a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

a new 9 point sixth order accurate compact finite difference method for the helmholtz equation a new 9 point sixth order accurate compact finite difference method for solving the helmholtz equation in one and two dimensions is developed and analyzed this scheme is based on sixth order approximation to the derivative calculated from the helmholtz equation a sixth order accurate symmetrical representation for the neumann boundary condition was also developed the efficiency and accuracy of the scheme is validated by its application to two test problems which have exact solutions numerical results show that this sixth order scheme has the expected accuracy and behaves robustly with respect to the wave number

stochastic switching for partially observable dynamics and optimal asset allocation abstractin industrial applications optimal control problems frequently appear in the context of decision making under incomplete information in such framework decisions must be adapted dynamically to account for possible regime changes of the underlying dynamics using stochastic filtering theory markovian evolution can be modelled in terms of latent variables which naturally leads to high dimensional state space making practical solutions to these control problems notoriously challenging in our approach we utilise a specific structure of this problem class to present a solution in terms of simple reliable and fast algorithms the algorithms presented in this paper have already been implemented in an r package

a frequency space 2 d scalar wave extrapolator using extended 25 point finite difference operator finite difference frequency domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s the generation of synthetic seismograms with the time domain finite difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse time migration this became possible with the rapid development of high performance computers however the space frequency x finite difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory therefore finite difference time domain modeling is much more popular among exploration geophysicists a limitation of the space frequency domain is that the recently developed nine point scheme still requires that g the number of grid points per wavelength be 5 this value is greater than for most other numerical modeling techniques for example the pseudospectral scheme

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

optimal stopping under partial observation near value iteration we propose a new approximate value iteration method namely near value iteration nvi to solve continuous state optimal stopping problems under partial observation which in general cannot be solved analytically and also pose a great challenge to numerical solutions nvi is motivated by the expression of the value function as the supremum over an uncountable set of linear functions in the belief state after a smart manipulation of the operations in the updating equation for the value function we reduce the set to only two functions at every time step so as to achieve significant computational savings nvi yields a value function approximation bounded by the tightest lower and upper bounds that can be achieved by existing algorithms in the same class so the nvi approximation is closer to the true value function than at least one of these bounds we demonstrate the effectiveness of our approach on an example of pricing american options under stochastic volatility

compact 2d and 3d sixth order schemes for the helmholtz equation with variable wave number several studies have presented compact fourth order accurate finite difference approximation for the helmholtz equation in two or three dimensions several of these formulae allow for the wave number k to be variable other papers have extended this further to include variable coefficients within the laplacian which models non homogeneous materials in electromagnetism later papers considered more accurate compact sixth order methods but these were restricted to constant k in this paper we extend these compact sixth order schemes to variable k in both two and three dimensions results on 2d and 3d problems with known analytic solutions verify the sixth order accuracy we demonstrate that for large wave numbers the second order scheme cannot produce comparable results with reasonable grid sizes

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

a fourth order optimal finite difference scheme for the helmholtz equation with pml abstract in this paper 17 point and 25 point finite difference fd schemes for the helmholtz equation with perfectly matched layer pml in the two dimensional domain are presented it is shown that the 17 point fd scheme is inconsistent in the presence of pml however the 25 point fd scheme is pointwise consistent an error analysis for the numerical approximation of the exact wavenumber is also presented we present the global and refined 25 point finite difference schemes based on minimizing the numerical dispersion numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

an optimal 25 point finite difference scheme for the helmholtz equation with pml in this paper we present an optimal 25 point finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in two dimensional domain based on minimizing the numerical dispersion we propose the refined choice strategy for choosing optimal parameters of the 25 point finite difference scheme numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

pattern formation by boundary forcing in convectively unstable oscillatory media with and without differential transport convectively unstable open reactive flows of oscillatory media whose phase is fixed or periodically modulated at the inflow boundary are known to result in stationary and traveling waves respectively the latter are implicated in biological segmentation the boundary controlled pattern selection by this flow distributed oscillator fdo mechanism has been generalized to include differential flow difi and differential diffusion turing modes our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion to do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing and show how the solutions are affected by differential transport we find that the difi and fdo modes are closely related and lie in the same frequency range while the turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range finally we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes

optimal stopping of partially observable markov processes a filtering based duality approach in this note we develop a numerical approach to the problem of optimal stopping of discrete time continuous state partially observable markov processes pomps our motivation is to find approximate solutions that provide lower and upper bounds on the value function such that the gap between the bounds can provide a practical measure of the quality of the solutions to this end we develop a filtering based duality approach which relies on the martingale duality formulation of the optimal stopping problem and the particle filtering technique we show that this approach complements an asymptotic lower bound derived from a suboptimal stopping time with an asymptotic upper bound on the value function we carry out error analysis and illustrate the effectiveness of our method on an example of pricing american options under partial observation of stochastic volatility

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

comment on flow distributed oscillations stationary chemical waves in a reacting flow in a recent paper by kaern and menzinger phys rev e 60 r3471 1999 a successful verification of the stationary space periodic structures predicted by andresen et al phys rev e 60 297 1999 was reported kaern and menzinger suggest a mechanism for the formation of such structures that yields a linear relationship between the selected wavelength and the flow rate we find this mechanism too simple and produce numerical simulations that support the original interpretation of these structures

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

harmonic resonant excitation of flow distributed oscillation waves and turing patterns driven at a growing boundary we perform numerical studies of a reaction diffusion system that is both turing and hopf unstable and that grows by addition at a moving boundary which is equivalent by a galilean transformation to a reaction diffusion advection system with a fixed boundary and a uniform flow we model the conditions of a recent set of experiments which used a temporally varying illumination in the boundary region to control the formation of patterns in the bulk of the photosensitive medium the frequency of the illumination variations can select patterns from among the competing instabilities of the medium in the usual case the waves that are excited have frequencies as measured at a constant distance from the upstream boundary matching the driving frequency in contrast to the usual case we find that both turing patterns and flow distributed oscillation waves can be excited by forcing at subharmonic multiples of the wave frequencies the final waves with frequencies at integer multiples of the driving frequency are formed by a process in which transient wave fronts break up and reconnect we find ratios of response to driving frequency as high as 10

a dispersion minimizing finite difference scheme for the helmholtz equation based on point weighting abstract in this paper we develop a new dispersion minimizing finite difference scheme for the helmholtz equation with perfectly matched layer pml in two dimensional domain which is a second order 9 point scheme to discretize the second derivative operator we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme based on minimizing the numerical dispersion the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy the new scheme is simple rotation free and pointwise consistent with the equation which is different from the classical rotated 9 point difference scheme obtained by combining the cartesian coordinate system and the rotated system moreover it is a robust scheme even if the step sizes of different directions are not equal convergence analysis and dispersion analysis are given several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme

an optimal compact sixth order finite difference scheme for the helmholtz equation abstract in this paper we present an optimal compact finite difference scheme for solving the 2d helmholtz equation a convergence analysis is given to show that the scheme is sixth order in accuracy based on minimizing the numerical dispersion a refined optimization rule for choosing the scheme s weight parameters is proposed numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters