity such as momentum density energy density or chemical species density thustis a mass specific transport quantity such as velocity energy per unit mass temperature or species mass fraction a new flux formulation and gradient limiting procedure is presented here which eliminates these artificial extrema preserves the second order accuracy and preserves the monotone character of the van leer method even in the limit of vanishing mass density such a formulation is calledcompatible the method is noniterative i e explicit and can be employed in a general finite volume framework sample results for the transport of a square wave in one and two dimensions are provided

multigrid methods for convection diffusion problems discretized by a monotone scheme abstract we study multigrid mg methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection diffusion problems by an exponential fitting scheme the latter ensures the stability of the simplest possible coarse grid operators obtained from galerkin projections based on graph matching linear and nonlinear mg preconditioners are defined in the framework of algebraic multilevel iteration the option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two and three dimensional test problems

on central l derivative values of automorphic forms we study the average and nonvanishing of the central l derivative values of l s f and l s f k scriptscriptstyle d for f in an orthogonal hecke eigenbasis mathcal h 2k of weight 2k cusp forms of level 1 for large odd k here f k scriptscriptstyle d is the base change of f to an imaginary quadratic field k scriptscriptstyle d mathbb q sqrt d with fundamental discriminant d we prove asymptotic formulas for the first and second moments of l frac 1 2 f as well as the first moment of l frac 1 2 f k scriptscriptstyle d over mathcal h 2k as odd k rightarrow infty further we employ mollifiers to establish that for sufficiently large k there are positive proportion of hecke eigenforms f in mathcal h 2k with l frac 1 2 f ne 0 we also give applications of our results to heegner cycles of high weights of the modular curve x 0 1

multipreconditioned gmres for shifted systems an implementation of gmres with multiple preconditioners is proposed for solving shifted linear systems with shift and invert preconditioners with this type of preconditioner the krylov subspace can be built without requiring the matrix vector product with the shifted matrix furthermore the multipreconditioned search space is shown to grow only linearly with the number of preconditioners this allows for a more efficient implementation of the algorithm the proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions the numerical results indicate the effectiveness of the proposed approach

implementation of the jacobian free newton krylov method for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the first order ice sheet momentum equation in order to improve the numerical performance of the glimmer community ice sheet model glimmer cism the land ice component of the community earth system model cesm our jfnk implementation is based on significant re use of existing code for example our physics based preconditioner uses the original picard linear solver in glimmer cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 8 3 6 times more efficient than the standard picard solver in glimmer cism importantly this computational gain of jfnk over the picard solver increases when refining the grid global convergence of the jfnk solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

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

adaptive moment of fluid method a novel adaptive mesh refinement amr strategy based on the moment of fluid mof method for volume tracking of evolving interfaces is presented moment of fluid method is a new interface reconstruction and volume advection method using volume fractions as well as material centroids the mesh refinement criterion is based on the deviation of the actual centroid obtained by interface reconstruction from the reference centroid given by moment advection process the centroid error indicator detects not only high curvature regions but also regions with complicated subcell structures like filaments a new lagrange remap scheme is presented for advecting moments which includes lagrangian backtracking polygon intersection based remapping and forward tracking to define the material centroid the effectiveness and efficiency of amr mof method is demonstrated with classical test problems such as zalesak s disk and reversible vortex problem the comparison with previously published results for these problems shows the superior accuracy of the amr mof method over other methods in addition two new test cases with severe deformation rates are introduced namely droplet deformation and s shape deformation problems for further demonstration of the capabilities of the amr mof method

development of a 2 d algorithm to simulate convection and phase transition efficiently we develop a jacobian free newton krylov jfnk method for the solution of a two dimensional convection phase change model using the incompressible navier stokes equation set and enthalpy as the energy conservation variable the simple algorithm acts as a physics based preconditioner to jfnk this combined algorithm is compared to solutions using simple as the main solver algorithm performance is assessed for two benchmark problems of phase change convection of a pure material one melting and one freezing the jfnk simple method is shown to be more efficient per time step and more robust at larger time steps overall cpu savings of more than an order of magnitude are realized

two sided bounds for nonuniform rates of convergence in the central limit theorem two sided bounds are derived for nonuniform rates of convergence in the central limit theorem and are applied to solve several problems on rates of convergence in nonuniform metrics in particular an expression for the fastest rate of convergence is obtained and rates of convergence using different norming constants are compared a useful estimate of the rate of convergence in the 1 x 2 metric is derived which does not require the assumption of 2 th order moments

