thermodynamic model of social influence on two dimensional square lattice case for two features abstract we propose a thermodynamic multi state spin model in order to describe equilibrial behavior of a society our model is inspired by the axelrod model used in social network studies in the framework of the statistical mechanics language we analyze phase transitions of our model in which the spin interaction j is interpreted as a mutual communication among individuals forming a society the thermal fluctuations introduce a noise t into the communication which suppresses long range correlations below a certain phase transition point t t large scale clusters of the individuals who share a specific dominant property are formed the measure of the cluster sizes is an order parameter after spontaneous symmetry breaking by means of the corner transfer matrix renormalization group algorithm we treat our model in the thermodynamic limit and classify the phase transitions with respect to inherent degrees of freedom each individual is chosen to possess two independent features f 2 and each feature can assume one of q traits e g interests hence each individual is described by q 2 degrees of freedom a single first order phase transition is detected in our model if q 2 whereas two distinct continuous phase transitions are found if q 2 only evaluating the free energy order parameters specific heat and the entanglement von neumann entropy we classify the phase transitions t t q in detail the permanent existence of the ordered phase the large scale cluster formation with a non zero order parameter is conjectured below a non zero transition point t t q 0 5 in the asymptotic regime q

convergence analysis of the extended krylov subspace method for the lyapunov equation the extended krylov subspace method has recently arisen as a competitive method for solving large scale lyapunov equations using the theoretical framework of orthogonal rational functions in this paper we provide a general a priori error estimate when the known term has rank one special cases such as symmetric coefficient matrix are also treated numerical experiments confirm the proved theoretical assertions

quasi kernel polynomials and their use in non hermitian matrix iterations abstract some of the most efficient iterative algorithms for large sparse hermitian matrix computations are based on orthogonal or kernel polynomials for the case of non hermitian matrices methods based on orthogonal or kernel polynomials are less satisfactory in that the resulting algorithms involve long recurrences consequently it is usually too expensive to run the full algorithms and restarts are necessary a typical example is the generalized minimal residual method gmres for solving non hermitian linear systems where work and storage per iteration grow linearly with the iteration number recently two quasi minimal residual methods qmr for solving non hermitian linear systems have been proposed which unlike gmres are based on short recurrences and hence can be used as true iterative schemes without restarts in this paper the concept of quasi kernel polynomials is introduced some general theory for quasi kernel polynomials is developed such as recurrence relations and a characterization of roots of quasi kernel polynomials as generalized eigenvalues it is pointed out that the qmr approaches are based on two particular instances of quasi kernel polynomials also the use of quasi kernel polynomials for approximating eigenvalues or pseudospectra of large sparse non hermitian matrices is briefly discussed

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

solution of large scale evolutionary problems using rational krylov subspaces with optimized shifts we consider the computation of u t exp ta varphi using rational krylov subspace reduction for 0 le t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n the objective of this work is the optimization of the shifts for the rational krylov subspace rks we consider this problem in the frequency domain and reduce it to a classical zolotaryov problem the latter yields an asymtotically optimal solution with real shifts we also construct an infinite sequence of shifts yielding a nested sequence of the rkss with the same optimal cauchy hadamard convergence rate the effectiveness of the developed approach is demonstrated on an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration

analysis of the rational krylov subspace and adi methods for solving the lyapunov equation for large scale problems an effective approach for solving the algebraic lyapunov equation consists of projecting the problem onto a significantly smaller space and then solving the reduced order matrix equation although krylov subspaces have been used for a long time only more recent developments have shown that rational krylov subspaces can be a competitive alternative to the classical and very popular alternating direction implicit adi recurrence in this paper we develop a convergence analysis of the rational krylov subspace method rksm based on the kronecker product formulation and on potential theory moreover we propose new enlightening relations between this approach and the adi method our results provide solid theoretical ground for recent numerical evidence of the superiority of rksm over adi when the involved parameters cannot be computed optimally as is the case in many practical application problems

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

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

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

an error analysis for rational galerkin projection applied to the sylvester equation in this paper we suggest a new formula for the residual of galerkin projection onto rational krylov spaces applied to a sylvester equation and establish a relation to three different underlying extremal problems for rational functions these extremal problems enable us to compare the size of the residual for the above method with that obtained by adi in addition we deduce several new a priori error estimates for galerkin projection onto rational krylov spaces both for the sylvester and for the lyapunov equation

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

efficient estimation of covariance selection models summary a bayesian method is proposed for estimating an inverse covariance matrix from gaussian data the method is based on a prior that allows the off diagonal elements of the inverse covariance matrix to be zero and in many applications results in a parsimonious parameterisation of the covariance matrix no assumption is made about the structure of the corresponding graphical model so the method applies to both nondecomposable and decomposable graphs all the parameters are estimated by model averaging using an efficient metropolis hastings sampling scheme a simulation study demonstrates that the method produces statistically efficient estimators of the covariance matrix when the inverse covariance matrix is sparse the methodology is illustrated by applying it to three examples that are high dimensional relative to the sample size

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

an interpolatory approximation of the matrix exponential based on faber polynomials in this paper we introduce a method for the approximation of the matrix exponential obtained by interpolation in zeros of faber polynomials in particular we relate this computation to the solution of linear ivps numerical examples arising from practical problems are examined

bayesian semiparametric regression models for evaluating pathway effects on continuous and binary clinical outcomes many statistical methods for microarray data analysis consider one gene at a time and they may miss subtle changes at the single gene level this limitation may be overcome by considering a set of genes simultaneously where the gene sets are derived from prior biological knowledge limited work has been carried out in the regression setting to study the effects of clinical covariates and expression levels of genes in a pathway either on a continuous or on a binary clinical outcome hence we propose a bayesian approach for identifying pathways related to both types of outcomes we compare our bayesian approaches with a likelihood based approach that was developed by relating a least squares kernel machine for nonparametric pathway effect with a restricted maximum likelihood for variance components unlike the likelihood based approach the bayesian approach allows us to directly estimate all parameters and pathway effects it can incorporate prior knowledge into bayesian hierarchical model formulation and makes inference by using the posterior samples without asymptotic theory we consider several kernels gaussian polynomial and neural network kernels to characterize gene expression effects in a pathway on clinical outcomes our simulation results suggest that the bayesian approach has more accurate coverage probability than the likelihood based approach and this is especially so when the sample size is small compared with the number of genes being studied in a pathway we demonstrate the usefulness of our approaches through its applications to a type ii diabetes mellitus data set our approaches can also be applied to other settings where a large number of strongly correlated predictors are present

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

approximation of functions of large matrices with kronecker structure we consider the numerical approximation of f mathcal a b f a b where b in mathbb r n b rn and mathcal a a is the sum of kronecker products that is mathcal a m 2 otimes i i otimes m 1 in mathbb r n times n a m2 i i m1 rn n here f is a regular function such that f mathcal a f a is well defined we derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations with special emphasis on the exponential and on completely monotonic functions for which the new procedure becomes particularly advantageous our findings are illustrated by numerical experiments with typical functions used in applications

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

an error analysis of galerkin projection methods for linear systems with tensor product structure recent results on the convergence of a galerkin projection method for the sylvester equation are extended to more general linear systems with tensor product structure in the hermitian positive definite case explicit convergence bounds are derived for galerkin projection based on tensor products of rational krylov subspaces the results can be used to optimize the choice of shifts for these methods numerical experiments demonstrate that the convergence rates predicted by our bounds appear to be sharp

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

majority vote model on hyperbolic lattices we study the critical properties of a nonequilibrium statistical model the majority vote model on heptagonal and dual heptagonal lattices such lattices have the special feature that they only ca

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

computing functions of very large matrices with small tt qtt ranks by quadrature formulas abstract the computation of matrix functions using quadrature formulas and rational approximations of very large structured matrices using tensor trains tt and quantized tensor trains qtt is considered here the focus is on matrices with a small tt qtt rank some analysis of the error produced by the use of the tt qtt representation and the underlying approximation formula used is also provided promising experiments on exponential power mittag leffler and logarithm function of multilevel toeplitz matrices that are among those which generate a low tt qtt rank representation are also provided confirming that the proposed approach is feasible

error estimates for krylov subspace approximations of matrix exponentials the paper develops a better upper estimate of the error for the krylov subspace approximations to the exponential of a large matrix than that given by gallopoulos and saad 1992 this approach is based on chebyshev series approximation

phase transition of clock models on a hyperbolic lattice studied by corner transfer matrix renormalization group method two dimensional ferromagnetic n state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons the lattice lies on the hyperbolic plane with a constant negative scalar curvature we observe the spontaneous magnetization the internal energy and the specific heat at the center of sufficiently large systems where fixed boundary conditions are imposed for the cases n or 3 up to n 30 the model with n 3 which is equivalent to the three state potts model on the hyperbolic lattice exhibits a first order phase transition a mean field like phase transition of second order is observed for the cases n or 4 when n or 5 we observe a schottky type specific heat below the transition temperature where its peak height at low temperatures scales as n 2 from these facts we conclude that the phase transition of the classical xy model deep inside hyperbolic lattices is not of the berezinskii kosterlitz thouless type

computing a alpha log a and related matrix functions by contour integrals new methods are proposed for the numerical evaluation of f mathbf a or f mathbf a b where f mathbf a is a function such as mathbf a 1 2 or log mathbf a with singularities in infty 0 and mathbf a is a matrix with eigenvalues on or near 0 infty the methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving jacobi elliptic functions the convergence is geometric so that the computation of f mathbf a b is typically reduced to one or two dozen linear system solves which can be carried out in parallel

analysis of acceleration strategies for restarted minimal residual methods we provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual mr krylov subspace methods due to restarting we evaluate the popular practice of using nearly invariant subspaces to either augment krylov subspaces or to construct preconditioners which invert on these subspaces in the case where these spaces are exactly invariant the augmentation approach is shown to be superior we further show how a strategy recently introduced by de sturler for truncating the approximation space of an mr method can be interpreted as a controlled loosening of the condition for global mr approximation based on the canonical angles between subspaces for the special case of krylov subspace methods we give a concise derivation of the role of ritz and harmonic ritz values and vectors in the polynomial description of krylov spaces as well as of the use of the implicitly updated arnoldi method for manipulating krylov spaces

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration

convergence analysis of projection methods for the numerical solution of large lyapunov equations the numerical solution of large scale continuous time lyapunov matrix equations is of great importance in many application areas assuming that the coefficient matrix is positive definite but not necessarily symmetric in this paper we analyze the convergence of projection type methods for approximating the solution matrix under suitable hypotheses on the coefficient matrix we provide new asymptotic estimates for the error matrix when a galerkin method is used in a krylov subspace numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed