block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

iterative methods for a fourth order boundary value problem abstract we consider a fourth order nonlinear ordinary differential equation together with two point boundary conditions and provide a priori error estimates on the length of the interval b a so that the picard s iterative method the approximate picard s iterative method and the quasilinear iterative method convergence to the solution of the problem

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

signal to noise ratio gain of a noisy neuron that transmits subthreshold periodic spike trains the transmission properties of an integrate and fire neuron model that transmits coherent subthreshold spike trains in a shot noise environment are investigated by numerical simulation for very weak coherent couplings it is shown that the input output signal to noise ratio snr gain is easier to exceed unity while for stronger coherent couplings it is difficult to observe the snr gain larger than unity at the optimal noise intensity these observations are different from those acquired in the case of continuous noise our analysis further suggests that the larger snr gain in the very weak coherent coupling case should be due to the noise induced resonance it is also shown that there is more possibility of the snr gain above unity for slower periodic spike trains transmitted by the model the results may be useful in understanding the performance of real noisy neurons acting as signal processing elements

gain in stochastic resonance precise numerics versus linear response theory beyond the two mode approximation in the context of the phenomenon of stochastic resonance sr we study the correlation function the signal to noise ratio snr and the ratio of output over input snr i e the gain which is associated to the nonlinear response of a bistable system driven by time periodic forces and white gaussian noise these quantifiers for sr are evaluated using the techniques of linear response theory lrt beyond the usually employed two mode approximation scheme we analytically demonstrate within such an extended lrt description that the gain can indeed not exceed unity we implement an efficient algorithm based on work by greenside and helfand detailed in the appendix to integrate the driven langevin equation over a wide range of parameter values the predictions of lrt are carefully tested against the results obtained from numerical solutions of the corresponding langevin equation over a wide range of parameter values we further present an accurate procedure to evaluate the distinct contributions of the coherent and incoherent parts of the correlation function to the snr and the gain as a main result we show for subthreshold driving that both the correlation function and the snr can deviate substantially from the predictions of lrt and yet the gain can be either larger or smaller than unity in particular we find that the gain can exceed unity in the strongly nonlinear regime which is characterized by weak noise and very slow multifrequency subthreshold input signals with a small duty cycle this latter result is in agreement with recent analog simulation results by gingl et al icnf 2001 edited by g bosman world scientific singapore 2002 pp 545 548 fluct noise lett 1 l181 2001

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

updating constraint preconditioners for kkt systems in quadratic programming via low rank corrections this work focuses on the iterative solution of sequences of kkt linear systems arising in interior point methods applied to large convex quadratic programming problems this task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method constraint preconditioners are very effective in this context nevertheless their computation may be very expensive for large scale problems and resorting to approximations of them may be convenient here we propose a procedure for building inexact constraint preconditioners by updating a seed constraint preconditioner computed for a kkt matrix at a previous interior point iteration these updates are obtained through low rank corrections of the schur complement of the 1 1 block of the seed preconditioner the updated preconditioners are analyzed both theoretically and computationally the results obtained show that our updating procedure coupled with an adaptive strategy f

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

partitioned versus global krylov subspace iterative methods for fe solution of 3 d biot s problem finite element analysis of 3 d biot s consolidation problem needs fast solution of discretized large 2 2 block symmetric indefinite linear systems in this paper partitioned iterative methods and global krylov subspace iterative methods are investigated and compared the partitioned iterative methods considered include stationary partitioned iteration and non stationary prevost s pcg procedure the global krylov subspace methods considered include minres and symmetric qmr sqmr two efficient preconditioners are proposed for global methods numerical experiments based on a pile group problem and simple footing problems with varied soil profiles are carried out numerical results show that when used in conjunction with suitable preconditioners global krylov subspace iterative methods are more promising for large scale computations and further improvement could be possible if significant differences in the solid material properties are addressed in these preconditioned iterative methods 2007 elsevier b v all rights reserved

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

an augmented lagrangian based approach to the oseen problem we describe an effective solver for the discrete oseen problem based on an augmented lagrangian formulation of the corresponding saddle point system the proposed method is a block triangular preconditioner used with a krylov subspace iteration like bicgstab the crucial ingredient is a novel multigrid approach for the 1 1 block which extends a technique introduced by schoberl for elasticity problems to nonsymmetric problems our analysis indicates that this approach results in fast convergence independent of the mesh size and largely insensitive to the viscosity we present experimental evidence for both isop2 p0 and isop2 p1 finite elements in support of our conclusions we also show results of a comparison with two state of the art preconditioners showing the competitiveness of our approach

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

linear transmitter with correlated noises abstract a linear transmitter with correlated gaussian white additive and multiplicative noises and a periodic signal coupled either additively or multiplicatively is considered the correlations have a weak destructive effect in case of the additive signal and a strong constructive effect in case of the multiplicative one analytical results for the linear transmitter with the additive signal agree with those obtained previously by a different approach we also show how to analytically calculate certain expectation values involving exponentials of gaussian processes

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

boundary value problems for higher order integro differential equations where r t x t x t x j t and a is a continuous operator map c l al a c ar a the function f is continuous in all of its arguments the problem l l 1 2 includes several particular cases for example the boundary value problems for the differential equations if a 0 considered in 7 8 and the references therein for the integro differential equations of the volterra type if

an inexact primal dual path following algorithm for convex quadratic sdp we propose primal dual path following mehrotra type predictor corrector methods for solving convex quadratic semidefinite programming qsdp problems of the form min x frac 1 2 x bullet mathcal q x c bullet x mathcal a x b x succeq 0 where mathcal q is a self adjoint positive semidefinite linear operator on mathcal s n b r m and mathcal a is a linear map from mathcal s n to r m at each interior point iteration the search direction is computed from a dense symmetric indefinite linear system called the augmented equation of dimension m n n 1 2 such linear systems are typically very large and can only be solved by iterative methods we propose three classes of preconditioners for the augmented equation and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions numerical experiments on a variety of qsdps with n up to 1600 are performed and the computational results show that our methods are efficient and robust

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

boundary value problems for second order differential equations of sobolev type abstract in this paper we provide necessary and sufficient conditions for the existence and uniqueness of solutions of second order differential equations of sobolev type satisfying some boundary conditions we also give sufficient conditions so that the picard and approximate picard method converge to the unique solution of these boundary value problems

block preconditioners for symmetric indefinite linear systems this paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a krylov subspace method for solving symmetric indefinite linear systems the focus is on large scale real world problems where block approximations are a practical necessity the main illustration is the performance of the block diagonal constrained and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large scale finite element discretization of biot s consolidation equations this system of equations is of fundamental importance to geomechanics numerical studies show that simple diagonal approximations to the 1 1 block k and inexpensive approximations to the schur complement matrix s may not always produce the most spectacular time savings when k is explicitly available but is able to deliver reasonably good results on a consistent basis in addition the block diagonal preconditioner with a negative 2 2 block appears to be reasonably competitive when compared to the more complicated ones these observation are expected to remain valid for coefficient matrices whereby the 1 1 block is sparse diagonally significant a notion weaker than diagonal dominance moderately well conditioned and has a much larger block size than the 2 2 block copyright 2004 john wiley sons ltd

a note on the positive stable block triangular preconditioner for generalized saddle point problems abstract in this paper we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by cao z h cao positive stable block triangular preconditioners for symmetric saddle point problems appl numer math 57 2007 899 910 the intervals containing these eigenvalues of the preconditioned matrix remove the origin which is benefit for further study numerical experiments of a model stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners

a modified ssor preconditioner for sparse symmetric indefinite linear systems of equations the standard ssor preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the biot s consolidation equations in this paper a modified block ssor preconditioner combined with the eisenstat trick implementation is proposed for actual implementation a pointwise variant of this modified block ssor preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness numerical experiments show that the proposed modified ssor preconditioned symmetric qmr solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime moreover the proposed modified ssor preconditioners can be generalized to non symmetric biot s systems copyright 2005 john wiley sons ltd

optimal accelerated sor like asor method for singular symmetric saddle point problems abstract in a recent paper a new iterative method for the solution of the nonsingular symmetric saddle point problem was proposed njeru and guo 2016 the asor method belongs to the family of the sor like methods and uses two parameters and convergence intervals for the parameters involved were found in the present work we analyze and study an extension of the above problem to the singular case and determine optimal values for the two parameters as well as for the semi convergence factor of the asor method numerical results are presented to show the efficiency of the optimal singular asor method

role of fluctuations in the response of coupled bistable units to weak time periodic driving forces we analyze the stochastic response of a finite set of globally coupled noisy bistable units driven by rather weak time periodic forces we focus on the stochastic resonance and phase frequency synchronization of the collective variable defined as the arithmetic mean of the variable characterizing each element of the array for single unit systems stochastic resonance can be understood with the powerful tools of linear response theory proper noise induced phase frequency synchronization for a single unit system in this linear response regime does not exist for coupled arrays our numerical simulations indicate an enhancement of the stochastic resonance effects leading to gains larger than unity as well as genuine phase frequency synchronization the nonmonotonicity of the response with the strength of the coupling strength is investigated comparison with simplifying schemes proposed in the literature to describe the random response of the collective variable is carried out

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

on augmentation block triangular preconditioners for regularized saddle point problems two augmentation block triangular preconditioners were introduced by shen et al 2012 for the regularized saddle point problem however the spectral analysis of the preconditioner based on the augmentation of the 2 2 block was not throughly derived there in this short paper we give a detailed spectral analysis on this preconditioner a numerical example is employed to validate the efficiency of the presented theoretical results

novel preconditioners for the iterative solution to fe discretized coupled consolidation equations a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by an inexact constraint preconditioner icp preserving the same block structure as the native fe matrix the conditioning number of the preconditioned coupled problem depends on the quality of the approximation of the block corresponding to the structural stiffness matrix an efficient algorithm to implement icp into a krylov subspace method is developed numerical tests performed on realistic 3d problems reveal that icp typically outperforms standard ilut preconditioners and proves much more robust in severely ill conditioned problems

performance and robustness of block constraint preconditioners in finite element coupled consolidation problems block constraint preconditioners are a most recent development for the iterative solution to large scale often ill conditioned coupled consolidation problems a major limitation to their practical use however is the somewhat difficult selection of a number of user defined parameters at least 4 in a more or less optimal way the present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters a theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters a number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner mcp combined with the symmetric quasi minimal residual sqmr solver and the mcp triangular variant t mcp combined with the bi conjugate gradient stabilized bi cgstab are efficient and robust tools for the solution to difficult finite element discretized coupled consolidation problems moreover the practical selection of the user defined parameters is relatively easy as a stable behavior is observed for a wide range of fill in degree values the theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination copyright 2009 john wiley sons ltd

signal amplification factor in stochastic resonance an analytic non perturbative approach abstract the response of an overdamped bistable system driven by a gaussian white noise and perturbed by a weak monochromatic force signal is studied analytically in order to get amplitude dependent signal amplification factor a non perturbative scheme is put forward by taking into account all the terms of a perturbation series with amplitude of the signal as an expansion parameter an approximate analytic expression of the signal amplification factor is derived and compared with the numerical results the contributions of infinite number of relaxation modes of the stochastic dynamics to the response are also taken into account in the final expression the calculation of the response based on the derived expression requires only the knowledge of the first non trivial eigenvalue and the lowest eigenfunction of the unperturbed fokker planck operator