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

the repair paradigm new algorithms and applications to compressible flow the repair paradigm leads to several algorithms for redistributing mass momentum and energy while adhering to local maximum principles as an adjunct to the remapping step in certain compressible flow codes that use remapping such as arbitrary lagrangian eulerian codes or for just redistributing mass in advection codes in the case of advection of a concentration repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells thus guaranteeing at least that the new concentration is between zero and one for compressible flow density velocity and internal energy are similarly constrained while maintaining conservation of mass momentum and total energy in this way positive density and internal energy are achieved as a side effect we propose a new algorithm combining both local and global repair that maintains causality and is efficient in a parallel computational setting the local global algorithm is independent of the order in which the distribution is performed and it maintains 1d symmetry this is applied to advection in two dimensions and to among others the leblanc problem the sedov problem and an interacting 2d blast wave problem the latter is done with a lagrangian code for which rezoning remapping and repair are essential

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

edgeworth expansion for u statistics under minimal conditions berry esseen bounds for u statistics under the optimal moment conditions were derived by koroljuk and borovskich and friedrich under the same optimal moment assumptions with an additional nonlattice condition we establish a one term edgeworth expansion with remainder o n 1 2 for u statistics

compatible energy conserving bounds preserving remap of hydrodynamic fields for an extended ale scheme abstract from the very origins of numerical hydrodynamics in the lagrangian work of von neumann and richtmyer 83 the issue of total energy conservation as well as entropy production has been problematic because of well known problems with mesh deformation lagrangian schemes have evolved into arbitrary lagrangian eulerian ale methods 39 that combine the best properties of lagrangian and eulerian methods energy issues have persisted for this class of methods we believe that fundamental issues of energy conservation and entropy production in ale require further examination the context of the paper is an ale scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity the principal design goals for a remap method then consist of total energy conservation bounded internal energy and compatibility of kinetic energy and momentum we also have secondary objectives of limiting velocity and stress in a non directional manner keeping primitive variables monotone and providing a higher than second order reconstruction of remapped variables in particular the new contributions fall into three categories associated with energy conservation and entropy production reconstruction and bounds preservation of scalar and tensor fields and conservative remap of nonlinear fields the paper presents a derivation of the methods details of implementation and numerical results for a number of test problems the methods requires volume integration of polynomial functions in polytopal cells with planar facets and the requisite expressions are derived for arbitrary order

constrained interpolation remap of divergence free fields abstract a novel constrained interpolation algorithm for remapping of solenoidal face finite element vector fields is presented the algorithm is based on explicit recovery postprocessing and interpolation of a potential for the original vector field and a subsequent application of a curl operator to obtain the desired divergence free finite element field on the new mesh the use of interpolation instead of advection in the remap process offers valuable computational advantages old and new meshes are neither required to have the same connectivity nor to be close to each other slope limiting and upwinding which can be sensitive to grid structure are avoided and replaced by local optimization to control energy of the remapped field the new method is validated using a suite of cyclic remap problems on random and tensor product mesh sequences a comparison with a local remapper based on a constrained transport advection algorithm is also included

asymptotically optimal multistage tests of simple hypotheses a family of variable stage size multistage tests of simple hypotheses is described based on efficient multistage sampling procedures using a loss function that is a linear combination of sampling costs and error probabilities these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero a numerical study shows significant improvement over group sequential tests in a binomial testing problem

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

a multistep flux corrected transport scheme a multistep flux corrected transport mfct scheme is developed to achieve conservative and monotonic tracer transports for multistep dynamical cores mfct extends zalesak two time level scheme to any multistep time differencing schemes by including multiple high order fluxes in the antidiffusive flux while computing the two time level low order monotone solution the multistep time differencing scheme used in this study is the third order adams bashforth ab3 scheme implemented in a finite volume icosahedral shallow water model the accuracy of ab3 mfct is quantified by the shape preserving advection experiments in non divergent flow as well as a cosine bell whose shape changes during advection in shear flow ab3 mfct has been shown to be insensitive to time step size this make ab3 mfct an attractive transport scheme for explicit high resolution model applications with small time step mfct is tested in shallow water model simulations to demonstrate that the use of mfct maintains positive definite tracer transport while at the same time conserving both fluid mass and tracer mass within round off errors in the ab3 dynamic core

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

pressure corrected implicit θ schemes for the incompressible navier stokes equations abstract in this note the crank nicolson and the fractional step scheme are applied on the semi discretised incompressible navier stokes equations in a first step the formulation of the methods is modified such that the methods have optimal order for the pressure p i e the pressure p 0 is included in the scheme the main idea of this paper is that the fractional step scheme can be written as a diagonally implicit runge kutta method dirk method in this context it can easily be shown that the fractional step schemes have only stage order q 1 whereas the crank nicolson scheme has stage order q 2 hence the fractional step scheme may have order reduction if the method is applied on stiff odes and daes i e the semi discretised incompressible navier stokes equations some theoretical results and numerical examples illustrate this phenomena moreover it is shown that it is impossible to improve the fractional step method such that the scheme has the stage order q 2 and is strongly a stable or has the order p 3

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

reprint of atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

nonvanishing of central l values of maass forms abstract with the method of moments and the mollification method we study the central l values of gl 2 maass forms of weight 0 and level 1 and establish a positive proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals which is qualitatively optimal in view of weyl s law as an application of this result and a formula of katok sarnak we give a nonvanishing result on the first fourier coefficients of maass forms of weight 1 2 and level 4 in the kohnen plus space

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

tracer transport within an unstructured grid ocean model using characteristic discontinuous galerkin advection abstract in a previous article a characteristic discontinuous galerkin cdg advection scheme was presented for tracer transport lee et al 2016 the scheme is conservative unconditionally stable with respect to time step and scales sub linearly with the number of tracers being advected here we present the implementation of the cdg advection scheme for tracer transport within mpas ocean a boussinesque unstructured grid ocean model with an arbitrary lagrangian eulerian vertical coordinate the scheme is implemented in both the vertical and horizontal dimensions and special care is taken to ensure that the scheme remains conservative in the context of moving vertical layers consistency is ensured with respect to the dynamics by a renormalization of the fluxes with respect to the volume fluxes derived from the continuity equation for spherical implementations the intersection of the flux swept regions and the eulerian grid are determined for great circle arcs and the fluxes and element assembly are performed on the plane via a length preserving projection solutions are presented for a suite of test cases and comparisons made to the existing flux corrected transport scheme in mpas ocean

un critere sur les petites boules dans le theoreme limite central we show that a banach space valued random variablex such that mathop lim limits x to infty t 2 mathbb p left left x right t right 0 satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled mathop lim limits x to infty t 2 mathbb p left left x right t right 0

a forward trajectory global semi lagrangian transport scheme a forward trajectory semi lagrangian scheme for advection on the surface of the sphere is proposed the advection scheme utilizes the forward downstream trajectory originating at eulerian grid points and cascade interpolation a sequence of 1d interpolations to transfer data from the downstream lagrangian points to the eulerian points a new and more accurate algorithm determines pole values the resulting forward trajectory semi lagrangian scheme can easily incorporate high order trajectory integration methods this avoids the standard iterative process in a typical backward trajectory scheme two third order accurate schemes and a second order accurate scheme are presented a mass conservative version of the forward trajectory semi lagrangian scheme is also derived within the cascade interpolation framework mass from a lagrangian cell is transferred to the corresponding eulerian cell with two 1d remappings through an intermediate cell system mass in the polar region is redistributed by way of an efficient local approximation the resulting scheme is globally conservative but restricted to meridional courant number c 1

implementation of the jacobian free newton krylov method for solving the for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the rst order ice sheet momentum equation in order to improve the numerical performance of the community ice sheet model cism the land ice component of the community earth system model cesm our jfnk implementation is based on signicant re use of existing code for example our physics based preconditioner uses the original picard linear solver in cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 84 3 62 times more efficient than the standard picard solver in cism importantly this computational gain of jfnk over the picard solver increases when rening the grid global convergence of the jfnk solver has been signicantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

a newton krylov solver for remapping based volume of fluid methods a new conservative remapping method is presented here for generating multi dimensional fluxes used for the time integration of hyperbolic equations through single stage unsplit advection a solution to the lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one dimensional flux definitions at their corresponding cell faces the method is most feasible within localized subdomains of an eulerian solution domain the main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass in generating solutions to lagrangian mesh location by solving a nonlinear equation system the new remap requires the anchoring of the lagrangian mesh removal of degrees of freedom to make the problem definite the method is based on characteristic tracing along trajectories assumed to be linear this paper investigates the convergence and sensitivity of the lagrangian mesh solver to mesh and timestep size in assessing its merit for application in transient cfd codes for single stage unsplit advection volume of fluid methods the new remap suppresses the introduction of undershoots overshoots and wisps in combination with a conservative remapping phase the new method for fixing the location of the lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal

global fom and gmres algorithms for a class of complex matrix equations abstract in this paper a generalized global arnoldi process is given to produce an f orthonormal basis of complex matrix space then we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving a class of complex matrix equations the new methods can be implemented by the original coefficient matrices by using schur complement we give the expressions of the approximate solutions and the corresponding residuals some convergence results of the proposed methods are also derived finally numerical examples are given to illustrate the effectiveness with comparison to other existing methods

on simplifying incremental remap based transport schemes the flux form incremental remapping transport scheme introduced by dukowicz and baumgardner 1 converts the transport problem into a remapping problem this involves identifying overlap areas between quadrilateral flux areas and regular square grid cells which is non trivial and leads to some algorithm complexity in the simpler swept area approach originally introduced by hirt et al 2 the search for overlap areas is eliminated even if the flux areas overlap several regular grid cells the resulting simplified scheme leads to a much simpler and robust algorithm we show that for sufficiently small courant numbers approximately cfl 1 2 the simplified or swept area scheme can be more accurate than the original incremental remapping scheme this is demonstrated through a von neumann stability analysis an error analysis and in idealized transport test cases on the sphere using the incremental remapping based scheme called ff cslam flux form version of the conservative semi lagrangian multi tracer scheme on the cubed sphere

efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method abstract radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes for example in fluid structure interaction or design optimization problems this is known to be a robust methodology that results in high quality deformed meshes the applicability of this method to large problems is currently hampered by its prohibitive computational cost however which is caused by the need to solve a dense system of equations the computation time grows as o n b 3 if a direct solver is employed for the solution where n b denotes the number of boundary nodes in the mesh while alternative iterative solvers often suffer from an unfavorable convergence behavior in this paper we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as o n b the linear complexity is achieved by transforming the dense system into an extended sparse system along with the compression of certain matrix blocks into low rank factorizations the solver is inexact although the error can be controlled and made as small as needed a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme numerical benchmarks are presented demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly without jeopardizing their robustness and quality

exact convergence rate and leading term in central limit theorem for student s t statistic the leading term in the normal approximation to the distribution of student s t statistic is derived in a general setting with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law the form of the leading term is shown to have its origin in the way in which extreme data influence properties of the studentized sum the leading term approximation is used to give the exact rate of convergence in the central limit theorem up to order n 1 2 where n denotes sample size it is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points moreover the exact rate is identical to that for the non studentized sum when the latter is normalized for scale using a truncated form of variance but when the corresponding truncated centering constant is omitted examples of characterizations of convergence rates are also given it is shown that in some instances their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points

second order accurate volume tracking based on remapping for triangular meshes this paper presents a second order accurate piecewise linear volume tracking based on remapping for triangular meshes this approach avoids the complexity of extending unsplit second order volume of fluid algorithms advection methods on triangular meshes the method is based on lagrangian eulerian le methods therefore it does not deal with edge fluxes and corner fluxes flux corrections as is typical in advection algorithms the method is constructed of three parts a lagrangian phase a reconstruction phase and a remapping phase in the lagrangian phase the original eulerian grid is projected along trajectories to obtain lagrangian grids in practice this projection is handled through the time integration of velocity field for grid vertices at each time step the reconstruction is based on truncating the volume material polygon for each lagrangian mixed grid since in piecewise linear approximation the interface is represented by a segment line the polygon material truncation is mainly finding the segment interface finding the segment interface is calculating the line normal and line constant at each multi fluid cell details of applying two normal calculation methods differential and geometric least squares gls methods are given while the gls method exhibits second order accurate approximation in reproducing circular interfaces the differential least squares dls method results in a first order accurate representation of the interface the last part of the algorithm which is remapping of the volume materials from the lagrangian grid to the original one is performed by a series of polygon intersection procedures the behavior of the algorithm is investigated for flow fields with constant interface topology and flow fields inducing large interfacial stretching and tearing second order accuracy is obtained if the velocity integration as well as the reconstruction steps are at least second order accurate

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

on the asymptotic equivalence of l p metrics for convergence to normality this paper is concerned with the rate of convergence to zero of the l p metrics np l p constructed out of differences between distribution functions for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance it is shown that the np are under broad conditions asymptotically equivalent in the strong sense that for l p p np np is universally bounded away from zero and infinity as n

the repair paradigm and application to conservation laws repair is a conservative post processing procedure to be used in numerical methods for hyperbolic conservation laws in order to preserve certain qualitative characteristics of the numerical solution such as positivity of density and internal energy by means of redistribution of conserved quantities such as mass momentum and total energy among the cells of the mesh in this paper we describe the repair paradigm and prove several theorems which form a theoretical foundation for the repair procedures we consider two applications of repair and present corresponding numerical results the first application deals with improving properties of the remapping conservative interpolation stage of arbitrary lagrangian eulerian ale methods for the gas dynamics equations in which the solution is conservatively transferred from one mesh to another one requirement for remapping is that the interpolated density and internal energy on the new mesh have to stay positive another desirable property is that the remapping procedure should not create new extrema for the velocity field for various reasons these properties may not be satisfied especially for high order methods repair plays a supplemental role by bringing gas dynamics quantities into physically justified bounds another application of repair is to improve the quality of numerical methods for advection of some scalar tracer field with prescribed divergence free velocity field in which case the advection equation can be written as a conservation law and therefore the total amount of tracer is conserved we show how the repair procedure allows us to reduce oscillations in a numerical solution obtained by a formally high order method repair offers an alternative to more classical methods of reducing oscillations and maintaining positivity

a semi lagrangian discontinuous galerkin method for scalar advection by incompressible flows a new conservative semi lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form the approach combines the accuracy and conservation properties of the discontinuous galerkin dg method with the computational efficiency and robustness of semi lagrangian sl techniques unconditional stability in the von neumann sense is proved for the proposed discretization in the one dimensional case a monotonization technique is then introduced based on the flux corrected transport approach this yields a multi dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard dg methods the accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests in the case of incompressible flows the comparison with results obtained by standard sl and dg methods highlights several advantages of the new technique

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

solving slp recourse problems with arbitrary multivariate distributions the dependent case bounding the expectation of a convex function of a multivariate random variable is of great importance for solving stochastic linear programming problems slp with recourse the classic bounds are those of jensen and edmundson madansky whereby jensen s inequality is available in the independent as well as in the dependent case in contrast with the edmundson madansky e m inequality madansky a 1959 bounds on the expectation of a convex function of multivariate random variable ann math statist 30 743 746 which is valid only in the independent case this paper presents a straightforward extension of the e m inequality for the dependent case yielding a discrete distribution that is extremal with respect to a partial ordering of a set of distribution functions further we prove that the bounds for the expectation behave monotonically by applying the obtained inequality to smaller and smaller subintervals as this requires the evaluation of conditional probabilities and expectations we replace the given distribution by a discrete one resulting from sampling and determine the sampling size that keeps the statistical error negligible finally we conclude with the application to slp recourse problems and state some computational results illustrating the effort for solving recourse problems

second invariant preserving remap of the 2d deviatoric stress tensor in ale methods abstract for numerical simulations of impact problems or fluid solid interactions the ale arbitrary lagrangian eulerian approach is a useful tool due to its ability to keep the computational mesh smooth and moving with the fluid the elastic plastic extension of the compressible fluid model requires tensor variables for the description of non volumetric deviatoric mechanical stress while lagrangian numerical schemes based on the evolution equation of the stress tensor are well developed tensor remap is still a relatively unexplored territory we propose a new approach to deviatoric stress remapping where the second invariant j 2 a conservative scalar quantity related to the strain energy is remapped independently of the tensor components these are re scaled to match the remapped invariant value effectively using only the principal directions and eigenvalue ratio from the component wise remap this approach is frame invariant preserves j 2 invariant bounds and conserves the total invariant we compare our method with component based remapping using a simple synchronized limiter or a specialized stress tensor limiter described in the literature

a second order accurate material order independent interface reconstruction technique for multi material flow simulations a new second order accurate volume conservative material order independent interface reconstruction method for multi material flow simulations is presented first materials are located in multi material computational cells using a piecewise linear reconstruction of the volume fraction function these material locator points are then used as generators to reconstruct the interface with a weighted voronoi diagram that matches the volume fractions the interfaces are then improved by minimizing an objective function that smoothes interface normals while enforcing convexity and volume constraints for the pure material subcells convergence tests are shown demonstrating second order accuracy static and dynamic examples are shown illustrating the superior performance of the method over existing material order dependent methods

the ugc hardness threshold of the lp grothendieck problem for p 2 we consider the problem of given an n n matrix a aij whose diagonal entries vanish approximating in polynomial time the number rm opt p a max left sum i j 1 na ij x ix j x 1 ldots x n in mathbb r n wedge left sum i 1 n x i p right 1 p le 1 right when p 2 this is simply the problem of computing the maximum eigenvalue of a whereas for p actually it suffices to take p log n it is the grothendieck problem on the complete graph which was shown to have a o log n approximation algorithm in nemirovski et al nemirovski a c roos t terlaky 1999 on maximization of quadratic form over intersection of ellipsoids with common center math program ser a86 3 463 473 megretski megretski a 2001 relaxations of quadratic programs in operator theory and system analysis systems approximation singular integral operators and related topics bordeaux 2000 vol 129 operator theory advances and applications birkhauser basel 365 392 charikar and wirth charikar m a wirth 2004 maximizing quadratic programs extending grothendieck s inequality proc 45th annual sympos foundations comput sci ieee computer society 54 60 and was used in the work of charikar and wirth noted above to design the best known algorithm for the problem of computing the maximum correlation in correlation clustering thus the problem of approximating optp a interpolates between the spectral p 2 case and the correlation clustering p case from a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard wall potential well we design a polynomial time algorithm which given p 2 and an n n matrix a aij with zeros on the diagonal computes optp a up to a factor p e 30 log p on the other hand assuming the unique games conjecture ugc we show that it is np hard to approximate optp a up to a factor smaller than p e hence as p the ugc hardness threshold for computing optp a is exactly p e 1 o 1

a lax wendroff type theorem for unstructured quasi uniform grids a well known theorem of lax and wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a e as the mesh parameter goes to zero then the limit is a weak solution of the system moreover if the scheme satisfies a discrete entropy inequality as well the limit is an entropy solution the original theorem applies to uniform cartesian grids this article presents a generalization for quasi uniform grids with lipschitz boundary cells uniformly continuous inhomogeneous numerical fluxes and nonlinear inhomogeneous sources the added generality allows a discussion of novel applications like local time stepping grids with moving vertices and conservative remapping a counterexample demonstrates that the theorem is not valid for arbitrary non quasi uniform grids

a class of deformational flow test cases for linear transport problems on the sphere a class of new benchmark deformational flow test cases for the two dimensional horizontal linear transport problems on the sphere is proposed the scalar field follows complex trajectories and undergoes severe deformation during the simulation however the flow reverses its course at half time and the scalar field returns to its initial position and shape this process makes the exact solution available at the end of the simulation and facilitates assessment of the accuracy of the underlying transport scheme a procedure to eliminate possible cancellations of errors when the flow reverses is proposed the test suite consists of four cases three are based on non divergent flow fields and one on a divergent flow the initial conditions are prescribed in terms of regular latitude longitude spherical coordinates and are easy to implement the divergent flow is specifically aimed for conservative global transport schemes to test for conservation consistency and monotonicity or positivity of limiters filters in a challenging flow environment in the context of semi lagrangian schemes the time varying flow fields can be used to test trajectory algorithms where the exact trajectories do not follow great circle arcs the characteristics of the test cases are demonstrated with two different transport schemes

a low rank in time approach to pde constrained optimization the solution of time dependent pde constrained optimization problems is a challenging task in numerical analysis and applied mathematics all at once discretizations and corresponding solvers provide efficient methods to robustly solve the arising discretized equations one of the drawbacks of this approach is the high storage demand for the vectors representing the discrete space time cylinder here we introduce a low rank in time technique that exploits the low rank nature of the solution the theoretical foundations for this approach originate in the numerical treatment of matrix equations and can be carried over to pde constrained optimization we illustrate how three different problems can be rewritten and used within a low rank krylov subspace solver with appropriate preconditioning

