application of the arnoldi algorithm to the solution of the advection dispersion equation the solution to the finite element matrix differential equations resulting from the discretization of the contaminant transport equation is normally carried out by a finite difference approximation to the time derivative the total computational effort in simulating a contaminant plume is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions an alternative method is the arnoldi algorithm which uses orthogonal matrix transformations to reduce the finite element equations to a much smaller upper hessenberg system of first order differential equations this new system can be solved by a standard crank nicolson algorithm with very little computational effort a matrix vector multiplication is then used to obtain the original solution at desired time steps the algorithm is used to simulate accurately the contaminant plumes for a strip source areal aquifer a cross sectional problem and the borden landfill in ontario canada the arnoldi algorithm shows an impressive 613 increase in speed over the conventional crank nicolson scheme for this latter case the method affords an efficient means of solving large problems particularly when time durations are long

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

node and element resequencing using the laplacian of a finite element graph part i general concepts and algorithm a finite element graph feg is defined here as a nodal graph g a dual graph g or a communication graph g associated with a generic finite element mesh the laplacian matrix l g l g or l g used for the study of spectral properties of an feg is constructed from usual vertex and edge connectivities of a graph an automatic algorithm based on spectral properties of an feg g g or g is proposed to reorder the nodes and or elements of the associated finite element mesh the new algorithm is called spectral peg resequencing sfr this algorithm uses global information in the graph it does not depend on a pseudoperipheral vertex in the resequencing process and it does not use any kind of level structure of the graph moreover the sfr algorithm is of special advantage in computing environments with vector and parallel processing capabilities nodes or elements in the mesh can be reordered depending on the use of an adequate graph representation associated with the mesh if g is used then the nodes in the mesh are properly reordered for achieving profile and wavefront reduction of the finite element stiffness matrix if either g or g is used then the elements in the mesh are suitably reordered for a finite element frontai solver a unified approach involving fegs and finite element concepts is presented conclusions are inferred and possible extensions of this research are pointed out in part ii of this work 1 the computational implementation of the sfr algorithm is described and several numerical examples are presented the examples emphasize important theoretical numerical and practical aspects of the new resequencing method

an improved lanczos algorithm for solving ill conditioned linear equations abstract in the existing lanczos algorithms for solving systems of linear equations the estimate for the residual is effective for well conditioned systems however in actual implementation on a computer we find that the estimate is no longer reliable for ill conditioned cases we first analyze in theory this observation then develop an improved lanczos algorithm numerical examples are also given to demonstrate the effectiveness of the present algorithm

improving variational methods via pairwise linear response identities inference methods are often formulated as variational approximations these approximations allow easy evaluation of statistics by marginalization or linear response but these estimates can be inconsistent we show that by introducing constraints on covariance one can ensure consistency of linear response with the variational parameters and in so doing inference of marginal probability distributions is improved for the bethe approximation and its generalizations improvements are achieved with simple choices of the constraints the approximations are presented as variational frameworks iterative procedures related to message passing are provided for finding the minima

formulation and solution of hierarchical finite element equations the paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems static and eigenvalue linear elastic problems as well as convection diffusion problems are studied the hierarchical formulation is well suited for adaptive procedures for the convection diffusion problem the hierarchical approximation is made in time only different hierarchical functions are proposed for different types of problems both weighted residual and least squares formulations are applied a combination of these two gives a penalty method with a constraint equation corresponding to the least squares method a whole class of time integration formulae is obtained these are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain if a linear discontinuous hierarchical base function is used in the galerkin weak formulation the method so obtained corresponds to the discontinuous galerkin method in time and is especially suited for convection dominated problems the streamline diffusion method is found to be the aforementioned penalty method this paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation properties of these systems arising from static problems are investigated the paper presents some new possibilities for iterative solution of hierarchic element equations and different procedures are compared in a numerical example finally a simple id convection diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow

critical behaviour of dyson s hierarchical model with a random field the critical behaviour of dyson s hierarchical ising model is studied in the presence of a random field simple scaling relations determine the exponents eta eta exactly for random fields with either short or long range correlations the schwartz soffer inequality eta or 2 eta is satisfied as an equality provided the random field correlations are not too long ranged for short range random fields the exponent v is calculated to order in lambda 4 3 where lambda is the range parameter of the dyson model the exponent v is also computed to o in 2 with in 2 lambda 3 for the hierarchical o n model in zero field the results reveal striking similarities with the exponents of the corresponding one dimensional model with long range interactions

the three dimensional dilute ising magnet the dilute ising model with a p 0 8 fraction of the sites occupied by spins is simulated on l3 systems for l up to 300 using a single cluster algorithm of wolff in the range of reduced temperature 0 002 t tc tc 0 03 the susceptibility appears to fit a simple power with an effective exponent eff 1 36

generalized eigenvalue problems lanczos algorithm with a recursive partitioning method abstract in this paper the computation of the smallest eigenvalues and the corresponding eigenvectors of the generalized eigenvalue problem using lanczos algorithm with a recursive partitioning method as well as the sturm sequence bisection method have been discussed we have also presented the comparison of the numerical results and the cpu time between the above two methodologies our comparative study indicates that the lanczos with a recursive partitioning method takes relatively less computing time than that of the sturm sequence bisection method

critical behaviour of the random field ising model we present an analytic real space renormalization group calculation for the random field ising model we apply the migdal kadanoff approximation for the renormalization of a cubic cell in dimensions d introducing a new field partitioning scheme which allows us to treat the random field fluctuations in a coherent manner our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality we expand analytically in further we show that and the magnitude of the inequality go to zero exponentially with we calculate the crossover exponent from pure to the random field system and find surprisingly good agreement with experimental values we find that satisfies the schwartz soffer inequality the susceptibility exponent of the pure system we expand in and find that the magnitude of the inequality varies exponentially in finally we find that dimensional reduction is satisfied to first order in with the reduced dimension

on the missing scaling relation in random field systems abstract a detailed proof is given that for random field systems the susceptibility and correlation function are related at the transition this yields an additional scaling relation 2

self consistent ornstein zernike approximation for three dimensional spins abstract an ornstein zernike approximation for the two body correlation function embodying thermodynamic consistency is applied to a system of classical heisenberg spins on a three dimensional lattice the consistency condition determined in a previous work is supplemented by introducing a simplified expression for the mean square spin fluctuations the thermodynamics and the correlations obtained are then compared with approximants based on extrapolation of series expansions and with monte carlo simulations many properties of the model including the critical temperature are very well reproduced by this simple version of the theory but it shows substantial quantitative error in the critical region both above the critical temperature and with respect to its rendering of the spontaneous magnetization curve a less simple but conceptually more satisfactory version of the scoza is then developed but not solved in which the effects of transverse correlations on the longitudinal susceptibility is included yielding a more complete and accurate description of the spin wave properties of the model

loop calculus in statistical physics and information science considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series the leading term in the series is the bethe peierls belief propagation bp contribution the rest are expressed as loop contributions on the factor graph and calculated directly using the bp solution the series unveils a small parameter that often makes the bp approximation so successful applications of the loop calculus in statistical physics and information science are discussed

cccp algorithms to minimize the bethe and kikuchi free energies convergent alternatives to belief propagation this article introduces a class of discrete iterative algorithms that are provably convergent alternatives to belief propagation bp and generalized belief propagation gbp our work builds on recent results by yedidia freeman and weiss 2000 who showed that the fixed points of bp and gbp algorithms correspond to extrema of the bethe and kikuchi free energies respectively we obtain two algorithms by applying cccp to the bethe and kikuchi free energies respectively cccp is a procedure introduced here for obtaining discrete iterative algorithms by decomposing a cost function into a concave and a convex part we implement our cccp algorithms on two and three dimensional spin glasses and compare their results to bp and gbp our simulations show that the cccp algorithms are stable and converge very quickly the speed of cccp is similar to that of bp and gbp unlike cccp bp will often not converge for these problems gbp usually but not always converges the results found by cccp applied to the bethe or kikuchi free energies are equivalent or slightly better than those found by bp or gbp respectively when bp and gbp converge note that for these and other problems bp and gbp give very accurate results see yedidia et al 2000 and failure to converge is their major error mode finally we point out that our algorithms have a large range of inference and learning applications

numerical comparison of iterative eigensolvers for large sparse symmetric positive definite matrices abstract the jacobi davidson jd method has been recently proposed for the evaluation of the partial eigenspectrum of large sparse matrices in this work we report a set of numerical experiments that compare this method with other previously proposed techniques deflation accelerated conjugate gradient dacg and lanczos arpack on large sparse symmetric matrices the results obtained by jd and dacg are benchmarked against those obtained with arpack in terms of computational time for the evaluation of s s 100 leftmost eigenpairs the comparison is performed on a number of matrices with dimensions up to a few hundred thousands arising from the discretization of the diffusion equation by means of finite element mixed finite element and finite difference techniques the results show that dacg and jd are equally more efficient than arpack for a small number of eigenpairs with dacg more performing for small s and particularly for s 1 while arpack tends to become competitive when 40 or more eigenpairs are sought for the largest and worst conditioned matrix dacg proves the only viable alternative

new approach to percolation lattice system results results of an integral equation approach to pair connectedness are given for lattice models with emphasis on site percolation the lowest order approximation is expressible in terms of the lattice green function and is isomorphic to a polya random walk it yields percolation exponents gamma 2 nu 1 for d 3 and gamma 1 nu 1 2 for d or 4 d dimension for simple hypercubic lattices it also yields usefully sharp estimates for the site percolation threshold probability pp a first correction is described

two dimensional mesh generation adaptive remeshing and refinement this paper discusses the generation of unstructured triangular meshes by the advancing front technique we also employ the technique to discretize surfaces a simple and effective algorithm for the specification of element sizes is described a similar idea is then applied to adaptive remeshing where a completely new mesh is created on the basis of the isolines of a certain important variable the isolines are obtained from the fe calculation based on the previous mesh in this paper attention is focused on the poisson s equation the optimal a priori error estimate for the problem is used to estimate the number of isolines needed for the remeshing when the estimated error exceeds the tolerance in only a small part of the domain the adaptive procedure is switched automatically to an adaptive refinement

characterization of electromagnetic devices via reduced order models abstract efficient procedures are presented for simultaneously characterizing the time and frequency domain behavior of 3d electromagnetic devices the procedures work in the complex frequency domain on either the finite element or boundary element formulation of maxwell s equations various approximation techniques are used to derive reduced order models describing the system transfer functions of the 3d device two different methods for evaluating reduced order models are presented one is called asymptotic waveform evaluation awe and is combined with the finite element method the other is called adaptive lanzcos pade sweep alps and is combined with the boundary element method the resulting reduced order models provide the frequency domain behavior of the device over a broad bandwidth using the inverse laplace transform these reduced order models can also provide the time domain behavior of the device several numerical examples have been run using commercial eda software to demonstrate that this solution procedure is a highly efficient and accurate way to characterize the electromagnetic performance of real life devices

scoza for d dimensional spins abstract the self consistent ornstein zernike approach scoza developed earlier by the authors for the lattice gas and ising model as well as simple continuum fluids is extended to the d vector spin model that includes the classical rotator model d 2 the classical heisenberg model d 3 and the spherical model limit d in which the scoza becomes exact the approach entails first formulating an exact treatment of the model that includes its response to the presence of a field h with one primary component h and a transverse component h

fourier multiplier theorems for triebel lizorkin spaces in this paper we study sharp generalizations of dot f p 0 q multiplier theorem of mikhlin hormander type the class of multipliers that we consider involves herz spaces k u s t plancherel s theorem proves widehat l s 2 k 2 s 2 and we study the optimal triple u t s for which sup k in mathbb z vert m 2 k cdot varphi vee vert k u s t infty implies dot f p 0 q boundedness of multiplier operator t m where varphi is a cutoff function our result also covers the bmo type space dot f infty 0 q

development of a block lanczos algorithm for free vibration analysis of spinning structures this paper is concerned with the development of an efficient eigenproblem solution algorithm and an associated computer program for the economical solution of the free vibration problem of complex practical spinning structural systems thus a detailed description of a newly developed block lanczos procedure is presented in this paper that employs only real numbers in all relevant computations and also fully exploits sparsity of associated matrices the procedure is capable of computing multiple roots and proves to be most efficient compared to other existing similar techniques

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

multiscale multifractal dcca and complexity behaviors of return intervals for potts price model abstract to investigate the characteristics of extreme events in financial markets and the corresponding return intervals among these events we use a potts dynamic system to construct a random financial time series model of the attitudes of market traders we use multiscale multifractal detrended cross correlation analysis mm dcca and lempel ziv complexity lzc perform numerical research of the return intervals for two significant china s stock market indices and for the proposed model the new mm dcca method is based on the hurst surface and provides more interpretable cross correlations of the dynamic mechanism between different return interval series we scale the lzc method with different exponents to illustrate the complexity of return intervals in different scales empirical studies indicate that the proposed return intervals from the potts system and the real stock market indices hold similar statistical properties

adaptive eigenfrequency analysis by superconvergent patch recovery an adaptive technique of the h version for vibration problems utilizing a new error indicator and mesh refinement strategy is proposed it is based on improvement of the eigenmodes by superconvergent patch recovery a modification of the termination criterion of the adaptive strategy is proposed the error indicator is tested by a numerical example and discussed the adaptive technique is tested by some engineering examples