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

weakly noisy chaotic scattering received 30 april 2013 published 23 september 2013 the effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest we investigate how a weak source of additive uncorrelated gaussian noise affects both the dynamics and thetopology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regimeof the h enon heiles hamiltonian system we have found long transients for the time escape distributions forcritical values of the noise intensity for which the particles escape slower as compared with the noiseless case an analysis of the survival probability of the scattering function versus the gaussian noise intensity shows asmooth curve with one local maximum and with one local minimum which are related to those long transientsand with the basin structure in phase space on the other hand the computation of the exit basins in phase spaceshows a quadratic curve for which the basin boundaries lose their fractal like structure as noise turned on doi 10 1103 physreve 88 032914 pacs number s 05

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

finite difference methods for boundary value problems of differential equations with deviating arguments abstract we propose a convergent finite difference method to construct an approximate solution of the boundary value problems with deviating arguments the method developed takes into account the continuity limitations of the solutions of these problems to demonstrate the effectiveness of the proposed method numerical evidence is also included

two state markovian theory of input output frequency and phase synchronization abstract a markovian dichotomic system driven by a deterministic time periodic force is analyzed in terms of the statistical properties of the switching events between the states the consideration of the counting process of the switching events leads us to define a discrete phase we obtain expressions for the instantaneous output frequency and phase diffusion associated to the dichotomic process as well as for their cycle averages these expressions are completely determined by the rates of escape from both states they are a convenient starting point for the study of the stochastic frequency and phase synchronization in a wide range of situations both classical and quantum in which two metastable states are involved

an efficient diagonal preconditioner for finite element solution of biot s consolidation equations finite element simulations of very large scale soil structure interaction problems e g excavations tunnelling pile rafts etc typically involve the solution of a very large ill conditioned and indefinite biot system of equations the traditional preconditioned conjugate gradient solver coupled with the standard jacobi sj preconditioner can be very inefficient for this class of problems this paper presents a robust generalized jacobi gj preconditioner that is extremely effective for solving very large scale biot s finite element equations using the symmetric quasi minimal residual method the gj preconditioner can be formed inverted and implemented within an element by element framework as readily as the sj preconditioner it was derived as a diagonal approximation to a theoretical form which can be proven mathematically to possess an attractive eigenvalue clustering property the effectiveness of the gj preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three dimensional footing problem this paper casts a new perspective on the potentialities of the simple diagonal preconditioner which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix copyright 2002 john wiley sons ltd

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

mixed constraint preconditioners for the iterative solution of fe coupled consolidation equations the finite element fe integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix because of ill conditioning the repeated solution in time of the fe equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of krylov subspace methods in the present paper a mixed constraint preconditioner mcp is developed combining implicit and explicit approximations of the inverse of the structural sub matrix with the performance investigated in some representative examples an upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner the mcp is efficiently implemented into a krylov subspace method with the performance obtained in 2d and 3d examples compared to that of inexact constraint preconditioners and least square logarithm scaled ilut preconditioners two variants of mcp t mcp and d mcp developed with the aim at reducing the cost of the preconditioner application are also tested the results show that the mcp variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation fe models and especially so in severely ill conditioned problems

minres qlp a krylov subspace method for indefinite or singular symmetric systems cg symmlq and minres are krylov subspace methods for solving symmetric systems of linear equations when these methods are applied to an incompatible system that is a singular symmetric least squares problem cg could break down and symmlq s solution could explode while minres would give a least squares solution but not necessarily the minimum length pseudoinverse solution this understanding motivates us to design a minres like algorithm to compute minimum length solutions to singular symmetric systems minres uses qr factors of the tridiagonal matrix from the lanczos process where r is upper tridiagonal minres qlp uses a qlp decomposition where rotations on the right reduce r to lower tridiagonal form on ill conditioned systems singular or not minres qlp can give more accurate solutions than minres we derive preconditioned minres qlp new stopping rules and better estimates of the solution and residual norms the matrix norm and the condition number