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

edgeworth expansion of a function of sample means many important statistics can be written as functions of sample means of vector variables a fundamental contribution to the edgeworth expansion for functions of sample means was made by bhattacharya and ghosh in their work the crucial cramer c condition is assumed on the joint distribution of all the components of the vector variable however in many practical situations only one or a few of the components satisfy conditionally this condition while the rest do not such a case is referred to as satisfying the partial cramer c condition the purpose of this paper is to establish edgeworth expansions for functions of sample means when only the partial cramer c condition is satisfied

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

distributed schur complement techniques for general sparse linear systems this paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments these techniques utilize the schur complement system for deriving the preconditioning matrix in a number of ways two of these preconditioners consist of an approximate solution process for the global system which exploits approximate lu factorizations for diagonal blocks of the schur complement another preconditioner uses a sparse approximate inverse technique to obtain certain local approximations of the schur complement comparisons are reported for systems of varying difficulty

optimal lossless data compression non asymptotics and asymptotics this paper provides an extensive study of the behavior of the best achievable rate and other related fundamental limits in variable length strictly lossless compression in the non asymptotic regime the fundamental limits of fixed to variable lossless compression with and without prefix constraints are shown to be tightly coupled several precise quantitative bounds are derived connecting the distribution of the optimal code lengths to the source information spectrum and an exact analysis of the best achievable rate for arbitrary sources is given fine asymptotic results are proved for arbitrary not necessarily prefix compressors on general mixing sources nonasymptotic explicit gaussian approximation bounds are established for the best achievable rate on markov sources the source dispersion and the source varentropy rate are defined and characterized together with the entropy rate the varentropy rate serves to tightly approximate the fundamental nonasymptotic limits of fixed to variable compression for all but very small block lengths

on the effect of random norming on the rate of convergence in the central limit theorem it is shown that studentizing i e normalizing by the sample standard deviation rather than the population standard deviation can improve the rate of convergence in the central limit theorem this provides concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the case of infinite population standard deviation is also examined 1 introduction our purpose in this paper is to show that studentizing the operation of replacing a population standard deviation in a norming constant by its sample estimate can improve the rate of convergence in the central limit theorem thus we provide concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum the reason is that large deviations of summands which are responsible for slow rates of convergence when ordinary norming constants are used cancel from the sample mean and sample standard deviation remark 3 7 in section 3 will give an intuitive explanation of the mechanism behind this cancellation note that the large deviations have a particularly subtle influence since under the condition that the distribution is attracted to the normal law the largest summand is negligible in comparison with the sum itself section 2 will briefly survey results on rates of convergence in the ordinary i e nonstudentized central limit theorem and will discuss their connection with our work section 3 will state our main results and summarize their consequences and section 4 will give proofs

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

a conservative multi tracer transport scheme for spectral element spherical grids atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers for computational efficiency conservative multi tracer semi lagrangian type transport schemes are appropriate global models based on high order galerkin approach employ highly non uniform spectral element grids and semi lagrangian transport is a challenge on those grids a conservative semi lagrangian scheme spelt spectral element lagrangian transport employing a multi moment compact reconstruction procedure is developed for non uniform quadrilateral grids the scheme is based on a characteristic semi lagrangian method that avoids complex and expensive upstream area computations the spelt scheme has been implemented in the high order method modeling environment homme which is based on a cubed sphere grid with spectral element spatial discretization additionally we show the strong scalability and multi tracer efficiency using several benchmark tests the spelt solution can be made monotonic positivity preserving by combining the flux corrected transport algorithm which is demonstrated on a uniform resolution grid in particular spelt can be efficiently used for non uniform grids and provides accurate and stable results for high resolution meshes

convergence determining sets in the central limit theorem a complete characterisation is given of the class of all doublets which determine the rate of convergence in the central limit theorem this enables a number of important properties of convergence determining sets to be deduced in particular it is shown that no singleton can be convergence determining and any set consisting of four or more distinct points is convergence determining numerical and analytic methods are used to derive the geometry of the class of all convergence determining doublets

efficient preconditioning of sequences of nonsymmetric linear systems we present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems this approach is algebraic and it is theoretically motivated it generalizes diagonal updates introduced by benzi and bertaccini bit 43 2003 pp 231 244 and bertaccini electron trans numer anal 18 2004 pp 49 64 it is shown experimentally that this approach can be very beneficial for example it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence in some cases the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners since the updates are typically cheap and straightforward their use is of practical interest they can replace recomputing preconditioners which is often expensive especially in parallel and matrix free environments