superior properties of the presb preconditioner for operators on two by two block form with square blocks matrices or operators in two by two block form with square blocks arise in numerous important applications such as in optimal control problems for pdes the problems are normally of very large scale so iterative solution methods must be used thereby the choice of an efficient and robust preconditioner is of crucial importance since some time a very efficient preconditioner the preconditioned square block presb method has been used by the authors and coauthors in various applications in particular for optimal control problems for pdes it has been shown to have excellent properties such as a very fast and robust rate of convergence that outperforms other methods in this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs under certain conditions the condition number of the preconditioned matrix is bounded by 2 or even smaller furthermore under certain assumptions the rate of convergence is superlinear

preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

superlinearly convergent cg methods via equivalent preconditioning for nonsymmetric elliptic operators the convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators with focus on elliptic convection diffusion operators in sobolev space superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a hilbert space then the same convergence result is verified for elliptic convection diffusion equations using different preconditioning operators the convergence factor involves the eigenvalues of the corresponding operators for which an estimate is also given the above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection diffusion problems

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

statistical properties of the indonesian stock exchange index abstract using the tools developed for statistical physics we have analyzed statistical properties of the indonesian stock exchange index ihsg in spite of the small number of available data used in the analysis the result still shows the universal behavior of complex systems previously found in the leading stock indices we also found that the fluctuation of the index return becomes more random after the crisis

mesh independent superlinear pcg rates via compact equivalent operators the subject of the paper is the mesh independent convergence of the preconditioned conjugate gradient pcg method for nonsymmetric elliptic problems the approach of equivalent operators is involved in which one uses the discretization of another suitable elliptic operator as preconditioning matrix by introducing the notion of compact equivalent operators it is proved that for a wide class of elliptic problems the superlinear convergence of the obtained pcg method is mesh independent under finite element discretizations that is the rate of superlinear convergence is given in the form of a sequence which is mesh independent and is determined only by the elliptic operators

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

newton ll method for the second order semi linear elliptic partial differential equations newton s method with first order system least squares fosls finite element method has been widely used to approximately solve a system of nonlinear partial differential equations adler et al 2010 9 codd et al 2003 10 manteuffel et al 2006 11 in this paper we propose to use the first order system ll method to find a correction in each newton s iteration which provides an l2 approximation of the second order semi linear elliptic partial differential equations while the typical newton fosls method providesh1 approximations the numerical tests have been conducted to validate the theory

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

preconditioning nonconforming finite element methods for treating dirichlet boundary conditions ii this work deals with theh 1 condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are finite element discretizations of second order elliptic operators a andb respectively b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval goldstein also has derived results on the spectral distribution ofb h 1 a h using a different approach as a generalization of the results of parter and wong the current work includes nonconforming finite element methods which deal with dirichlet boundary conditions it will be shown that in this more general setting the singular values also cluster in a positive finite interval in particular if the leading part ofb is the same as the leading part ofa then the singular values cluster about the point 1 two specific methods are given as applications of this theory they are the penalty method of babuska and the method of nearly zero boundary conditions of nitsche finally it will be shown that the same results can be proven by an approach generalized from the work of goldstein

generation of stable limit cycles in controllable linear systems in this paper we prove that any controllable linear systems x ax bu x n u m admits a polynomial feedback u u x such that the closed loop system x ax bu x admits an orbitally asymptotically stable limit cycle moreover we prove that for any positive integer n there exists an nth order polynomial autonomous ordinary differential equation with a unique limit cycle

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

preconditioningp1 nonconforming finite elements condition numbers and singular value distributions this work deals with the condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are discretizations of second order elliptic operatorsa andb usingp 1 nonconforming finite elements of crouzeix and raviart b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval these reults are being extended to the aforementioned nonconforming finite elements it will be shown that for quasiuniform grids theb h singular values are bounded above and below by positive constants which are independent of the grid sizeh moreover they also cluster in a smaller but usually estimable interval issues of implementation are also discussed

preconditioning isoparametric finite element methods taking into account numerical integration this paper is concerned with the h 1 condition numbers and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h and b h are isoparametric finite element discretizations of second order elliptic operators a and b by taking into account numerical integration respectively where a is nonselfad joint and possibly indefinite and b is selfadjoint and positive definite it is proved that the h 1 condition numbers of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally the implementation of b h 1 is discussed

superlinearly convergent pcg algorithms for some nonsymmetric elliptic systems a preconditioned conjugate gradient method is applied to finite element discretizations of some nonsymmetric elliptic systems mesh independent superlinear convergence is proved which is an extension of a similar earlier result from a single equation to systems the proposed preconditioning method involves decoupled preconditioners which yields small and parallelizable auxiliary problems

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

an analysis of equivalent operator preconditioning for equation free newton krylov methods we consider the computation of a fixed point of a time stepper using newton krylov methods and propose and analyze equivalent operator preconditioning for the resulting linear systems for a linear scalar advection reaction diffusion equation we investigate in detail how the convergence rate depends upon the choice of preconditioner parameters and upon the time discretization the results are especially valuable when computing fixed points of a coarse time stepper in the equation free multiscale framework in which one simulates an unavailable coarse scale model by wrapping a set of computational routines around appropriately initialized fine scale simulations both analytical results and numerical experiments are presented showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

optimal equivalent preconditioners preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem the technique considered here makes use of a selfadjoint positive definite operator b as the preconditioner for the nonselfadjoint convection diffusion operator a preconditioning by a s leading term plus a positive zeroth order term sigma i and optimizing over sigma is proposed the analysis of a model problem indicates that when a has large first order terms both the spectral and norm condition of b 1 a are minimized by sigma gg 0

statistical properties of share volume traded in financial markets we quantitatively investigate the ideas behind the often expressed adage it takes volume to move stock prices and study the statistical properties of the number of shares traded q deltat for a given stock in a fixed time interval deltat we analyze transaction data for the largest 1000 stocks for the two year period 1994 95 using a database that records every transaction for all securities in three major us stock markets we find that the distribution p q deltat displays a power law decay and that the time correlations in q deltat display long range persistence further we investigate the relation between q deltat and the number of transactions n deltat in a time interval deltat and find that the long range correlations in q deltat are largely due to those of n deltat our results are consistent with the interpretation that the large equal time correlation previously found between q deltat and the absolute value of price change g deltat related to volatility are largely due to n deltat