perturbation corrections in approximate inference mixture modelling applications bayesian inference is intractable for many interesting models making deterministic algorithms for approximate inference highly desirable unlike stochastic methods which are exact in the limit the accuracy of these approaches cannot be reasonably judged in this paper we show how low order perturbation corrections to an expectation consistent ec approximation can provide the necessary tools to ameliorate inference accuracy and to give an indication of the quality of approximation without having to resort to monte carlo methods further comparisons are given with variational bayes and parallel tempering pt combined with thermodynamic integration on a gaussian mixture model to obtain practical results we further generalize pt to temper from arbitrary distributions rather than a prior in bayesian inference

susceptibility propagation by using diagonal consistency a susceptibility propagation that is constructed by combining a belief propagation and a linear response method is used for approximate computation for markov random fields herein we formulate an improved susceptibility propagation by using the concept of a diagonal matching method that is based on mean field approaches to inverse ising problems the proposed susceptibility propagation is robust for various network structures and it is reduced to the ordinary susceptibility propagation and to the adaptive thouless anderson palmer equation in special cases

the ferromagnetic q state potts model on three dimensional lattices a study for real values of q abstract we study the phase diagram of the ferromagnetic q state potts model on the various three dimensional lattices for integer and non integer values of q 1 our approach is based on a thermodynamically self consistent ornstein zernike approximation for the two point correlation functions we calculate the transition temperatures and when the transition is first order the jump discontinuities in the magnetization and the internal energy as well as the coordinates of the critical endpoint in external field our predictions are in very good agreement with the best available estimates from the numerical study of the region of weak first order transition we estimate the critical value qc for which the transition changes from second to first order the q 1 limit that describes the bond percolation problem is also investigated

a self consistent ornstein zernike approximation for the site diluted ising model we propose a theory for the site diluted ising model which is an extension to disordered systems of the self consistent ornstein zernike approximation of hoye and stell by using the replica method in the context of liquid state theory we treat the concentration of impurities as an ordinary thermodynamic variable this approach is not limited to the weak disorder regime or to the vicinity of the percolation point a preliminary analysis using series expansion shows that it can predict accurately the dependence of the critical temperature on dilution and can reproduce the nonuniversal behavior of the effective exponents the theory also gives a reasonable estimate of the percolation threshold

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

necessary conditions for vector valued operator inequalities in harmonic analysis via a random construction we establish necessary conditions for l p ell q inequalities for certain families of operators arising in harmonic analysis in particular we consider dilates of a convolution kernel with compactly supported fourier transform vector maximal functions acting on classes of entire functions of exponential type and a characterization of sobolev spaces by square functions and pointwise moduli of smoothness

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

partial eigensolution of damped structures by non linear optimization the partial solution of damped structures involves the determination of only a few of the resonant frequencies and corresponding damping characteristics and mode shapes the procedure outlined here uses a modification of the variable metric solution algorithm in which the dicision variables are elements of the complex eigenvector and the eigenvalue the quadratic eigenvalue problem is reformulated as a sequential unconstrained minimization problem both normality and orthogonality constraints are incorporated in the objective function the method is compared to an adaptation of the subspace iteration technique also applicable to the quadratic eigenvalue problem

lanczos method for heat conduction analysis an analysis technique that uses the lanczos vectors to construct a reyleigh ritz solution for the transient heat conduction problem is presented the lanczos vectors are derived from the heat supply vector and incorporate all the information peculiar to a given problem they are orthonormal with respect to the heat capacitance matrix and are generated with a minimum of computational effort furthermore an error bound is derived that can be used to terminate the lanczos algorithm as soon as the number of vectors required to obtain a desired degree of accuracy has been generated a summary of the algorithm is presented and the accuracy and efficiency of the method is illustrated using a numerical example

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

linear response algorithms for approximate inference in graphical models belief propagation bp on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph however it does not prescribe a way to compute joint distributions over pairs of distant nodes in the graph in this article we propose two new algorithms for approximating these pairwise probabilities based on the linear response theorem the first is a propagation algorithm that is shown to converge if bp converges to a stable fixed point the second algorithm is based on matrix inversion applying these ideas to gaussian random fields we derive a propagation algorithm for computing the inverse of a matrix

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

the use of a refined error bound when updating eigenvalues of tridiagonals abstract the lanczos algorithm is used to compute some eigenvalues of a given symmetric matrix of large order at each step of the lanczos algorithm it is valuable to know which eigenvalues of the associated tridiagonal matrix have stabilized at eigenvalues of the given symmetric matrix we present a robust algorithm which is fast 20 j to 40 j operations at the j th lanczos step uses about 30 words of extra storage and has a fairly short program approxiamately 200 executable statements

on some hp adaptive finite element method for natural vibrations model eigenproblem results obtained by some hp adaptive fem are presented the fem is based on the texas 3 step strategy and the equilibrated residual method the focus is on numerical experiments the practical issues related to the above mentioned techniques are raised the potentials of the method for engineering applications are demonstrated

approximations of mean spherical type for lattice percolation models the authors develop a general class of approximations of mean spherical msa type as a method for studying lattice percolation problems they review the msa and test certain extensions of it on lattice spin models the relations between thermal spin models and percolation models are then reviewed in order to identify natural extensions of the msa to percolation models these extensions are used to treat both site and bond percolation models in one low density formulation of msa the threshold for bond percolation on a lattice is found to equal the value at the origin of the corresponding lattice green s function this formula gives accurate results for all lattices studied and in all space dimensions d or 3 an accurate treatment is also given of the general site bond problem the entire percolation locus for this problem agrees closely with the results of simulation they also introduce a new method for studying percolation transitions which is a hybrid of the kikuchi cluster approximation scheme and the msa the method is shown to give good values for percolation thresholds while preserving the valuable features of the standard msa in particular it provides a convenient means of computing the pair connectedness function they also explore extensions of their approximations to treat directed site and bond percolation

an error indicator for the generalized eigenvalue problem using the hierarchical finite element method the hierarchical concept applied to eigenvalue problems is considered an error indicator is derived from the pertinent rayleigh quotient the indicator serves as an estimation a posteriori of the relative change in an eigenvalue for a hierarchical refinement a numerical example is included

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

real space renormalization group analysis of the random field ising model we present an analytic migdal kadanoff renormalization group analysis of the random field ising model the renormalization flows to a zero temperature critical point from which we calculate independently three critical exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality expanding analytically in we find that and the distance from the upper bound of the equality go to zero exponentially with

three dimensional coons macroelements application to eigenvalue and scalar wave propagation problems this paper discusses the matter of using higher order approximations in three dimensional problems through coons macroelements recently we have proposed a global functional set based on coons interpolation formula for the construction of large two dimensional macroelements with degrees of freedom appearing at the boundaries only of the domain after successive application in many engineering problems this paper extends the methodology to large three dimensional hexahedral macroelements with the degrees of freedom appearing at the 12 edges of the entire domain in case of smooth box like structures closed form expressions of the global shape functions are presented for the first time it is shown that these global shape functions can be automatically constructed in a systematic way by arbitrarily choosing univariate approximations such as piecewise linear cubic b splines lagrange polynomials etc along the control lines moreover the mechanism of adding facial and internal nodes is presented relationships with higher order p methods are discussed following to excellent results previously derived for the solution of the laplace equation as well as static and eigenvalue extraction analysis of structures the paper investigates the performance of coons macroelements in 3 d eigenvalue and scalar wave propagation problems by implementing standard time integration schemes copyright 2005 john wiley sons ltd

an explicit formula for lanczos polynomials abstract the lanczos algorithm for tridiagonalizing a given matrix a generates a sequence of approximating matrices a p that can naturally be obtained as restrictions to subspaces the eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of a in this paper we produce explicit formulas for the characteristics polynomials of the a p in terms of the eigenvalues of a these formulas can be used to explain heuristically why these approximations are often quite good at present we have no concrete analytic argument that explains the quality of the approximation the main result draws on the formal relationship between the lanczos algorithm and pade approximations to the moment generating function of a this result is one of the few analytic results available for the unsymmetric lanczos algorithm

some exact inequalities for the random field ϕ4 system exact lower bound on the correlation function and susceptibility are obtained in terms of the quantity r r0 12u 2 d ddl l l where l l are the exact averages further lower bounds are obtained on r relating it to an exactly soluble model

