fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

an algorithm using the finite volume element method and its splitting extrapolation this paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation this method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation such as a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than a richardson extrapolation because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

entropy extropy and information potential in stochastic systems far from equilibrium the relations between information entropy and energy which are well known in equilibrium thermodynamics are not clear far from equilibrium moreover the usual expression of the classical thermodynamic potentials is only valid near equilibrium in previous publications we showed for a chemical system maintained far from equilibrium that a new thermodynamic potential the information potential can be de4ned by using the stochastic formalism of the master equation here we extend this theory to a completely general discrete stochastic system for this purpose we use the concept of extropy which is de4ned in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration we show that the statistical equivalent of the thermodynamic extropy is the relative information if a coarse grained description by means of the thermodynamic extensive variables is available which is the case for many macroscopic systems the coarse grained statistical extropy allows one to de4ne the information potential in the thermodynamic limit using this potential we study the evolution of such systems towards a non equilibrium stationary state we derive a general thermodynamic inequality between energy dissipation and information dissipation which sharpens the law of the maximum available work for non equilibrium systems c 2002 published by elsevier science b v

asymptotic expansion for the discretization error in linear elliptic boundary value problems on general regions to define a discretization we introduce a set f h of grid points see 2 2 and replace the derivatives in 1 1 by symmetric divided differences with respect to those grid points this device is unproblematic as long as the direct grid neighbours of interior grid points are either interior grid points or boundary points by a proper choice of h that may be obtained e g for rectangular shape regions f2 for all inner grid points for general domains f2 this assumption is no longer satisfied and direct grid neighbours will not always be in and therefore symmetric divided differences are not directly applicable in some points so called irregular mesh points there is a long history to solve this problem 1930 gerschgorin 153 has proposed to use for the unknown function value in a grid neighbour outside of f2 next to an irregular grid point the boundary value of the next boundary point on that grid line obtaining a method of order 1 compared with order 2 available in cases without irregular grid points in 1933 collatz 11 improved this method by using linear interpolation between the irregular grid point and its neighbouring boundary point obtaining a method of order 2 wasow 22 showed that for the methods in 11 15 no asymptotic expansions of the discretization error can exist the 1938 approach of shortley weller 20 may be interpreted as using polynomials of degree 2 interpolating in the

high effective finite element algorithm for elliptic partial differential equation in this paper we will discuss elliptic partial differential equation whose coefficients are constants on basis of a new computing technique for green function we will propose a new effective finite element algorithm for that problem finally we will give some numerical examples to investigate that finite element algorithm

discrimination between states of a quantum system by observations let m be a von neumann algebra with normal states and and let i ai m be a net of positive unital mappings of finite dimensional algebras the paper studies the existence of a net i m ai such that i i and i i a necessary and sufficient condition is given for the existence of i which may fail to hold when ai are commutative and does not commute with reformulation in terms of relative entropy is given and the study of the entropic version is extended to the case when m and are infinite tensor products all these mathematical problems are motivated by quantum statistical considerations

asymptotic error rates in quantum hypothesis testing we consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero this leads to the identification of the quantum generalisation of the classical chernoff distance which is the corresponding quantity in classical symmetric hypothesis testing

optimal sequence of quantum measurements in the sense of stein s lemma in quantum hypothesis testing we derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing we discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0 as an asymptotically optimal measurement we propose a projection measurement characterized by the irreducible representation theory of the special linear group sl especially in the spin 1 2 system it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction as a by product we obtain another proof of quantum stein s lemma in addition an asymptotically optimal measurement is constructed in the quantum gaussian case and it is physically meaningful

