comparison of several iterative techniques in the solution of symmetric banded equations on a two pipe cyber 205 the effectiveness of several iterative techniques for solving matrix equations resulting from finite difference approximations to self adjoint parabolic and elliptic partial differential equations is investigated the strongly implicit procedure sip the incomplete cholesky conjugate gradient method iccg the vectorized iccg viccg the modified iccg miccg the diagonally scaled conjugate gradient method dscg the polynomial preconditioned conjugate gradient method polcg and the polynomial form of iccg piccg the comparison is made on a vector machine two pipe cyber 205 where vectorization of the code is done primarily by the vector compiler available it is found that of the methods studied polcg and miccg appear to require the least amount of cpu time an advantage of miccg over polcg is that it is less sensitive to increasing matrix size its disadvantages are that it requires an iteration parameter has a greater setup time and needs more storage than polcg

first passage time problems and resonant behavior on a fluctuating lattice abstract we study the mean residence time of a random walker on a line segment of a lattice which randomly switches between two states in each of the states only motions in a one way direction are allowed when the particle can leave the line segment through both of its ends the residence time is a monotonous increasing function of the switching rate and does not depend on the initial state of the lattice nevertheless if one of the boundaries of the segment is reflecting it goes through a minimum for a given value of the flipping rate showing a resonancelike behavior the results are compared with previous studies of first passage time problems in time dependent fields

infinite dimensional features of matrices and pseudospectra abstract given a hilbert space operator t the level sets of function t z t z 1 1 determine the so called pseudospectra of t we set t to be zero on the spectrum of t after giving some elementary properties of t which as it seems were not noticed before we apply them to the study of the approximation we prove that for any operator t there is a sequence t n of finite matrices such that t n z tends to t z uniformly on c in this proof quasitriangular operators play a special role this is merely an existence result we do not give a concrete construction of this sequence of matrices one of our main points is to show how to use infinite dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size in particular we get a result which means in a sense that the pseudospectrum of a nilpotent matrix can be anything one can imagine we also study the norms of the multipliers in the context of cowen douglas class operators we use these results to show that to the opposite to the function s the function s z for certain finite matrices s may oscillate arbitrarily fast even far away from the spectrum

truncations of a class of pseudo hermitian tridiagonal matrices abstract we consider the class of non hermitian operators represented by infinite tridiagonal matrices selfadjoint in an indefinite inner product space with one negative square we approximate them with their finite truncations both infinite and truncated matrices have eigenvalues of nonpositive type either a single one on the real axis or a couple of complex conjugate ones as a tool to evaluate the reliability of the use of truncations in numerical simulations we give bounds for the rate of convergence of their eigenvalues of nonpositive type numerical examples illustrate our results

universal analytic properties of noise introducing the j matrix formalism we propose a new method in the spectral analysis of noisy time series data for damped oscillators from the jacobi three terms recursive relation for the denominators of the pade approximations built on the well known z transform of an infinite time series we build a hilbert space operator a j operator where each bound state inside the unit circle in the complex plane is simply associated with one damped oscillator while the essential spectrum of the j operator which lies on the unit circle itself is shown to represent the noise signal and noise are thus clearly separated in the complex plane for a finite time series of length 2n the j operator is replaced by a finite order j matrix jn having n eigenvalues which are time reversal covariant different classes of input noise such as blank white and uniform gaussian and pink are discussed in detail the j matrix formalism allowing us to efficiently calculate hundreds of poles of the z transform evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the z transform is shown in particular the poles and zeros tend when the length of the time series goes to infinity to a uniform angular distribution on the unit circle therefore at finite order the roots of unity in the complex plane appear to be noise attractors we show that the z transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property a few basic examples are given to suggest the power of the proposed method

spectral problems for generalized jacobi matrices a new class of generalized jacobi matrices is introduced every proper real rational function is proved to be the m function of a unique finite generalized jacobi matrix moreover every generalized nevanlinna function m which is a solution of a determinate indefinite moment problem turns out to be the m function of a unique infinite generalized jacobi matrix the method we use is based on the step by step schur process of solving the indefinite moment problem the convergence of the sequence of subdiagonal pade approximants for the corresponding hamburger series is investigated

solving groundwater flow problems by conjugate gradient methods and the strongly implicit procedure the performance of the preconditioned conjugate gradient method with three preconditioners is compared with the strongly implicit procedure sip using a scalar computer the preconditioners considered are the incomplete cholesky iccg and the modified incomplete cholesky miccg which require the same computer storage as sip as programmed for a problem with a symmetric matrix and a polynomial preconditioner polcg which requires less computer storage than sip although polcg is usually used on vector computers it is included here because of its small storage requirements in this paper published comparisons of the solvers are evaluated all four solvers are compared for the first time and new test cases are presented to provide a more complete basis by which the solvers can be judged for typical groundwater flow problems based on nine test cases the following conclusions are reached 1 sip is actually as efficient as iccg for some of the published linear two dimensional test cases that were reportedly solved much more efficiently by iccg 2 sip is more efficient than other published comparisons would indicate when common convergence criteria are used and 3 for problems that are three dimensional nonlinear or both and for which common convergence criteria are used sip is often more efficient than iccg and is sometimes more efficient than miccg

matrix methods for pade approximation numerical calculation of poles zeros and residues abstract a representation of the pade approximation of the z transform of a signal as a resolvent of a tridiagonal matrix j n is given several formulas for the poles zeros and residues of the pade approximation in terms of the matrix j n are proposed their numerical stability is tested and compared methods for computing forward and backward errors are presented

global and local behavior of zeros of nonpositive type abstract a generalized nevanlinna function q z with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane the fractional linear transformation defined by q z q z 1 q z r is a generalized nevanlinna function with one negative square its generalized zero of nonpositive type as a function of is being studied in particular it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated

a new approach to spectral approximation a new technique for approximating eigenvalues and eigenvectors of a self adjoint operator is presented the method does not incur spectral pollution uses trial spaces from the form domain has a self adjoint algorithm and exhibits superconvergence

the mechanism of stochastic resonance it is shown that a dynamical system subject to both periodic forcing and random perturbation may show a resonance peak in the power spectrum which is absent when either the forcing or the perturbation is absent

a conjugate gradient finite element model of flow for large multiaquifer systems a quasi three dimensional finite element model based on the conjugate gradient technique is developed and applied to the analysis of subsurface flow occurring in complex heterogeneous multiaquifer systems transient leakage across the aquitards is fully taken into consideration using the convolution approach originally derived by herrera 1970 the convolution integrals enhance the favorable features of the modified conjugate gradients mcg which prove to be a very efficient tool for the solution of the integrodifferential model the high performance of mcg results from two combined factors 1 the diagonal dominance of the assembled finite element matrix which is emphasized by the integral contributions 2 the essential reducibility of the matrix for a wide range of time step values which is due to the virtual decoupling of the aquifer equations the preconditioning of the conjugate gradient scheme is therefore improved and a good convergence occurs after a number of iterations smaller than n n n and n being the overall number of nodes and aquifers respectively the computational cost of the model is generally low and its relative efficiency tends to increase for small to moderate time steps and for complex multiaquifer systems where several interconnected aquifers are simulated

solving three dimensional hexahedral finite element groundwater models by preconditioned conjugate gradient methods practical aspects of three dimensional modeling of groundwater flow in heterogeneous aquifer systems are investigated using a finite element approach particular attention is given to the properties of the conductance matrix and the efficiency of the conjugate gradient method with different preconditioners diagonal scaling incomplete cholesky decomposition incomplete factorization and modified incomplete factorization it is shown that for hexahedral trilinear finite elements the resulting matrix is except for cube shaped elements never diagonally dominant which restricts the existence of several preconditioners numerical comparison of several test problems including hypothetical and field applications with different degrees of heterogeneity show that the incomplete cholesky and the incomplete factorization preconditioners if they exist are more efficient than diagonal scaling with respect to both rate of convergence and overall computing time but diagonal scaling can be considered superior because it is always possible an m matrix transformation is proposed which guarantees the existence of all preconditioners numerical comparison of the test problems shows that this technique is very effective from the resulting preconditioners the incomplete cholesky and the incomplete factorization are shown to be the most efficient but the latter is superior from the point of view of computer storage and is recommended for all practical applications

a comparison of the incomplete cholesky conjugate gradient method with the strongly implicit method as applied to the solution of two dimensional groundwater flow equations this paper compares the efficiency of the strongly implicit procedure sip and the incomplete cholesky conjugate gradient method iccg applied to the solution of the finite difference approximating equations for groundwater flow results for five isotropic two dimensional test problems are presented three are linear confined aquifer problems and two are nonlinear water table aquifer problems in all but one of the test problems the aquifer was considered to be nonhomogeneous both sip and iccg as applied to water table aquifer problems make use of iteration parameters these were varied for each of the test problems to reduce the amount of computational work needed to find a solution iccg was usually substantially more efficient than sip when applied to the confined aquifer test problems for the water table aquifer test problems sip and iccg performed equally well

a transition in a noisy linear system driven by a periodic signal we consider a one dimension linear random walk between two trapping points in which the transition probabilities vary periodically in time an earlier analysis of this system showed that the mean time to trapping of a particle in this system exhibits a minimum when considered as a function of frequency in this note we show that this parameter makes a transition in behavior from a monotonic decrease with increasing amplitude of the periodic term to a monotonic increase with this parameter depending on the frequency a physical argument is suggested to explain this behavior confirmation of this crossover can also be derived from a diffusion model

conjugate gradient type methods and preconditioning in this paper we consider various iterative methods for the numerical solution of very large sparse linear systems of equations which arise in the discretization of partial differential equations as the performance of the methods generally depends on the characteristics of the problems to be solved a judicious choice between the methods will require knowledge about the system the aim of this paper is to review the properties of the various iteration methods in order to assist the user in making a deliberate selection