effect of the cyber 205 on the choice of method for solving the eigenvalue problem a λm x 0 abstract we consider the generalized eigenvalue problem a m x 0 where a and m are large sparse symmetric matrices for large problems finding only a few eigenpairs involves a major computational task in a typical example from structural dynamic analysis with matrices of order 8000 o 10 9 operations are required to compute 50 eigenpairs it is therefore interesting to examine the advantage that vector computers such as cyber 205 can offer we adopted our best versions of the subspace iteration method and the simple lanczos method in order to take advantage of the special vector processor of the cyber 205 both techniques lend themselves to vectorization our extensive comparisons support the following general statements both methods require the triangular factorization of the same large n by n matrix this factorization dominates the total computation as n provided that the number of wanted eigenpairs p remains fixed independent of n however simple lanczos is at least an order of magnitude more efficient in cpu time for the remainder of the computation for p 40 n 500 the factorization time is not important and the full order of magnitude difference is seen in the total cpu time when p 40 n 8000 simple lanczos is only 4 times faster than subspace iteration on the cyber 205 this confirms experience on serial computers for problems that cannot fit into primary storage input output becomes increasingly important we found that the cost of input output dominated over the cpu cost for a problem that required twice the available primary storage on our cyber 205 however this will depend on the billing algorithm of the computer center we conclude that problems which have a substantial overhead in reading and writing the matrices should not be solved by the simple lanczos method but by a block lanczos method

tchebychev acceleration technique for large scale nonsymmetric matrices the acceleration by tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed a simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue new criteria are established to identify the optimal ellipse of the eigenspectrum the algorithm is fast reliable and does not require a search for all possible ellipses which enclose the spectrum the procedure is applicable to nonsymmetric linear systems as well

lack of self averaging in random systems liability or asset abstract the study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings provided that the system is large enough careful examination suggests that this idea might have a flaw when the correlation length becomes of the order of the size of the system we find that certain bounded quantities are not self averaging when the correlation length becomes of the order of the size of the system this suggests that the lack of self averaging expressed in terms of properly chosen signal to noise ratios may serve to identify phase boundaries this is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase of the random field ising system and compare the findings with more traditional measures

a variational basis for error analysis in finite element elastodynamic problems in this paper the variational basis for finite element analysis of elastodynamic problems has been examined using the principle of virtual work using the principle of virtual work two fundamental important theorems on errors in variationally correct formulations in computational elastodynamics have been discussed and illustrated with simple one dimensional elements a geometric interpretation of the behavior of these errors in approximate solutions from a variationally correct formulation has been presented using the frequency error hyperboloid it has been shown that derivation of a complete and accurate mathematical description of the nature of errors in free vibration analysis involves a simultaneous consideration of errors in both displacement and strains this is in sharp contrast to the error analysis in elastostatic problems where the variational basis involves only strains furthermore it has been observed that variationally correct and conforming formulations satisfy the projection theorems that result from the weak forms of elastodynamic problems by virtue of the virtual work principle these formulations involve consistent mass matrices and yield eigenfrequencies that are always higher than the analytical values independent of domain discretisation this is not necessarily true for variationally incorrect lumped mass formulations in which no guarantee of the boundedness of the eigenvalues with respect to the exact ones can be given with the help of sweep tests it has been demonstrated that the computed eigenvalues with lumped mass formulations can be higher than equal to or lower than the exact values depending on the finite element mesh

on the boundedness of pseudo differential operators on triebel lizorkin and besov spaces abstract in this work we show endpoint boundedness properties of pseudo differential operators of type 0 1 on triebel lizorkin and besov spaces our results are sharp and they also cover operators defined by compound symbols

complex interpolation and fourier multipliers for the spacesb p q s andf p q s of besov hardy sobolev type the case 0 p 0 q

scoza for continuous spins abstract our earlier work applying the self consistent ornstein zernike approach scoza to the d vector model is extended to a more general class of continuous spin models for which the constraint of a single spin magnitude is relaxed

a self consistent ornstein zernike approximation for the random field ising model we extend the self consistent ornstein zernike approximation scoza first formulated in the context of liquid state theory to the study of the random field ising model within the replica formalism we treat the quenched random field just as another spin variable thereby avoiding the usual average over the random field distribution this allows us to study the influence of the distribution on the phase diagram in finite dimensions the thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization temperature and disorder strength as independent variables a preliminary analysis based on high temperature and 1 d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength no sharp transition however occurs for d 4 for the bimodal distribution we find a tricritical point which moves to weaker fields as the dimension is reduced for the gaussian distribution a tricritical point may appear for d around 4