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

universal coding for classical quantum channel we construct a universal code for a stationary and memoryless classical quantum channel as a quantum version of the universal coding by csiszar and korner our code is constructed utilizing a combination of irreducible representations a decoder introduced through the quantum information spectrum and the packing lemma

metastable lifetime for a class of chemical systems near criticality abstract we consider a n dimensional autocatalytic chemical system with a finite metastable state which tends to become critical away from the critical situation the relaxation time from the metastable state to the final absorbing state can be computed by using inverse of the stationary probability distribution of the system at its minimal value it is difficult to obtain explicit results on the stationary probability distribution in general for several variables nevertheless we can estimate this quantity near criticality and we show that its critical exponent is 2 under rather general conditions this result is verified explicitly on an example for which a direct calculation can be done

implicit extrapolation methods for multilevel finite element computations extrapolation methods for the solution of partial differential equations are commonly based on the existence of error expansions for the approximate solution implicit extrapolation by contrast is based on applying extrapolation indirectly by using it on quantities like the residual in the context of multigrid methods a special technique of this type is known as tau extrapolation for finite element systems this algorithm can be shown to be equivalent to higher order finite elements the analysis is local and does not use global expansions so that the implicit extrapolation technique may be used on unstructured meshes and in cases where the solution fails to be globally smooth furthermore the natural multilevel structure can be used to construct efficient multigrid and multilevel preconditioning techniques the effectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory

classical capacities of compound and averaged quantum channels we determine the capacity of compound classical quantum channels as a consequence we obtain the capacity formula for the averaged classical quantum channels the capacity result for compound channels demonstrates as in the classical setting the existence of reliable universal classical quantum codes in scenarios where the only a priori information about the channel used for the transmission of information is that it belongs to a given set of memoryless classical quantum channels our approach is based on a universal classical approximation of the quantum relative entropy which in turn relies on a universal hypothesis testing result

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

finite size scaling for mean field percolation by studying transfer matrix eigenvalues correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime near criticality finite size scaling behavior is derived and an analytic technique is provided for obtaining the finite size scaling function our methods involve the generating function matched asymptotic expansions and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

on the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators the standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed they are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds further the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues a new method is presented which combines multigrid techniques with the lanczos process the resulting scheme requires o mn arithmetic operations and o n storage requirement where n is the number of unknowns and m the number of needed eigenvalues the discretization of the considered differential operators is realized by means of p finite elements and is applicable on general geometries numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods at least on current workstations the ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a pc shows the potential of this method practical applications are found e g in the numerical simulation of quantum billiards

existence uniqueness and asymptotics of steady jets we consider the three dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition we discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity moreover we show that each solution with a bounded dirichlet integral which has a certain weak additional decay behaves like o r 2 as r x and admits a wide jet region these investigations are based on the solvability properties of the linear stokes system in a half space 3 to investigate the stokes problem in 3 we apply the mellin transform technique and reduce the stokes problem to the determination of the spectrum of the corresponding invariant stokes beltrami operator on the hemisphere

asymptotics of quantum relative entropy from a representation theoretical viewpoint in this paper it is proved that the quantum relative entropy d can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive operator valued measures povms the sequence of povms depends on but is independent of the choice of

implicit extrapolation methods for variable coefficient problems implicit extrapolation methods for the solution of partial differential equations are based on applying the extrapolation principle indirectly multigrid tau extrapolation is a special case of this idea in the context of multilevel finite element methods an algorithm of this type can be used to raise the approximation order even when the meshes are nonuniform or locally refined the implicit extrapolation multigrid algorithm converges to the solution of a higher order finite element system this is obtained without explicitly constructing higher order stiffness matrices but by applying extrapolation in a natural form within the algorithm the algorithm requires only a small change of a basic low order multigrid method

numerical simulation of 2d square driven cavity using fourth order compact finite difference schemes abstract fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large reynolds numbers the governing navier stokes equation is linearized in streamfunction and vorticity formulation the fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries numerical solutions are obtained for square driven cavity flow at high reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods

typical support and sanov large deviations of correlated states discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support the exponential growth rate of which is given by the maximal ergodic entropy in the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy according to sanov s theorem we give an extension to the correlated case referring to the newly introduced class of hp states

information potential and transition to criticality for certain two species chemical systems using the general results of the stochastic theory of chemical systems we consider a certain model of chemical reactions with two species for which it is possible to calculate the rst passage time explicitly and study the transition to criticality our method uses a non standard hamilton jacobi theory for the master equation introduced initially by kubo et al j stat phys 9 1973 51 which leads to solvable hamiltonian systems c 2000 elsevier science b v all rights reserved