coherent stochastic resonance in one dimensional diffusion with one reflecting and one absorbing boundaries it is shown that the single step periodic signal periodic telegraph signal can not produce coherent stochastic resonance for diffusion on a segment with one absorbing and one reflecting end points while the multi step periodic signal does the general features of this process are exihibited the resonant frequency is found to decrease and the mean first passage time at resonant frequency increases linearly as we increase the length of the medium the cycle variable is shown to be the proper argument to express the first passage probability at resonance a formula for first passage probability at resonance is derived in terms of two universal functions which clearly isolates its dependence on the length of the medium

nonconventional stochastic resonance it is argued on the basis of linear response theory lrt that new types of stochastic resonance sr are to be anticipated in diverse systems quite different from the one most commonly studied to date which has a static double well potential and is driven by a net force equal to the sum of periodic and stochastic terms on this basis three new nonconventional forms of sr are predicted sought found and investigated both theoretically and by analogue electronic experiment a in monostable systems b in bistable systems with periodically modulated noise and c in a system with coexisting periodic attractors in each case it is shown that lrt can provide a good quantitative description of the experimental results for sufficiently weak driving fields it is concluded that sr is a much more general phenomenon than has hitherto been appreciated

finite element algorithms for simulating three dimensional groundwater flow and solute transport in multilayer systems a finite element formulation is developed for the simulation of groundwater flow and solute transport in multilayer systems of several aquifers and aquitards this formulation is general flexibile and capable of taking full advantage of the nature of flow in such multilayer systems a fully three dimensional spatial representation can be performed for certain aquifers or for the entire flow system if needed those parts of the system that require three dimensional spatial discretization are handled effectively by combining two dimensional basis functions in the x y space and one dimensional basis functions in the z space furthermore the formulation has a desirable option to perform one dimensional representation of flow and transport in aquitards and areal representation of flow and transport in certain aquifers that do not require the fully three dimensional discretization when the one dimensional representation of an aquitard is used coupling of adjacent aquifer and aquitard layers is handled using a convolution integral approach a general solution strategy is also developed to allow systematic time stepping and cost effective matrix handling schemes for those parts of the system that require three dimensional discretization an algorithm referred to as the alternate sublayer and line sweep procedure is presented for decomposing three dimensional matrix equations this algorithm can accomodate several thousand nodal unknowns without requiring excessive core storage and cpu time four examples of transient flow and transport problems are provided these examples show verification of numerical results and demonstrate that the present finite element models are far more cost effective than earlier three dimensional models that are based on the conventional galerkin approach and direct matrix solution algorithms

stochastic resonance and symmetry breaking in a one dimensional system we derive and discuss properties of an exact solution for the average time for trapping of a brownian particle driven by a random asymmetric but unbiased telegraph signal the particle moves along a line segment terminated by either two traps or a trap and a reflecting point numerical results suggest that stochastic resonance defined as a nonmonotonic behavior of the mean trapping time is absent in the first case but present in the second this generalizes a result obtained earlier by doering and gadoua phys rev lett 69 2318 1992 and implies that symmetry breaking alone does not necessarily create stochastic resonance

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

an algorithmic test for diagonalizability of finite dimensional pt invariant systems a non hermitian operator does not necessarily have a complete set of eigenstates contrary to a hermitian one an algorithm is presented which allows one to decide whether the eigenstates of a given pt invariant operator on a finite dimensional space are complete or not in other words the algorithm checks whether a given pt symmetric matrix is diagonalizable the procedure neither requires to calculate any single eigenvalue nor any numerical approximation

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

zeros of nonpositive type of generalized nevanlinna functions with one negative square abstract a generalized nevanlinna function q z with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane the fractional linear transformation defined by q z q z 1 q z r is a generalized nevanlinna function with one negative square its generalized zero of nonpositive type as a function of defines a path in the closed upper halfplane various properties of this path are studied in detail

saturated flow in a single fracture evaluation of the reynolds equation in measured aperture fields fracture transmissivity and detailed aperture fields are measured in analog fractures specially designed to evaluate the utility of the reynolds equation the authors employ a light transmission technique with well defined accuracy approximately 1 error to measure aperture fields at high spatial resolution approximately 0 015 cm a hele shaw cell is used to confirm the approach by demonstrating agreement between experimental transmissivity simulated transmissivity on the measured aperture field and the parallel plate law in the two rough walled analog fractures considered the discrepancy between the experimental and numerical estimates of fracture transmissivity was sufficiently large approximately 22 47 to exclude numerical and experimental errors 2 as a source they conclude that the three dimensional character of the flow field is important for fully describing fluid flow in the two rough walled fractures considered and that the approach of depth averaging inherent in the formulation of the reynolds equation is inadequate they also explore the effects of spatial resolution aperture measurement technique and alternative definitions for link transmissivities in the finite difference formulation including some that contain corrections for tortuosity perpendicular to the mean fracture plane and stokes flow various formulations for link transmissivity are shown to converge at high resolution approximately 1 5 the spatial correlation length in the smoothly varying fracture at coarser resolutions the solution becomes increasingly sensitive to definition of link transmissivity and measurement technique aperture measurements that integrate over individual grid blocks were less sensitive to measurement scale and definition of link transmissivity than point sampling techniques