large deviation eigenvalue density for the soft edge laguerre and jacobi β ensembles we analyze the eigenvalue density for the laguerre and jacobi ensembles in the cases that the corresponding exponents are extensive in particular we obtain the asymptotic expansion up to terms o 1 in the large deviation regime outside the limiting interval of support as found in recent studies of the large deviation density for the gaussian ensemble and laguerre ensemble with fixed exponent there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge

optimal and near optimal advection diffusion finite difference schemes vi two dimensional alternating directions computationally efficient alternating direction implicit finite difference schemes with high accuracy are derived for the numerical solution on a regular rectangular grid of the two dimensional constant coefficient diffusion equation with decay advection off diagonal diffusivity and sources t c c u 1 x 1 c u 2 x 2 c k 11 x 1 2 c 2 k 12 x 1 x 2 c k 22 x 2 2 c q the 3 3 2 compact computational module involves three consecutive spatial points in each of the perpendicular spatial directions at two successive time levels the schemes are tested for three variants of an exactly solvable test case corresponding to the fractionation of an initial mixture of sinking neutral and rising particles released together in a vertically well mixed turbulent open channel flow with mean current not aligned with either x 1 or x 2

finite difference schemes on regular triangular grids the phase error and isotropy properties of various finite difference schemes on grids consisting of regular triangles are compared with similar schemes on square grids the comparisons are based on a fourier analysis of semidiscrete solutions to the two dimensional linear convection equation the finite difference schemes presented on the triangular grid include a second order method a compact fourth order method and a modified compact method designed to extend the accurate wave number range of the numerical approximation all of the schemes considered are centered and hence nondissipative in each case the finite difference scheme on the triangular grid reduces the anisotropy of the phase error in comparison with a similar scheme on the square grid

direct simulations of turbulent flow using finite difference schemes abstract this paper presents finite difference solutions to the evolution of small amplitude disturbances and incompressible fully developed turbulent channel flow the main objective of the paper is to provide a comprehensive comparison between the results obtained using finite difference and spectral methods an advantage of finite difference schemes over the highly accurate spectral methods lies in the ease with which they can be applied to complex geometries the finite difference methods used include a kinetic energy conserving type of central difference scheme and a high order accurate upwind difference scheme unlike the central difference scheme the upwind difference scheme was found not to require a kinetic energy conservation property to control aliasing error the dissipative nature of the upwind scheme results in a damping of the higher frequency content as a result very little energy is aliased back the computed data including first and second order statistics for the turbulent channel flow case are found to compare well with experimental data and earlier spectral simulations it appears that the high order accurate upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries

applications of high order optimized upwind schemes for computational aeroacoustics high order optimized upwind dispersion relation preserving finite difference schemes are used for solving multidimensional acoustic wave propagation problems the accuracy and the robustness of the schemes are tested for acoustic problems with various geometry and boundary conditions excellent agreements between the numerical and the analytical solutions indicate that the high order optimized upwind schemes minimize not only the dispersion error but also the dissipation error while retaining the numerical stability the simulation results have also demonstrated that the optimized upwind schemes damp out the spurious short waves automatically without the need of filter or explicit dissipation terms the overall effectiveness of the high order optimized upwind schemes is verified the good numerical result of the problem of acoustic scattering by a sphere suggests that the optimized upwind schemes may be used to solve acoustic problems involving a relatively complex geometry

multidimensional optimization of finite difference schemes for computational aeroacoustics because of the long propagation distances computational aeroacoustics schemes must propagate the waves at the correct wave speeds and lower the isotropy error as much as possible the spatial differencing schemes are most frequently analyzed and optimized for one dimensional test cases therefore in multidimensional problems such optimized schemes may not have isotropic behavior in this work optimized finite difference schemes for multidimensional computational aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes the derivation is performed based on both taylor series expansion and fourier analysis various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed the isotropy corrector factor a parameter of the schemes can be determined by minimizing the integrated error between the phase or group velocities on different spatial directions the order of accuracy of the optimized schemes is the same as that of the classical schemes the advantage being in reducing the isotropy error the present schemes are restricted to equally spaced cartesian grids so the generalized curvilinear transformation method and cartesian grid methods are good candidates the optimized schemes are tested by solving various multidimensional problems of aeroacoustics

