arbitrarily high order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws we design arbitrarily high order accurate entropy stable schemes for systems of conservation laws the schemes termed tecno schemes are based on two main ingredients i high order accurate entropy conservative fluxes and ii suitable numerical diffusion operators involving eno reconstructed cell interface values of scaled entropy variables numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the tecno schemes

copula based geostatistical models for groundwater quality parameters 1 groundwater quality parameters exhibit considerable spatial variability geostatistical methods including the assessment of variograms are usually used to characterize this variability copulas offer an interesting opportunity to describe dependence structures for multivariate distributions bivariate empirical copulas can be used as an alternative to variograms and covariance functions for the description of the spatial variability rank correlations of these copulas express the strength of the dependence independently of the marginal distributions and thus offer an alternative to the variograms empirical copulas for four quality parameters chloride sulfate ph and nitrate obtained from a large scale groundwater quality measurement network in baden wurttemberg germany are calculated they indicate that the spatial dependence structure of the investigated parameters is not gaussian two theoretical copula based models are presented in this paper a gaussian and a non gaussian bootstrap based statistical tests using stochastic simulation of the multivariate distributions are used to investigate the appropriateness of the models according to the test results the gaussian copula is rejected for most of the parameters while the non gaussian alternative is not rejected in most cases

affordable entropy conserving and entropy stable flux functions for the ideal mhd equations in this work we design an entropy stable finite volume approximation for the ideal magnetohydrodynamics mhd equations the method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system to guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal mhd system exact entropy conserving schemes cannot dissipate energy at shocks thus to compute accurate solutions to problems that may develop shocks we determine a dissipation term to guarantee entropy stability for the numerical scheme numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness

stability of stochastic approximation under verifiable conditions in this paper we address the problem of the stability and convergence of the stochastic approximation procedure theta n 1 theta n gamma n 1 h theta n xi n 1 the stability of such sequences theta n is known to heavily rely on the behavior of the mean field h at the boundary of the parameter set and the magnitude of the stepsizes used the conditions typically required to ensure convergence and in particular the boundedness or stability of theta n are either too difficult to check in practice or not satisfied at all this is the case even for very simple models the most popular technique for circumventing the stability problem consists of constraining theta n to a compact subset mathcal k in the parameter space this is obviously not a satisfactory solution as the choice of mathcal k is a delicate one in this paper we first prove a deterministic stability result which relies on simple conditions on the sequences xi n and gamma n we then propose and analyze an algorithm based on projections on adaptive truncation sets which ensures that the aforementioned conditions required for stability are satisfied we focus in particular on the case where xi n is a so called markov state dependent noise we establish both the stability and convergence with probability 1 w p 1 of the algorithm under a set of simple and verifiable assumptions we illustrate our results with an example related to adaptive markov chain monte carlo algorithms

multivariate type g matern stochastic partial differential equation random fields for many applications with multivariate data random field models capturing departures from gaussianity within realizations are appropriate for this reason we formulate a new class of multivariate non gaussian models based on systems of stochastic partial differential equations with additive type g noise whose marginal covariance functions are of matern type we consider four increasingly flexible constructions of the noise where the first two are similar to existing copula based models in contrast with these the last two constructions can model non gaussian spatial data without replicates computationally efficient methods for likelihood based parameter estimation and probabilistic prediction are proposed and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications less

staggered grid entropy stable multidimensional summation by parts discretizations on curvilinear coordinates abstract the entropy conservative stable staggered grid tensor product algorithm of parsani et al 1 is extended to multidimensional sbp discretizations the required sbp preserving interpolation operators are proven to exist under mild restrictions and the resulting algorithm is proven to be entropy conservative stable as well as elementwise conservative for 2 dimensional simplex elements the staggered grid algorithm is shown to be more accurate and have a larger maximum time step restriction as compared to the collocated algorithm the staggered algorithm significantly reduces the number of computationally expensive two point flux evaluations which is potentially important for both explicit and implicit time marching schemes furthermore the staggered algorithm requires fewer degrees of freedom for comparable accuracy which has favorable implications for implicit time marching schemes

matern cross covariance functions for multivariate random fields we introduce a flexible parametric family of matrix valued covariance functions for multivariate spatial random fields where each constituent component is a matern process the model parameters are interpretable in terms of process variance smoothness correlation length and colocated correlation coefficients which can be positive or negative both the marginal and the cross covariance functions are of the matern type in a data example on error fields for numerical predictions of surface pressure and temperature over the north american pacific northwest we compare the bivariate matern model to the traditional linear model of coregionalization

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

an explicit link between gaussian fields and gaussian markov random fields the stochastic partial differential equation approach continuously indexed gaussian fields gfs are the most important ingredient in spatial statistical modelling and geostatistics the specification through the covariance function gives an intuitive interpretation of the field properties on the computational side gfs are hampered with the big n problem since the cost of factorizing dense matrices is cubic in the dimension although computational power today is at an all time high this fact seems still to be a computational bottleneck in many applications along with gfs there is the class of gaussian markov random fields gmrfs which are discretely indexed the markov property makes the precision matrix involved sparse which enables the use of numerical algorithms for sparse matrices that for fields in r 2 only use the square root of the time required by general algorithms the specification of a gmrf is through its full conditional distributions but its marginal properties are not transparent in such a parameterization we show that using an approximate stochastic weak solution to linear stochastic partial differential equations we can for some gfs in the matern class provide an explicit link for any triangulation of r d between gfs and gmrfs formulated as a basis function representation the consequence is that we can take the best from the two worlds and do the modelling by using gfs but do the computations by using gmrfs perhaps more importantly our approach generalizes to other covariance functions generated by spdes including oscillating and non stationary gfs as well as gfs on manifolds we illustrate our approach by analysing global temperature data with a non stationary model defined on a sphere less

optimized finite difference operator for broadband seismic wave modeling high resolution image and waveform inversion of smallscale targets requires the handling of high frequency seismic wavefields however conventional finite difference fd methods have strong numerical dispersions in the presence of high frequency components to reduce these numerical dispersions we optimized the constant coefficients of the fd operator by maximizing the wavenumber coverage within a given error limitation we set up three general criteria to enhance the convergence of the algorithm and reduce the optimization effort we selected the error limitation to be 0 0001 this being the smallest in the literature which led to perfect agreement between theoretical analyses and numerical experiments the accuracy of our optimized fd methods can even reach that of much higher order unoptimized fd methods which means great savings of computational efforts and memory demand these advantages become even more apparent with 3d modeling especially for saving memory demand

a new time space domain high order finite difference method for the acoustic wave equation a new unified methodology was proposed in finkelstein and kastner 2007 39 to derive spatial finite difference fd coefficients in the joint time space domain to reduce numerical dispersion the key idea of this method is that the dispersion relation is completely satisfied at several designated frequencies we develop this new time space domain fd method further for 1d 2d and 3d acoustic wave modeling using a plane wave theory and the taylor series expansion new spatial fd coefficients are frequency independent though they lead to a frequency dependent numerical solution we prove that the modeling accuracy is 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation however under the same discretization the new 1d method can reach 2m th order accuracy and is always stable the 2d method can reach 2m th order accuracy along eight directions and has better stability similarly the 3d method can reach 2m th order accuracy along 48 directions and also has better stability than the conventional fd method the advantages of the new method are also demonstrated by the results of dispersion analysis and numerical modeling of acoustic wave equation for homogeneous and inhomogeneous acoustic models in addition we study the influence of the fd stencil length on numerical modeling for 1d inhomogeneous media and derive an optimal fd stencil length required to balance the accuracy and efficiency of modeling a new time space domain high order staggered grid fd method for the 1d acoustic wave equation with variable densities is also developed which has similar advantages demonstrated by dispersion analysis stability analysis and modeling experiments the methodology presented in this paper can be easily extended to solve similar partial difference equations arising in other fields of science and engineering

on the construction and efficiency of staggered numerical differentiators for the wave equation finite difference fd techniques have established themselves as viable tools for the numerical modeling of wave propagation the accuracy and the computational efficiency of numerical modeling can be enhanced by using high order spatial differential operators dablain 1986

determining the optimal coefficients of the explicit finite difference scheme using the remez exchange algorithm explicit finite difference fd schemes are widely used in the seismic exploration field due to their simplicity in implementation and low computational cost however they suffer from strong artifacts caused by using coarse grids for high frequency applications the optimization of constant coefficients is popular in reducing spatial dispersions but current methods could not guarantee that the bandwidth of the tolerable dispersion error is the widest we have applied the remez exchange algorithm to optimize the constant coefficients of the explicit fd schemes for conventional and staggered grids the resulting dispersion errors are distributed alternately between the maxima and minima in the passband of the filter which is consistent with the most important equal ripple property of the error magnitude for the optimal solution according to the chebyshev criterion the remez exchange algorithm can determine the optimal coefficients of the fd method with only a few iterations and the resulting operator has a wider bandwidth compared with previous solutions it can handle arbitrary orders without the influence of local minima its computational cost for solving the objective function is comparable to that of the least squares method but its bandwidth is wider its accuracy is also higher than that of the maximum norm solved by the simulated annealing algorithm but its computational cost is much lower theoretically the equal ripple error can offer the widest bandwidth for suppressing numerical dispersions among all solutions obtained by the constant coefficient optimization in other words we can obtain a smaller error limitation than traditional methods under the same bandwidth this superiority over traditional methods is essential for reducing the total error accumulation which is helpful to avoid rapid error accumulations especially for large scale models and long term problems

a valid matern class of cross covariance functions for multivariate random fields with any number of components we introduce a valid parametric family of cross covariance functions for multivariate spatial random fields where each component has a covariance function from a well celebrated matern class unlike previous attempts our model indeed allows for various smoothnesses and rates of correlation decay for any number of vector components we present the conditions on the parameter space that result in valid models with varying degrees of complexity we discuss practical implementations including reparameterizations to reflect the conditions on the parameter space and an iterative algorithm to increase the computational efficiency we perform various monte carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging the application of the proposed multivariate matern model is illustrated on two meteorological datasets temperature pressure over the pacific northwest bivariate and wind temperature pressure in oklahoma trivariate in the latter case our flexi

notes on the eigensystem of magnetohydrodynamics the eigenstructure of the equations governing one dimensional ideal magnetohy drodynamics is examined motivated by the wish to exploit it for construction of high resolution computational algorithms the results are given in simple forms that avoid indeterminacy or degeneracy whenever possible the unavoidable indeterminacy near the magnetosonic or triple umbilic state is analysed and shown to cause no difficulty in evaluating a numerical flux function the structure of wave paths close to this singularity is obtained and simple expressions are presented for the structure coefficients that govern wave steepening

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

multivariate max stable spatial processes summary max stable processes allow the spatial dependence of extremes to be modelled and quantified so they are widely adopted in applications for a better understanding of extremes it may be useful to study several variables simultaneously to this end we study the maxima of independent replicates of multivariate processes both in the gaussian and student t cases we define a poisson process construction and introduce multivariate versions of the smith gaussian extreme value the schlather extremal gaussian and extremal t and the brown resnick models we develop inference for the models based on composite likelihoods we present results of monte carlo simulations and an application to daily maximum wind speed and wind gust

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

optimal staggered grid finite difference schemes by combining taylor series expansion and sampling approximation for wave equation modeling high order staggered grid finite difference sfd schemes have been universally used to improve the accuracy of wave equation modeling however the high order sfd coefficients on spatial derivatives are usually determined by the taylor series expansion te method which just leads to great accuracy at small wavenumbers for wave equation modeling some conventional optimization methods can achieve high accuracy at large wavenumbers but they hardly guarantee the small numerical dispersion error at small wavenumbers in this paper we develop new optimal explicit sfd esfd and implicit sfd isfd schemes for wave equation modeling we first derive the optimal esfd and isfd coefficients for the first order spatial derivatives by applying the combination of the te and the sampling approximation to the dispersion relation and then analyze their numerical accuracy finally we perform elastic wave modeling with the esfd and isfd schemes based on the te method and the optimal method respectively when the appropriate number and interval for the sampling points are chosen these optimal schemes have extremely high accuracy at small wavenumbers and can also guarantee small numerical dispersion error at large wavenumbers numerical accuracy analyses and modeling results demonstrate the optimal esfd and isfd schemes can efficiently suppress the numerical dispersion and significantly improve the modeling accuracy compared to the te based esfd and isfd schemes

maximizing the cfl number of stable time space domain explicit finite difference modeling abstract the time space domain explicit finite difference fd method numerically solves wave equation by applying fd to approximate its spatial and temporal derivatives this method always faces the stability problem i theoretically analyze stability condition of this method for the acoustic wave equation and find that the maximum courant friedrichs lewy cfl number of stable modeling depends on the maximum value of the spatial fd dispersion relation the conventional method commonly determines spatial fd coefficients by approximating the spatial fd dispersion to the exact relation within the given wavenumber area however outside this area the dispersion relation and hence the maximum cfl number is uncontrollable to make it controllable i propose to simultaneously approximate the spatial fd dispersion relation to a given function outside this area further i develop an algorithm to maximize the maximum cfl number stability analysis and modeling examples demonstrate that the proposed method has better stability than the conventional method especially when the maximum effective wavenumber frequency is small for the 2d case the conventional method performs stable simulation using a cfl number of no more than 0 707 but the new method can perform stable simulation using a large cfl number for example 1 6 4 8 and 9 0 and non long spatial fd stencils

acoustic and elastic numerical wave simulations by recursive spatial derivative operators anewschemeforthecalculationofspatialderivativeshas been developed the technique is based on recursive derivativeoperatorsthataregeneratedbyanlfitinthespectraldomain the use of recursive operators enables us to extend acousticandelasticwavesimulationstoshorterwavelengths the method is applied to the numerical solution of the 2d acousticwaveequationandtothesolutionoftheequationsof 2ddynamicelasticityinanisotropicmedium anexampleof reverse time migration of a synthetic data set shows that the numerical dispersion can be significantly reduced with respect to schemes that are based on finite differences the methodistestedforthesolutionsoftheequationsofdynamic elasticity by comparing numerical and analytic solutions to lamb sproblem

optimized chebyshev fourier migration a wide angle dual domain method for media with strong velocity contrasts a wide angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one way waveequation migration we have developed a dual domain one way propagator using truncated chebyshev polynomials and a globally optimized scheme our method increases the accuracy of the expanded square root operator by adding two high order terms to the traditional split step fourier propagator first we approximate the square root operator using taylor expansion around the reference background velocity then we apply the first kindchebyshevpolynomialstoeconomizetheresultsofthe taylor expansion finally we optimize the constant coefficients using the globally optimized scheme which are fixed and feasible for arbitrary velocity models theoretical analysis and numericalexperimentshavedemonstratedthatthemethodhasvery highaccuracyandexceedstheunoptimizedfourierfinite difference propagator for the entire range of practical velocity contrasts the accurate propagation angle of the method is always about 60 under the relative error of 1 for complex media with weak moderate and even strong lateral velocity contrasts the method allows us to handle wide angle propagations and strong lateral velocity contrast simultaneously by using fourier transform alone only four 2d fourier transforms are required for each step of 3d wavefield extrapolation and the computing cost is similar to that of the fourier finite difference method compared with the finite difference method our method has no twowaysplittingerrori e azimuthal anisotropyerrorfor3dcases and almost no numerical dispersion for coarse grids in addition it has strong potential to be accelerated when an enhanced fast fouriertransformalgorithmemerges

explicit 3 d migration using equiripple polynomial expansion and laplacian synthesis in f x explicit finite difference depth migration schemes wavefield downward extrapolation is carried out through spatial convolution using finite length filters existing methods for computing these filters are based on nonlinear least squares with a high computational cost or on taylor series expansion which is suboptimal in the 3 d case the physics of wavefield extrapolation requires 2 d extrapolation filters with circular symmetry recently mcclellan transformation has been used to design circularly symmetric extrapolation operators but this approach exhibits artifacts when the data are not spatially oversampled we describe an alternative method to take advantage of the circular symmetry the radial response of the filter is expanded as a polynomial in the laplacian which is synthesized as the sum of two 1 d second derivative filters using the laplacian rather than the mcclellan transform yields an artifact free impulse response for wavenumbers much closer to the nyquist wavenumber at the s

prefactored optimized compact finite difference schemes for second spatial derivatives we have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives first instead of emphasizing high resolution of a single monochromatic wave we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers this leads to the development of the optimized compact schemes whose coefficients will be determined by fourier analysis and the least squares optimization in the wavenumber domain the resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations however solving for each spatial derivative using these compact schemes requires the inversion of a band matrix to resolve this issue we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes which can be solved explicitly we achieve this by ensuring a property that the real numerical wavenumbers o

a positive conservative method for magnetohydrodynamics based on hll and roe methods the exact riemann problem solutions of the usual equations of ideal magnetohydrodynamics mhd can have negative pressures if the initial data has b 0 this creates a problem for numerical solving because in a first order finite volume conservative godunov type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three dimensional sense we show that by allowing magnetic monopoles in mhd equations and properly taking into account the magnetostatic contribution to the lorentz force an additional source term appears in faraday s law only using the harten lax vanleer hll riemann solver and discretizing the source term in a specific manner we obtain a method which is positive and conservative we show positivity by extensive numerical experimentation this mhd hll method is positive and conservative but rather diffusive thus we show how to hybridize this method with the roe method to obtain a much higher accuracy while still retaining positivity the result is a fully robust positive conservative scheme for ideal mhd whose accuracy and efficiency properties are similar to the first order roe method and which keeps b small in the same sense as powell s method as a special case a method with similar characteristics for accuracy and robustness is obtained for the euler equations as well

ideal glm mhd about the entropy consistent nine wave magnetic field divergence diminishing ideal magnetohydrodynamics equations abstract the paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics first we show how to extend the ideal magnetohydrodynamics mhd equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics as a byproduct of these derivations we show that not all of the commonly used divergence cleaning extensions of the ideal mhd equations are thermodynamically consistent secondly we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy it includes a mechanism to control the discrete divergence error of the magnetic field by construction and is galilean invariant we implement the new high order mhd solver in the adaptive mesh refinement code flash where we compare the divergence cleaning efficiency to the constrained transport solver available in flash unsplit staggered mesh scheme

factor copula models for replicated spatial data abstractwe propose a new copula model that can be used with replicated spatial data unlike the multivariate normal copula the proposed copula is based on the assumption that a common factor exists and affects the joint dependence of all measurements of the process moreover the proposed copula can model tail dependence and tail asymmetry the model is parameterized in terms of a covariance function that may be chosen from the many models proposed in the literature such as the matern model for some choice of common factors the joint copula density is given in closed form and therefore likelihood estimation is very fast in the general case one dimensional numerical integration is needed to calculate the likelihood but estimation is still reasonably fast even with large datasets we use simulation studies to show the wide range of dependence structures that can be generated by the proposed model with different choices of common factors we apply the proposed model to spatial temperature data and com

a novel averaging technique for discrete entropy stable dissipation operators for ideal mhd entropy stable schemes can be constructed with a specific choice of the numerical flux function first an entropy conserving flux is constructed secondly an entropy stable dissipation term is added to this flux to guarantee dissipation of the discrete entropy present works in the field of entropy stable numerical schemes are concerned with thorough derivations of entropy conservative fluxes for ideal mhd however as we show in this work if the dissipation operator is not constructed in a very specific way it cannot lead to a generally stable numerical scheme the two main findings presented in this paper are that the entropy conserving flux of ismail roe can easily break down for certain initial conditions commonly found in astrophysical simulations and that special care must be taken in the derivation of a discrete dissipation matrix for an entropy stable numerical scheme to be robust we present a convenient novel averaging procedure to evaluate the entropy jacobians of the ideal mhd and the compressible euler equations that yields a discretization with favorable robustness properties

a uniquely defined entropy stable matrix dissipation operator for high mach number ideal mhd and compressible euler simulations we describe a unique averaging procedure to design an entropy stable dissipation operator for the ideal magnetohydrodynamic mhd and compressible euler equations often in the derivation of an ent

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

a staggered grid finite difference scheme optimized in the time space domain for modeling scalar wave propagation in geophysical problems abstract for modeling scalar wave propagation in geophysical problems using finite difference schemes optimizing the coefficients of the finite difference operators can reduce numerical dispersion most optimized finite difference schemes for modeling seismic wave propagation suppress only spatial but not temporal dispersion errors we develop a novel optimized finite difference scheme for numerical scalar wave modeling to control dispersion errors not only in space but also in time our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil we design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers dispersion analysis and numerical examples demonstrate that our optimized finite difference scheme is computationally up to 2 5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2d or 3d problem compared with the high order finite difference scheme using the same new stencil our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy this new optimized finite difference scheme is particularly useful for large scale 3d scalar wave modeling and inversion

globally optimal finite difference schemes based on least squares abstractspatial finite difference fd coefficients are usually determined by the taylor series expansion te or optimization methods the former can provide high accuracy on a smaller wavenumber or frequency zone and the latter can give moderate accuracy on a larger zone present optimization methods applied to calculate fd coefficients are generally gradient like or global optimization like algorithms and thus iterations are involved they are more computationally expensive and sometimes the global solution may not be found i examined second order spatial derivatives and computed the optimized spatial fd coefficients over the given wavenumber range using the least squares ls method the results indicated that the fd accuracy increased with increasing operator length and decreasing wavenumber range therefore for the given error and operator length globally optimal spatial fd coefficients can be easily obtained some optimal fd coefficients were given i developed schemes to obtain optimized ls

optimal staggered grid finite difference schemes based on the minimax approximation method with the remez algorithm abstractfinite difference fd schemes especially staggered grid fd sfd schemes have been widely implemented for wave extrapolation in numerical modeling whereas the conventional approach to compute the sfd coefficients is based on the taylor series expansion te method which leads to unignorable great errors at large wavenumbers in the solution of wave equations we have developed new optimal explicit sfd esfd and implicit sfd isfd schemes based on the minimax approximation ma method with a remez algorithm to enhance the numerical modeling accuracy starting from the wavenumber dispersion relations we derived the optimal esfd and isfd coefficients by using the ma method to construct the objective functions and solve the objective functions with the remez algorithm we adopt the ma based esfd and isfd coefficients to solve the spatial derivatives of the elastic wave equations and perform numerical modeling numerical analyses indicated that the ma based esfd and isfd schemes can overcome th

entropy stable discontinuous interfaces coupling for the three dimensional compressible navier stokes equations non linear entropy stability and a summation by parts sbp framework are used to derive entropy stable interior interface coupling for the semi discretized three dimensional 3d compressible navier stokes equations a complete semidiscrete entropy estimate for the interior domain is achieved combining a discontinuous entropy conservative operator of any order 1 2 with an entropy stable coupling condition for the inviscid terms and a local discontinuous galerkin ldg approach with an interior penalty ip procedure for the viscous terms the viscous penalty contributions scale with the inverse of the reynolds number re so that for re their contributions vanish and only the entropy stable inviscid interface penalty term is recovered this paper extends the interface couplings presented 1 2 and provides a simple and automatic way to compute the magnitude of the viscous ip term the approach presented herein is compatible with any diagonal norm summation by parts sbp spatial operator including finite element finite volume finite difference schemes and the class of high order accurate methods which include the large family of discontinuous galerkin discretizations and flux reconstruction schemes this note relies on the formalism introduced in 1 3 and complements the new class of interior entropy stable sbp operators of any order for the 3d compressible navier stokes equations on unstructured grids that was proposed in 1 2 to keep the notation as simple as possible a uniform cartesian grid is considered in the derivation however the extension to generalized curvilinear coordinates and unstructured grids follows immediately if the transformation from computational to physical space preserves the semi discrete geometric conservation 4 the proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two dimensional 2d and 3d viscous subsonic and supersonic flows presented in 1 5 6

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

uniform dispersion reduction schemes for the one dimensional wave equation in isotropic media finite difference fd methods for the wave equation in general suffer from numerical dispersion although fd methods based on accuracy give good dispersion at low frequencies waves tend to disperse for higher wavenumber in this work we will give a unified methodology to derive dispersion reduction fd schemes for the one dimensional wave equation and this new method can reduce dispersion error uniformly for all wavenumbers up to the nyquist stability criteria are given and stability analysis is done for each generated scheme

an entropy stable nodal discontinuous galerkin method for the resistive mhd equations part i theory and numerical verification abstract the first paper of this series presents a discretely entropy stable discontinuous galerkin dg method for the resistive magnetohydrodynamics mhd equations on three dimensional curvilinear unstructured hexahedral meshes compared to other fluid dynamics systems such as the shallow water equations or the compressible navier stokes equations the resistive mhd equations need special considerations because of the divergence free constraint on the magnetic field for instance it is well known that for the symmetrization of the ideal mhd system as well as the continuous entropy analysis a non conservative term proportional to the divergence of the magnetic field typically referred to as the powell term must be included as a consequence the mimicry of the continuous entropy analysis in the discrete sense demands a suitable dg approximation of the non conservative terms in addition to the ideal mhd terms this paper focuses on the resistive mhd equations our first contribution is a proof that the resistive terms are symmetric and positive definite when formulated in entropy space as gradients of the entropy variables which enables us to show that the entropy inequality holds for the resistive mhd equations this continuous analysis is the key for our dg discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality typically termed entropy stability our second contribution is a detailed derivation and analysis of the discretization on three dimensional curvilinear meshes the discrete analysis relies on the summation by parts property which is satisfied by the dg spectral element method dgsem with legendre gauss lobatto lgl nodes although the divergence free constraint is included in the non conservative terms the resulting method has no particular treatment of the magnetic field divergence errors which might pollute the solution quality our final contribution is the extension of the standard resistive mhd equations and our dg approximation with a divergence cleaning mechanism that is based on a generalized lagrange multiplier glm as a conclusion to the first part of this series we provide detailed numerical validations of our dgsem method that underline our theoretical derivations in addition we show a numerical example where the entropy stable dgsem demonstrates increased robustness compared to the standard dgsem