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

a new splitting to solve a large hermitian eigenproblem abstract a particular case of rohe s convergent splitting is presented which gives a rapidly converging method of estimating the lowest eigenvalues and corresponding vectors of a large hermitian matrix when a leading principal minor of that matrix provides a good approximation to the desired eigensolution this is particularly relevant for example to the total energy calculations of solids

replica symmetry breaking in the critical behaviour of the random ferromagnet we study the critical properties of the weakly disordered p component random heisenberg ferromagnet it is shown that if the specific heat critical exponent of the pure system is positive the traditional renormalization group rg flows at dimensions d 4 epsilon which are usually considered as describing the disorder induced universal critical behaviour are unstable with respect to replica symmetry breaking rsb potentials such as those found in spin glasses it is demonstrated that the rg flows involving rsb potentials lead to fixed points which have a structure known as the one step rsb and there exists a whole spectrum of such fixed points it is argued that spontaneous rsb can occur due to the interactions of the fluctuating fields with the local non perturbative degrees of freedom coming from the multiple local minima solutions of the mean field equations however it is not clear whether or not rsb occurs for infinitesimally weak disorder physical consequences of these conclusions are discussed

how to implement the spectral transformation the general linear eigenvalue equations h x m z 0 where h and m are real symmetric matrices with m positive semidefinite must be transformed if the lanczos algorithm is to be used to compute eigenpairs x z when the matrices are large and sparse but not diagonal some factorization must be performed as part of the transformation step if we are interested in only a few eigenvalues x near a specified shift then the spectral transformation of ericsson and ruhe 1 proved itself much superior to traditional methods of reduction the purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it it arises in the use of inverse iteration a second goal is to introduce a systematic modification of the computed ritz vectors which improves the accuracy when m is ill conditioned or singular we confine our attention to the simple lanczos algorithm although the first two sections apply directly to the block algorithms as well 1 overview this contribution is an addendum to the paper by ericsson and ruhe 1 and also 7 the value of the spectral transformation is reiterated in a later section here we outline our implementation of this transformation the equation to be solved for an eigenvalue a and eigenvector z is 1 h xm z 0 h and m are real symmetric n x n matrices and m is positive semidefinite a practical instance of 1 occurs in dynamic analysis of structures where h and m are the stiffness and mass matrices respectively we assume that a linear combination of h and m is positive definite it then follows that all eigenvalues x are real in addition one has a real scalar a distinct from any eigenvalue and we seek a few eigenvalues a close to a together with their eigenvectors z ericsson and ruhe replace 1 by a standard eigenvalue equation 2 c h am ct v y 0 where c is the choleski factor of m m ctc and y cz if m is singular then so is c but fortunately the eigenvector z can be recovered from y via z h am icty of course there is no intention to invert h a m explicitly the received may 14 1984 revised december 20 1985 1980 mathematics subject classification primary 65f15 this research was supported in part by the afosr contract f49620 84 c 0090 the third author was also supported in part by the swedish natural science research council the paper was written while this author was visiting the center for pure and applied mathematics university of california berkeley california 94720 1987 american mathematical society 0025 5718 87 1 00 25 per page

an adaptive procedure for eigenvalue problems using the hierarchical finite element method an adaptive procedure for the solution of the generalized linear eigenvalue problem within the hierarchical finite element method is described the problem of finding for a given discretization an upper limit eigenvalue that is accurate within a prescribed tolerance is especially studied an error estimator is presented and a recomputational scheme for improved solutions is proposed a numerical example is included

on the inclusion principle for the hierarchical finite element method the inclusion principle provides a qualitative characterization of the eigenvalues of a matrix the principle has been shown to apply to systems described by a single hermitian matrix the most important of which being the real symmetric matrix self adjoint distributed systems when discretized by either the classical rayleigh ritz method or by the finite element method lead to algebraic eigenvalue problems described in terms of two real symmetric matrices the algebraic eigenvalues problem derived by the classical rayleigh ritz method possesses the embedding feature required by the inclusion principle but that derived by the finite element method in general does not this paper demonstrates that the inclusion principle can be extended to discretized systems derived by the hierarchical finite element method

efficient learning in boltzmann machines using linear response theory the learning process in boltzmann machines is computationally very expensive the computational complexity of the exact algorithm is exponential in the number of neurons we present a new approximate learning algorithm for boltzmann machines based on mean field theory and the linear response theorem the computational complexity of the algorithm is cubic in the number of neurons in the absence of hidden units we show how the weights can be directly computed from the fixed point equation of the learning rules thus in this case we do not need to use a gradient descent procedure for the learning process we show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role finally we apply the method to a pattern completion task and show good performance for networks up to 100 neurons

high temperature series expansions for the q state potts model on a hypercubic lattice and critical properties of percolation we present results for the high temperature series expansions of the susceptibility and free energy of the q state potts model on a d dimensional hypercubic lattice zd for arbitrary values of q the series are up to order 20 for dimension d 1 limit of these series we estimate the percolation threshold pc and critical exponent gamma for bond percolation in different dimensions we also extend the 1 d expansion of the critical coupling for arbitrary values of q up to order d 9