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

a consistently formulated quick scheme for fast and stable convergence using finite volume iterative calculation procedures abstract previous applications of quick for the discretization of convective transport terms in finite volume calculation procedures have failed to employ a rigorous and systematic approach for consistently deriving this finite difference scheme instead earlier formulations have been established numerically by trial and error the new formulation for quick presented here is obtained by requiring that it satisfy four rules that guarantee physically realistic numerical solutions having overall balance careful testing performed for the wall driven square enclosure flow configuration shows that the consistently derived version of quick is more stable and converges faster than any of the formulations previously employed this testing includes the relative evaluation of boundary conditions approximated by second and third order finite difference schemes as well as calculations performed at higher reynolds numbers than previously reported

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

analysis of a new high resolution upwind compact scheme a new high resolution upwind compact scheme is presented and analyzed along with several previously proposed schemes for numerical dispersion dissipation anisotropy phase damping dispersion relation preservation property and numerical stability the schemes are tested on problems of the propagation of a initially discontinuous wave and of the transport of a sharp scalar cone a benchmark problem of aeroacoustics is solved with the present scheme giving satisfactory resolution the scheme compares well with the existing schemes for the range of properties considered desirable in high resolution schemes

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

a study of finite difference approximations to steady state convection dominated flow problems abstract five different finite difference schemes first order upwind skew upwind second order upwind second order central differencing and quick approximating the convection terms in the transport equation with fluid motion have been studied three simple test problems are used to compare the performances by the five schemes for high cell peclet number flows they are also used to demonstrate the restraints on the accuracy of the numerical approximations set by the types of the boundary conditions by the presence of the source term in the flow region and by the skewness of the numerical grid lines the basic reasons behind the spurious oscillations in a numerical solution are studied among all five schemes studied the second order upwind is found to be in general the most satisfactory

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

finding maps for belief networks is np hard given a probabilistic world model an important problem is to find the maximum a posteriori probability map instantiation of all the random variables given the evidence numerous researchers using such models employ some graph representation for the distributions such as a bayesian belief network this representation simplifies the complexity of specifying the distributions from exponential in n the number of variables in the model to linear in n in many interesting cases we show however that finding the map is np hard in the general case when these representations are used even if the size of the representation happens to be linear in n furthermore minor modifications to the proof show that the problem remains np hard for various restrictions of the topology of the graphs the same technique can be applied to the results of a related paper by cooper to further restrict belief network topology in the proof that probabilistic inference is np hard

analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods fourier analysis is used to quantitatively assess the resolution and in particular the isotropy of wave solution using finite difference spatial discretization schemes along with fourth order runge kutta temporal scheme aspect ratio of the grid in two dimension along with the angle of wave propagation are the parameters varied to qualitatively and quantitatively assess the anisotropy of the solutions for a a skewed one dimensional wave convecting in two dimensions following the standard convection equation and b a wave propagating following the two dimensional linearized rotating shallow water equations results show the effect of changing the aspect ratio and the propagation angle on the directional nature of the solution as obtained by different methods for the above non dispersive and dispersive wave system

residual distribution schemes for advection and advection diffusion problems on quadrilateral cells this paper provides a study of some difficulties arising when extending residual distribution schemes for scalar advection and advection diffusion problems from triangular grids to quadrilateral ones the fourier and truncation error analyses on a structured mesh are employed and a generalized modified wavenumber is defined which provides a general framework for the multidimensional analysis and comparison of different schemes it is shown that for the advection equation linearity preserving schemes for quadrilaterals provide lower dissipation with respect to their triangle based counterparts and very low or no damping for high frequency fourier modes on structured grids therefore they require an additional artificial dissipation term for damping marginally stable modes in order to be employed with success for pure advection problems in the case of advection diffusion problems a hybrid approach using an upwind residual distribution scheme for the convective fluctuation and any other scheme for the diffusion term is only first order accurate on the other hand distributing the entire residual by an upwind scheme provides second order accuracy however such an approach is unstable for diffusion dominated problems since residual distribution schemes are characterized by undamped modes associated with the discretization of the diffusive fluctuation the present analysis allows one to determine the conditions for a stable hybrid approach to be second order accurate and to design an optimal scheme having minimum dispersion error on a nine point stencil well documented testcases for advection and advection diffusion problems are used to compare the accuracy properties of several schemes

wavenumber extended high order upwind biased finite difference schemes for convective scalar transport this paper proposes some new wavenumber extended high order upwind biased schemes the dispersion and dissipation errors of upwind biased finite difference schemes are assessed and compared by means of a fourier analysis of the difference schemes up to 11th order upwind biased schemes are analyzed it is shown that both the upwind biased scheme of order 2n 1 and the corresponding centered differencing scheme of order 2nhave the same dispersion characteristics thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation the new second order wavenumber extended scheme is tested and compared with some well known schemes the range of wavenumbers that are accurately treated by the upwind biased schemes is improved by using additional constraints from the fourier analysis to construct the new schemes the anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber extended upwind biased finite difference schemes

a sponge layer damping technique for aeroacoustic time reversal abstract this paper presents the underlying theory associated mathematical modelling and analysis of a sponge layer damping technique termed the time reversal sponge layer trsl that significantly improves the performance of aeroacoustic time reversal tr the tr technique requires the use of multiple line arrays las in a time reversal mirror trm to accurately predict the source location and its characteristics however it is shown that when using multiple las the interference between the opposite propagating fluxes near the la boundaries results in the formation of spurious local maxima regions throughout the computational domain thereby reducing the capacity of tr to resolve acoustic sources the novel trsl technique proposed in this work minimises this unwanted interference by damping the flux normally incident on a la boundary and is implemented using the pseudo characteristic formulation pcf of the two dimensional linearised euler equations lee the performance of trsl is assessed by simulating a number of test cases such as an idealised time harmonic monopole dipole and lateral quadrupole sources as well as multiple two dipole sources of different strengths located in a nonuniform mean shear flow the use of trsl suppresses the formation of spurious maxima and significantly improves the source map thereby demonstrating the effectiveness of this damping technique the performance of trsl is compared with two other methods a tr superposition technique and conventional beamforming cb the tr superposition technique prevents the flux interference problem near the la boundaries by superposing the instantaneous time reversed acoustic pressure fields computed from individual las the source map obtained using the superposition technique was found to be identical to that obtained using the trsl damping technique however the computational cost was much higher a comparison with cb indicated that although cb accurately predicts the aeroacoustic source location the relative magnitudes of the side lobes in the cb source map s are substantially higher in comparison to the corresponding tr source map s it was also shown that tr with use of the trsl is better suited for resolving multiple coherent sources in a domain than the cb method especially when one source is much weaker than other sources in the domain

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

approximate belief updating in max 2 connected bayes networks is np hard a max 2 connected bayes network is one where there are at most 2 distinct directed paths between any two nodes we show that even for this restricted topology null evidence belief updating is hard to approximate

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

direct numerical simulation of scalar transport using unstructured finite volume schemes an unstructured finite volume method for direct and large eddy simulations of scalar transport in complex geometries is presented and investigated the numerical technique is based on a three level fully implicit time advancement scheme and central spatial interpolation operators the scalar variable at cell faces is obtained by a symmetric central interpolation scheme which is formally first order accurate or by further employing a high order correction term which leads to formal second order accuracy irrespective of the underlying grid in this framework deferred correction and slope limiter techniques are introduced in order to avoid numerical instabilities in the resulting algebraic transport equation the accuracy and robustness of the code are initially evaluated by means of basic numerical experiments where the flow field is assigned a priori a direct numerical simulation of turbulent scalar transport in a channel flow is finally performed to validate the numerical technique against a numerical dataset established by a spectral method in spite of the linear character of the scalar transport equation the computed statistics and spectra of the scalar field are found to be significantly affected by the spectral properties of interpolation schemes although the results show an improved spectral resolution and greater spatial accuracy for the high order operator in the analysis of basic scalar transport problems the low order central scheme is found superior for high fidelity simulations of turbulent scalar transport