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

high order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems high mixed order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed for the solution of these systems an efficient and robust multigrid algorithm is presented theoretical and experimental results show the advantages of higher order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang bang control

centroidal voronoi tessellation based finite element superconvergence in this article a finding on finite element superconvergence is reported the laplacian operator with dirichlet boundary condition is considered the linear finite element solutions have an o h2 0 5 superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal voronoi tessellation for an arbitrary 2d bounded domain extensive numerical examples are presented to demonstrate the superconvergence property copyright 2008 john wiley sons ltd

error estimates of finite element method about elliptic problems with singular righthand side abstract in this paper on basis of other related works see r scott optimal l estimates for the finite element method on irregular meshes math comp v 30 1976 681 697 qun lin qi ding zhu the preprocessing and postprocessing for fem shanghai science press china 1994 i babuska a miller the post processing in the finite element method parts i ii internat j numer methods engrg 20 1984 1085 1109 1111 1129 we will continue to study the theory of fem for elliptic problems with singular righthand side the main difference between this paper and other related works see scott 1976 lin and zhu 1994 babuska and miller 1984 is that it will present some local estimates for fem of elliptic problems with singular righthand side of course some numerical examples will be given to investigate those error estimates

goal oriented error estimation for fluid structure interaction problems in this work we present an adaptive finite element method for the numerical simulation of stationary fluid structure interaction problems the coupled system is given in a variational and monolithic arbitrary lagrangian eulerian framework we derive methods for goal oriented error estimation and mesh adaptation with the dual weighted residual method key to applying this error estimator is the underlying canonic variational formulation of the fluid structure interaction problem by mapping the flow problem to ale coordinates the developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible navier stokes equations with a nonlinear hyper elastic material law

superconvergence analysis and error expansion for the wilson nonconforming finite element in this paper the wilson nonconforming finite element is considered for solving a class of two dimensional second order elliptic boundary value problems superconvergence estimates and error expansions are obtained for both uniform and non uniform rectangular meshes a new lower bound of the error shows that the usual error estimates are optimal finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms

an information spectrum approach to classical and quantum hypothesis testing for simple hypotheses the information spectrum analysis made by han for classical hypothesis testing for simple hypotheses is extended to a unifying framework including both classical and quantum hypothesis testing the results are also applied to fixed length source coding when loosening the normalizing condition for probability distributions and for quantum states we establish general formulas for several quantities relating to the asymptotic optimality of tests codes in terms of classical and quantum information spectra

a parallel geometric multigrid method for finite elements on octree meshes in this article we present a parallel geometric multigrid algorithm for solving variable coefficient elliptic partial differential equations on the unit box with dirichlet or neumann boundary conditions using highly nonuniform octree based conforming finite element discretizations our octrees are 2 1 balanced that is we allow no more than one octree level difference between octants that share a face edge or vertex we describe a parallel algorithm whose input is an arbitrary 2 1 balanced fine grid octree and whose output is a set of coarser 2 1 balanced octrees that are used in the multigrid scheme also we derive matrix free schemes for the discretized finite element operators and the intergrid transfer operations the overall scheme is second order accurate for sufficiently smooth right hand sides and material properties its complexity for nearly uniform trees is mathcal o frac n n p log frac n n p mathcal o n p log n p where n is the number of octree nodes and n p is the number of processors our implementation uses the message passing interface standard we present numerical experiments for the laplace and navier linear elasticity operators that demonstrate the scalability of our method our largest run was a highly nonuniform 8 billion unknown elasticity calculation using 32 000 processors on the teragrid system ranger at the texas advanced computing center our implementation is publically available in the dendro library which is built on top of the petsc library from argonne national laboratory

a posteriori error estimates based on the polynomial preserving recovery superconvergence of order o h 1 rho for some rho 0 is established for the gradient recovered with the polynomial preserving recovery ppr when the mesh is mildly structured consequently the ppr recovered gradient can be used in building an asymptotically exact a posteriori error estimator

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

new expansions of numerical eigenvalues by wilson s element the paper explores new expansions of eigenvalues for email protected email protected ru in s with dirichlet boundary conditions by wilson s element the expansions indicate that wilson s element provides lower bounds of the eigenvalues by the extrapolation or the splitting extrapolation the o h 4 convergence rate can be obtained where h is the maximal boundary length of uniform rectangles numerical experiments are carried to verify the theoretical analysis made it is worth pointing out that these results are new compared with the recent book lin and lin q lin j lin finite element methods accuracy and improvement science press beijing 2006

new expansions of numerical eigenvalues for δ by nonconforming elements the paper explores new expansions of the eigenvalues for u pu in s with dirichlet boundary conditions by the bilinear element denoted q 1 and three nonconforming elements the rotated bilinear element denoted q 1 rot the extension of q 1 rot denoted eq 1 rot and wilson s elements the expansions indicate that q 1 and q 1 rot provide upper bounds of the eigenvalues and that eq 1 rot and wilson s elements provide lower bounds of the eigenvalues by extrapolation the o h 4 convergence rate can be obtained where h is the maximal boundary length of uniform rectangles numerical experiments are carried out to verify the theoretical analysis made

superconvergence analysis for the explicit polynomial recovery method a new recovery technique explicit polynomial recovery epr is analyzed for finite element methods epr reconstructs the value at edge centers by solving a local problem in combination with the finite element solution at the vertex a quadratic approximation is constructed besides improving the accuracy it can also be applied in building the epr based error estimator for the poisson equation the element center is a superconvergent point of the gradient of the epr recovered function on an equilateral triangulation numerical examples are presented to verify the theoretical results and to show the performance of the epr in the adaptive finite element method

mudpack multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations the solution regions are rectangles in two dimensions and boxes in three dimensions any combination of periodic dirichlet and mixed derivative boundary conditions is allowed the equations are automatically discretized using second order finite differencing the package will vectorize on cray computers examples are given which demonstrate ease of use efficiency and applicability to a wide range of problems

operational capacity and pseudoclassicality of a quantum channel we explore some basic properties of coding theory of a general quantum communication channel and its operational capacity including 1 adaptive measurement with feedback code 2 reconsideration of single letterized capacity formula and 3 pseudoclassicality of a channel

a finite element splitting extrapolation for second order hyperbolic equations splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel this article discusses a finite element splitting extrapolation for second order hyperbolic equations with time dependent coefficients this method possesses a higher degree of parallelism less computational complexity and more flexibility than richardson extrapolation while achieving the same accuracy by means of domain decomposition and isoparametric mapping some grid parameters are chosen according to the problem the multiparameter asymptotic expansion of the d quadratic finite element error is also established the splitting extrapolation formulas are developed from this expansion an approximation with higher accuracy on a globally fine grid can be computed by solving a set of smaller discrete subproblems on different coarser grids in parallel some a posteriori error estimates are also provided numerical examples show that this method is efficient for solving discontinuous problems and nonlinear hyperbolic equations

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

two quantum analogues of fisher information from a large deviation viewpoint of quantum estimation we discuss two quantum analogues of the fisher information the symmetric logarithmic derivative fisher information and kubo mori bogoljubov fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators as another comparison it is shown that the difference between them is characterized by the change of the order of limits

collocation coarse approximation in multigrid the two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation dca and petrov galerkin coarse approximation gca here a new approach called collocation coarse approximation cca is introduced which like gca is algebraically defined and able to cater to difficult features such as discontinuous coefficients but unlike gca allows explicit control over the coarse grid sparsity pattern stencil and therefore control over the computational complexity of the solver cca relies on certain basis functions for which the coarse approximation to the fine grid problem is exact numerical experiments for two dimensional diffusion problems including jumping coefficients demonstrate the potential of the resulting multigrid algorithms

residual scaling techniques in multigrid ii practical applications this paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper residual scaling techniques in multigrid i equivalence proof appl math comput 86 283 303 1997 we discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent a heuristic residual analysis technique based on the geometry of the grid points and the relaxation pattern is introduced to estimate the optimal residual scaling factor for a high order multigrid method we compare the performance of a typical pre optimization pre acceleration technique with a typical post optimization post acceleration technique and show that the pre optimization is preferable in both convergence and efficiency our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre optimization technique over the post optimization technique

hypothesis testing on invariant subspaces of the symmetric group part i quantum sanov s theorem and arbitrarily varying sources we report a proof of the quantum sanov theorem by the elementary application of basic facts about representations of the symmetric group together with a complete characterization of the optimal error exponent in a situation where the null hypothesis is given by an arbitrarily varying quantum source instead our approach differs from previous ones in two ways first it supports a reasoning inspired by the method of types second the measurement scheme we propose to distinguish the two alternatives not only does that job asymptotically perfectly but also yields additional information about the null hypothesis an example of that is given the measurement is composed of projections onto permutation invariant subspaces thus providing a direct link between one of the most basic tasks in quantum information on the one hand and fundamental objects in representation theory on the other we additionally connect to the representation theory by proving a relation between kostka numbers and quantum states and to state estimation via a generalization of a well known spectral estimation theorem to non i i d sequences

variational localizations of the dual weighted residual estimator the dual weighted residual method dwr and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated the contribution of this paper is twofold first we introduce a novel localization technique based on the introduction of a partition of unity this new technique is very easy to apply as neither strong residuals nor jumps over element edges are required second we compare and analyze theoretically and numerically different localization techniques used for mesh adaptivity with respect to their effectivity here we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation in our mathematical analysis we show for different localization techniques established methods and our new approach that the local error indicators used for mesh adaptivity converge with proper order in the error functional several numerical tests substantiate our theoretical investigations

fluctuations in mean field self organized criticality we present two models that exhibit self organized criticality at the mean field level these can be variously interpreted in epidemiological or chemical reaction terms by studying the master equation for these models we find however that only in one of them does the self organized critical behavior survive in the face of fluctuations for this model we show the spectrum of the evolution operator to have spectral collapse i e instead of a gap as would occur in noncritical behavior there are eigenvalues that approach zero as an inverse power of system size

asymptotic expansions and richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods in this paper asymptotic error expansions for mixed finite element approximations of general second order elliptic problems are derived under rectangular meshes and the richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation postprocessing technique the results of this paper provide new asymptotic expansions and new approximate solutions which are one order and a half order higher in accuracy than those obtained in j wang math comp 56 1991 pp 477 503 and h chen r e ewing and r lazarov asymptotic error expansion for the lowest order raviart thomas rectangular mixed finite elements technical report isc 97 01 1997 respectively as a by product we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators for the mixed finite element method

convergence of the combination technique for second order elliptic differential equations the combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization the advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points in this paper the combination technique is used with a bilinear finite element discretization depending on the smoothness of the solution and the coefficients it is proved for general second order elliptic differential equations on the unit square that the combined solution converges with order o h or o h log h 1 in the energy norm and with order o h2 log h 1 or o h3 2 in the l2 norm respectively this holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite the proof does not use an asymptotic error expansion but sobolev space techniques

extrapolation techniques for reducing the pollution effect or reentrant corners in the finite element method standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities for a model situation it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter this provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

a splitting extrapolation for solving nonlinear elliptic equations with d quadratic finite elements nonlinear elliptic partial differential equations are important to many large scale engineering and science problems for this kind of equations this article discusses a splitting extrapolation which possesses a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than richardson extrapolation according to the problems some domain decompositions are constructed and some independent mesh parameters are designed multi parameter asymptotic expansions are proved for the errors of approximations based on the expansions splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes