accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

a preconditioning approach to the pagerank computation problem abstract some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of euler richardson er iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph we choose the preconditioner from an algebra u of matrices thereby obtaining an er u method and we observe that er u can outperform er in terms of rate of convergence the proposed preconditioner can be updated at a very low cost whenever the graph changes as is the case when it represents a generic set of information the particular u utilized requires a surplus of operations per step and memory allocations which make er u superior to er for not too wide graphs however the observed high improvement in convergence rate obtained by preconditioning and the general theory developed are a reason for investigating different choices of u more efficient for huge graphs

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

on the equivalence of dynamically orthogonal and bi orthogonal methods theory and numerical simulations abstract the karhunen loeve kl decomposition provides a low dimensional representation for random fields as it is optimal in the mean square sense although for many stochastic systems of practical interest described by stochastic partial differential equations spdes solutions possess this low dimensional character they also have a strongly time dependent form and to this end a fixed in time basis may not describe the solution in an efficient way motivated by this limitation of standard kl expansion sapsis and lermusiaux 2009 26 developed the dynamically orthogonal do field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty lives but also the stochastic characteristics of uncertainty recently cheng et al 2013 28 introduced an alternative approach the bi orthogonal bo method which performs the exact same tasks i e it evolves the spatial basis and the stochastic characteristics of uncertainty in the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence in the sense that one method is an exact reformulation of the other we show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the bo and the do solutions we also examine a pathology of the bo equations that occurs when two eigenvalues of the solution cross resulting in an instantaneous infinite speed internal rotation of the computed spatial basis we demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the bo approach on the other hand it is observed that the bo is more stable in nonlinear problems involving a relatively large number of modes several examples linear and nonlinear are presented to illustrate the do and bo methods as well as their equivalence

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

an efficient elimination strategy for solving pagerank problems in web link structures similar link distributions often occur especially for pages from same hosts for pagerank problems similar in link distributions of pages result in similar row patterns of the transition matrix we demonstrate that common row patterns in the transition matrix determine identical sub rows thus the identical sub rows with a large proportion of nonzeros can be eliminated to decrease the density of pagerank problems we propose an elimination strategy that exploits such identical sub rows and generates an elimination operator for transferring the problem to an equivalent but more sparse problem numerical experiments are reported to illustrate the effectiveness of this strategy for decreasing the computational cost of solving pagerank problems

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

a low cost goal oriented compact proper orthogonal decomposition basis for model reduction of static systems a novel model reduction technique for static systems is presented the method is developed using a goal oriented framework and it extends the concept of snapshots for proper orthogonal decomposition pod to include sensitivity derivatives of the state with respect to system input parameters the resulting reduced order model generates accurate approximations due to its goal oriented construction and the explicit training of the model for parameter changes the model is less computationally expensive to construct than typical pod approaches since efficient multiple right hand side solvers can be used to compute the sensitivity derivatives the effectiveness of the method is demonstrated on a parameterized aerospace structure problem

a dynamically bi orthogonal method for time dependent stochastic partial differential equations i derivation and algorithms we propose a dynamically bi orthogonal method dybo to solve time dependent stochastic partial differential equations spdes the objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi orthogonal basis it is well known that the karhunen loeve expansion kle minimizes the total mean squared error and gives the sparsest representation of stochastic solutions however the computation of the kl expansion could be quite expensive since we need to form a covariance matrix and solve a large scale eigenvalue problem the main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the kl expansion unlike other reduced model methods our method constructs the reduced basis on the fly without the need to form the covariance matrix or to compute its eigendecomposition in the first part of our paper we introduce the derivation of the dynamically bi orthogonal formulation for spdes discuss several theoretical issues such as the dynamic bi orthogonality preservation and some preliminary error analysis of the dybo method we also give some numerical implementation details of the dybo methods including the representation of stochastic basis and techniques to deal with eigenvalue crossing in the second part of our paper 11 we will present an adaptive strategy to dynamically remove or add modes perform a detailed complexity analysis and discuss various generalizations of this approach an extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the dybo method

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

dual dynamically orthogonal approximation of incompressible navier stokes equations with random boundary conditions abstract in this paper we propose a method for the strong imposition of random dirichlet boundary conditions in the dynamical low rank dlr approximation of parabolic pdes and in particular incompressible navier stokes equations we show that the dlr variational principle can be set in the constrained manifold of all s rank random fields with a prescribed value on the boundary expressed in low rank format with rank smaller then s we characterize the tangent space to the constrained manifold by means of a dual dynamically orthogonal dual do formulation in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions consistent with the original problem the dual do formulation is also convenient to include the incompressibility constraint when dealing with incompressible navier stokes equations we show the performance of the proposed dual do approximation on two numerical test cases the classical benchmark of a laminar flow around a cylinder with random inflow velocity and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from mri data with random inflow conditions coming from doppler measurements

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

new look ahead lanczos type algorithms for linear systems a breakdown due to a division by zero can arise in the algorithms for implementing lanczos method because of the non existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate such a breakdown can be avoided by jumping over the polynomials involved this strategy was already used in some algorithms such as the mrz and its variants in this paper we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors independent of the length of the jump these new algorithms are based on horner s rule and on a different way for computing the coefficients of the recurrence relationships moreover these new algorithms seem to be more stable than the old ones and they provide better numerical results numerical examples and comparisons with other algorithms will be given

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

an observation driven time dependent basis for a reduced description of transient stochastic systems we present a variational principle for the extraction of a time dependent orthonormal basis from random realizations of transient systems the optimality condition of the variational principle leads to a closed form evolution equation for the orthonormal basis and its coefficients the extracted modes are associated with the most transient subspace of the system and they provide a reduced description of the transient dynamics that may be used for reduced order modelling filtering and prediction the presented method is matrix free it relies only on the observables of the system and ignores any information about the underlying system in that sense the presented reduction is purely observation driven and may be applied to systems whose models are not known the presented method has linear computational complexity and memory storage requirement with respect to the number of observables and the number of random realizations therefore it may be used for a large number of observations and samples the effectiveness of the proposed method is tested on four examples i stochastic advection equation ii stochastic burgers equation iii a reduced description of transient instability of kuramoto sivashinsky and iv a transient vertical jet governed by the incompressible navier stokes equation in these examples we contrast the performance of the time dependent basis versus static basis such as proper orthogonal decomposition dynamic mode decomposition and polynomial chaos expansion

a novel kernel fuzzy clustering algorithm for geo demographic analysis we proposed a novel kernel based fuzzy clustering for geo demographic analysis it relied on gaussian kernel function updated elements and standardized weights the algorithm was both theoretical and experimental validated its clustering quality was better than those of other relevant algorithms some properties of solutions were examined geo demographic analysis gda is a major concentration of various interdisciplinary researches and has been used in many decision making processes regarding the provision and distribution of products and services in society machine learning methods namely principal component analysis self organizing map k means fuzzy clustering and fuzzy geographically weighted clustering were proposed to enhance the quality of gda among them the state of the art method modified intuitionistic possibilistic fuzzy geographically weighted clustering mipfgwc has some drawbacks such as i using the euclidean similarity measure often results in high error rate and sensitivity to noises and outliers ii updating the membership matrix by the spatial interaction modification model sim2 model leads to new centers not being geographically aware in this paper we present a novel fuzzy clustering algorithm named as kernel fuzzy geographically clustering kfgc that utilizes both the kernel similarity function and the new update mechanism of the sim2 model to remedy the disadvantages of mipfgwc some supported properties and theorems of kfgc are also examined in the paper specifically the differences between solutions of kfgc and those of mipfgwc and of some variants of kfgc are theoretically validated lastly experimental analysis is performed to compare the performance of kfgc with those of the relevant algorithms in terms of clustering quality

comparison of krylov subspace methods on the pagerank problem pagerank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions the problem incorporates a parameter a that determines the difficulty of the problem in this paper the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of a we see that stationary methods are very reliable and more competitive when the problem is well conditioned that is for small values of a however for large values of the parameter a the problem becomes more difficult and methods such as preconditioned bicgstab or restarted preconditioned gmres become competitive with stationary methods in terms of mflops count as well as in number of iterations necessary to reach convergence

a multi class approach for ranking graph nodes after the phenomenal success of the pagerank algorithm many researchers have extended the pagerank approach to ranking graphs with richer structures in addition to the simple linkage structure indeed in some scenarios we have to deal with networks modeling multi parameters data where each node has additional features and there are important relationships between such features this paper addresses the need of a systematic approach to deal with multi parameter data we propose models and ranking algorithms that can be applied to a large variety of networks bibliographic data patent data twitter and social data healthcare data we focus on several aspects not previously addressed in the literature 1 we propose different models for ranking multi parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models 2 we analyze stability and convergence of the proposed numerical schemes and we derive a fast and stable ranking algorithm 3 we analyze the robustness of our models when data are incomplete the comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60 when just 10 of the links of the attributes are maintained

real time reduced order modeling of stochastic partial differential equations via time dependent subspaces abstract we present a new methodology for the real time reduced order modeling of stochastic partial differential equations called the dynamically bi orthonormal dbo decomposition in this method the stochastic fields are approximated by a low rank decomposition to spatial and stochastic subspaces each of these subspaces is represented by a set of orthonormal time dependent modes we derive exact evolution equations of these time dependent modes and the evolution of the factorization of the reduced covariance matrix we show that dbo is equivalent to the dynamically orthogonal do 1 and bi orthogonal bo 2 decompositions via linear and invertible transformation matrices that connect dbo to do and bo however dbo shows several improvements compared to do and bo i dbo performs better than do and bo for cases with ill conditioned covariance matrix ii in contrast to bo the issue of eigenvalue crossing is not present in the dbo formulation iii in contrast to do the stochastic modes are orthonormal which leads to more accurate representation of the stochastic subspace we study the convergence properties of the method and compare it to the do and bo methods for demonstration we consider three cases i stochastic linear advection equation ii stochastic burgers equation and iii stochastic incompressible flow over a bump in a channel overall we observe improvements in the numerical accuracy of dbo compared against do and bo

approximation of the time dependent electronic schrodinger equation by mctdhf the numerical approximation of the solution of the time dependent schrodinger equation arising in ultrafast laser dynamics is discussed the linear electronic schrodinger equation is reduced to a computationally tractable lower dimensional system of nonlinear partial differential equations by the multi configuration time dependent hartree fock method mctdhf this method uses an ansatz for the wave function on a nonlinear manifold taking into account the antisymmetry inherent in the model to reduce the dimension of the solution space we show that the ansatz used is consistent with the original equation and the solution can be represented exactly in principle using the solutions of the nonlinear pdes associated with mctdhf for the practical solution of the mctdhf equations several numerical techniques are discussed and it is demonstrated that physically relevant problems can be solved satisfactorily

an inner outer iteration for computing pagerank we present a new iterative scheme for pagerank computation the algorithm is applied to the linear system formulation of the problem using inner outer stationary iterations it is simple can be easily implemented and parallelized and requires minimal storage overhead our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved the same idea can be used as a preconditioning technique for nonstationary schemes numerical examples featuring matrices of dimensions exceeding 100 000 000 in sequential and parallel environments demonstrate the merits of our technique our code is available online for viewing and testing along with several large scale examples

setting and analysis of the multi configuration time dependent hartree fock equations in this paper we motivate formulate and analyze the multi configuration time dependent hartree fock mctdhf equations for molecular systems under coulomb interaction they consist in approximating the n particle schrodinger wavefunction by a time dependent linear combination of time dependent slater determinants the equations of motion express as a system of ordinary differential equations for the expansion coefficients coupled to nonlinear schrodinger type equations for mono electronic wavefunctions the invertibility of the one body density matrix full rank hypothesis plays a crucial role in the analysis under the full rank assumption a fiber bundle structure shows up and produces unitary equivalence between convenient representations of the equations we discuss and establish existence and uniqueness of maximal solutions to the cauchy problem in the energy space as long as the density matrix is not singular a sufficient condition in terms of the energy of the initial data ensuring the global in time invertibility is provided first result in this direction regularizing the density matrix breaks down energy conservation however a global well posedness for this system in l 2 is obtained with strichartz estimates eventually solutions to this regularized system are shown to converge to the original one on the time interval when the density matrix is invertible