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

an exactly solvable nonlinear model constructive effects of correlations between gaussian noises abstract a system with two correlated gaussian white noises is analysed this system can describe both stochastic localization and long tails in the stationary distribution correlations between the noises can lead to a nonmonotonic behaviour of the variance as function of the intensity of one of the noises and to a stochastic resonance a method for improving the transmission of external periodic signal by tuning parameters of the system discussed in this paper is proposed

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

stochastic resonance of collective variables in finite sets of interacting identical subsystems we explore stochastic resonance effects in the response of a complex stochastic system formed by a finite number of interacting identical subunits driven by a time periodic force the driving force alone cannot induce sustained oscillations between the different attractors of the dynamics in the absence of noise we focus on a global stochastic variable defined as the arithmetic mean of the relevant stochastic variable of each subunit we construct numerical approximations to its first two long time cumulant moments and its long time correlation function we also compute the output signal to noise ratio and the stochastic resonance gain for a wide range of parameter values and several types of driving forces the coupling between the subsystems leads within adequate ranges of the parameter values to global outputs with very large signal to noise ratios we have also observed gains larger than unity in the global response to subthreshold sinusoidal driving forces

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

subthreshold stochastic resonance rectangular signals can cause anomalous large gains the main objective of this work is to explore aspects of stochastic resonance sr in noisy bistable symmetric systems driven by subthreshold periodic rectangular external signals possessing a large duty cycle of unity using a precise numerical solution of the langevin equation we carry out a detailed analysis of the behavior of the first two cumulant averages the correlation function and its coherent and incoherent parts we also depict the nonmonotonic behavior versus the noise strength of several sr quantifiers such as the average output amplitude i e the spectral amplification the signal to noise ratio and the sr gain in particular we find that with subthreshold amplitudes and for an appropriate duration of the pulses of the driving force the phenomenon of stochastic resonance is accompanied by sr gains exceeding unity this analysis thus sheds light on the interplay between nonlinearity and the nonlinear response which in turn yields nontrivial unexpected sr gains above unity

theory of frequency and phase synchronization in a rocked bistable stochastic system we investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked overdamped bistable potential driven by white gaussian noise the archetype description of stochastic resonance we present an approach to the stochastic counting process of noise induced switching events starting from the markovian dynamics of the nonstationary continuous particle dynamics one finds upon contraction onto two states a non markovian renewal dynamics a proper definition of an output discrete phase is given and the time rate of change of its noise average determines the corresponding output frequency the phenomenon of noise assisted phase synchronization is investigated in terms of an effective instantaneous phase diffusion the theory is applied to rectangular shaped rocking signals versus increasing input noise strengths in this case for an appropriate choice of the parameter values the system exhibits a noise induced frequency locking accompanied by a very pronounced suppression of the phase diffusion of the output signal precise numerical simulations corroborate very favorably our analytical results the novel theoretical findings are also compared with prior ones

parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics we present new results on the relaxed dimensional factorization rdf preconditioner for solving saddle point problems from incompressible flow simulations first introduced in benzi et al 2011 this method contains a parameter 0 0 to be chosen by the user previous works provided an estimate of in the 2d case using local fourier analysis a novel algebraic estimation technique for finding a suitable value of the rdf parameter in both the 2d and the 3d case with arbitrary geometries is proposed this technique is tested on a variety of discrete saddle point problems arising from the approximation of the navier stokes equations using a marker and cell scheme and a finite element one we also show results for a large scale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores

stochastic resonance in biological nonlinear evolution models we investigate stochastic resonance in the nonlinear one dimensional fisher eigen model fem which represents an archetypal model for biological evolution based on a global coupling scheme in doing so we consider different periodically driven fitness functions which govern the evolution of a biological phenotype population for the case of a simple harmonic fitness function we are able to derive the exact analytic solution for the asymptotic probability density a distinct feature of this solution is a phase lag between the driving signal and the linear response of the system furthermore for more complex systems a general perturbation theory linear response approximation is put forward using the latter approach we investigate stochastic resonance in terms of the spectral amplification measure for a quadratic a quartic single peaked and for a bistable fitness function our analytical results are also compared with those of detailed numerical simulations our findings vindicate that stochastic resonance does occur in these nonlinear globally coupled biological systems