free vibration analysis of multilayered composite plates and shells with the natural approach abstract natural curved triangular shell element natcop developed in 1 is employed to vibration mode analysis of multilayered composite plates and shells the shells can be either symmetric or unsymmetric the spectral transformation lanczos method with selective or full orthogonalization is used to solve the eigenvalue problem of a pencil k m proper shifting of the eigenvalues is essential for the success of this method certain problems associated with the shift are discussed several numerical examples including symmetric and unsymmetric composite plates and unsymmetric composite conical shells are tested and show that element natcop is also efficient and reliable for free vibration analysis

mean field method with correlations determined by linear response we introduce a mean field approximation based on the reconciliation of maximum entropy and linear response for correlations in the cluster variation method within a general formalism that includes previous mean field methods we derive formulas improving on e g the bethe approximation and the sessak monasson result at high temperature applying the method to direct and inverse ising problems we find improvements over standard implementations

perturbative corrections for approximate inference in gaussian latent variable models expectation propagation ep provides a framework for approximate inference when the model under consideration is over a latent gaussian field with the approximation being gaussian we show how these approximations can systematically be corrected a perturbative expansion is made of the exact but intractable correction and can be applied to the model s partition function and other moments of interest the correction is expressed over the higher order cumulants which are neglected by ep s local matching of moments through the expansion we see that ep is correct to first order by considering higher orders corrections of increasing polynomial complexity can be applied to the approximation the second order provides a correction in quadratic time which we apply to an array of gaussian process and ising models the corrections generalize to arbitrarily complex approximating families which we illustrate on tree structured ising model approximations furthermore they provide a polynomial time assessment of the approximation error we also provide both theoretical and practical insights on the exactness of the ep solution

thermodynamically self consistent theory for the blume capel model we use a self consistent ornstein zernike approximation to study the blume capel ferromagnet on three dimensional lattices the correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations the theory provides a comprehensive and accurate description of the phase diagram in all regions including the wing boundaries in a nonzero magnetic field in particular the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion numerical and analytical analysis strongly suggest that the theory predicts a universal ising like critical behavior along the lambda line and the wing critical lines and a tricritical behavior governed by mean field exponents

scoza critical exponents and scaling in three dimensions abstract the critical behavior of a self consistent ornstein zernike approach scoza that describes the pair correlation function and thermodynamics of a classical fluid lattice gas or ising model is analyzed in three dimensions below the critical temperature complementing our earlier analysis of the supercritical behavior the scoza subcritical exponents describing the coexistence curve susceptibility compressibility and specific heat are obtained analytically 7 20 7 5 1 10 these are in remarkable agreement with the exact values 0 326 1 24 0 11 considering that the scoza uses no renormalization group concepts the scaling behavior that describes the singular parts of the thermodynamic functions as the critical point is approached is also analyzed the subcritical scaling behavior in the scoza is somewhat less simple than that expected in an exact theory involving two scaling functions rather than one

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

a comparison of lanczos and optimization methods in the partial solution of sparse symmetric eigenproblems in the present paper we analyse the computational performance of the lanczos method and a recent optimization technique for the calculation of the p p 40 leftmost eigenpairs of generalized symmetric eigenproblems arising from the finite element integration of elliptic pdes the accelerated conjugate gradient method is used to minimize successive rayleigh quotients defined in deflated subspaces of decreasing size the pointwise lanczos scheme is employed in combination with both the cholesky factorization of the stiffness matrix and the preconditioned conjugate gradient method for evaluating the recursive lanczos vectors the three algorithms are applied to five sample problems of varying size up to almost 5000 the numerical results show that the lanczos approach with cholesky triangularization is generally faster up to a factor of 5 for small to moderately large matrices while the optimization method is superior for large problems in terms of both storage requirement and cpu time in the large case the lanczos cholesky scheme may be very expensive to run even on modern quite powerful computers

real space renormalization group approach to the random field ising model the authors discuss results from two types of real space renormalization group rsrg calculations applied to the random field ising model in three dimensions starting from a lattice of size l the rsrg is used to reduce the lattice to a size l 2 on which the trace is done exactly in this way thermodynamic properties such as the magnetization and susceptibility can be determined approximately they find that for a given size the susceptibility increases as the temperature t is reduced down to the transition temperature tc and becomes essentially independent of temperature below tc both in the vicinity of tc and at lower temperatures there are large sample to sample fluctuations in the susceptibility which grow with increasing system size they interpret these results in terms of the droplet theory of the transition