spectral properties of a class of matrix splitting preconditioners for saddle point problems based on the accelerated hermitian and skew hermitian splitting iteration scheme bai and golub 2007 we propose a new two parameter matrix splitting preconditioner in this paper spectral properties of the preconditioned matrix are analyzed in detail furthermore based on this preconditioner an improved version of matrix splitting preconditioner is presented and analyzed finally performance of the preconditioners is compared by using gmres m as an iterative solver on linear systems arising from the discretization of stokes and navier stokes equations

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

theory of stochastic resonance in signal transmission by static nonlinear systems a deterministic periodic signal plus a stationary random noise is applied to a static nonlinearity taking the form of a monovariable arbitrary function on real numbers the property of noise enhanced signal transmission through stochastic resonance is studied for this class of static nonlinear systems a theory is developed that provides expressions for the output autocorrelation function power spectral density signal to noise ratio and input output phase shift in the presence of a periodic input a noise distribution and a static nonlinearity all three being arbitrary both white and colored input noises are successively considered for white input noise exact expressions are derived in a discrete time framework directly confrontable to simulations or experiments the theory is applied to describe stochastic resonance in various examples of static nonlinear systems for instance a diode nonlinearity in addition confrontations with experiments and simulations are given that support the theory in particular interesting effects are reported such as a signal to noise ratio larger at the output than at the input or stochastic resonance at zero frequency finally the validity of the theory is extended to dynamic nonlinear systems that can be decomposed into a static nonlinearity followed by an arbitrary dynamic linear system s1063 651x 97 12002 5

stochastic resonance in a periodically driven linear system with multiplicative and periodically modulated additive white noises abstract the stochastic resonance sr of periodically driven linear system with multiplicative white noise and periodically modulated additive white noise is studied exact expressions of output signal to noise ratio are obtained the sr characteristics features are shown in two different situations the driven frequency is equal to the modulative frequency 1 and the driven frequency is not equal to the modulative frequency 1 the result was as follows the exact expression of output signal to noise ratio snr exhibits sr in three aspects in the case of 1 1 the curve of snr versus noise strength q exists a resonance peak this is a standard form of sr 2 a resonance peak exists in the snr vs curve in the region of 0 3 a maximum appears in the curve of snr as a function of the characteristic of signal itself the signal frequency it is of interest to examine the snr in the case of 1 from the exact expression of snr we see for 1 that the snr vs q curve exhibits a minimum firstly then a maximum that is to say it exhibits a suppression firstly resonance later and is very different from the case of 1

simple like preconditioners for saddle point problems from the steady navier stokes equations the semi implicit method for pressure linked equations simple preconditioner is further generalized to a class of simple like sl preconditioners for solving saddle point problems from the steady navier stokes equations the sl preconditioners can be also viewed as a generalization of the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 2015 convergence analysis of the corresponding sl iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the sl iteration matrix moreover krylov subspace acceleration of the sl preconditioning is studied the sl preconditioned saddle point matrix is analyzed results about eigenvalue and eigenvector distributions and the minimal polynomial are derived numerical experiments from leaky two dimensional lid driven cavity problems are given which show the advantages of the sl preconditioned gmres over the dpss and rdpss preconditioned ones for solving saddle point problems

stochastic resonance induced by bounded noise and periodic signal in an asymmetric bistable system abstract stochastic resonance sr induced by the bounded noise in an asymmetric bistable system is investigated based on stochastic simulation the signal power amplification s p a is derived for the case of the additive modulated signal the simulation results indicate that 1 the sr phenomenon is weakened by the asymmetry of the system 2 increasing the frequency restricts the sr of the system 3 there is an optimum value of amplitude a matching the strongest sr phenomenon