the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

path integration of the duffing rayleigh oscillator subject to harmonic and stochastic excitations this paper is focused on the duffing rayleigh oscillator subject to harmonic and stochastic excitations using path integration based on the gauss legendre integration scheme first applying methods of harmonic balance and multiple scales to the deterministic case the stabilities of the responses can be analyzed then the steady state periodic solution of probability density can be captured via path integration at the same time the changes of probability density induced by the intensities of harmonic and stochastic excitations are discussed in three cases

computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods the numerical simulation of the deformation of vesicle membranes under simple shear external fluid flow is considered in this paper a saddle point approach is proposed for the imposition of the fluid incompressibility and the membrane inextensibility constraints through lagrange multipliers defined in the fluid and on the membrane respectively using a level set formulation the problem is approximated by mixed finite elements combined with an automatic adaptive mesh procedure at the vicinity of the membrane boundary numerical experiments show that this combination of the saddle point and adaptive mesh method enhances the robustness of the method the effect of inertia on the stability of the vesicle in a shear flow is also investigated

error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations this article considers a posteriori error estimation and anisotropic mesh refinement for three dimensional laminar aerodynamic flow simulations the optimal order symmetric interior penalty discontinuous galerkin discretization which has previously been developed for the compressible navier stokes equations in two dimensions is extended to three dimensions symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model using duality arguments an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients furthermore residual based indicators as well as adjoint based indicators for goal oriented refinement are derived these refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous galerkin dg discretization two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint based error estimation are presented the performance of the proposed discretization error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows

power law relaxation in a complex system omori law after a financial market crash we study the relaxation dynamics of a financial market just after the occurrence of a crash by investigating the number of times the absolute value of an index return is exceeding a given threshold value we show that the empirical observation of a power law evolution of the number of events exceeding the selected threshold a behavior known as the omori law in geophysics is consistent with the simultaneous occurrence of i a return probability density function characterized by a power law asymptotic behavior and ii a power law relaxation decay of its typical scale our empirical observation cannot be explained within the framework of simple and widespread stochastic volatility models

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

relation between volatility correlations in financial markets and omori processes occurring on all scales we analyze the memory in volatility by studying volatility return intervals defined as the time between two consecutive fluctuations larger than a given threshold in time periods following stock market crashes such an aftercrash period is characterized by the omori law which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1 a shock followed by such a power law decay in the rate is here called omori process we find self similar features in the volatility specifically within the aftercrash period there are smaller shocks that themselves constitute omori processes on smaller scales similar to the omori process after the large crash we call these smaller shocks subcrashes which are followed by their own aftershocks we also show that the omori law holds not only after significant market crashes as shown by lillo and mantegna phys rev e 68 016119 2003 but also after intermediate shocks by appropriate detrending we remove the influence of the crashes and subcrashes from the data and find that this procedure significantly reduces the memory in the records moreover when studying long term correlated fractional brownian motion and autoregressive fractionally integrated moving average artificial models for volatilities we find omori type behavior after high volatilities thus our results support the hypothesis that the memory in the volatility is related to the omori processes present on different time scales

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

a priori anisotropic mesh adaptation on implicitly defined surfaces mesh adaptation on surfaces demands particular care due to the important role played by the surface fitting we propose an adaptive procedure based on a new error analysis which combines a rigorous anisotropic estimator for the l 1 norm of the interpolation error with an anisotropic heuristic control of the geometric error we resort to a metric based adaptive algorithm which employs local operations to modify the initial mesh according to the information provided by the error analysis an extensive numerical validation corroborates the robustness of the error analysis as well as of the adaptive procedure

a local problem error estimator for anisotropic tetrahedral finite element meshes the poisson problem is solved by the finite element method on anisotropic tetrahedral or triangular meshes the focus is on adaptive algorithms and in particular on a posteriori error estimators based on the solution of a local problem on anisotropic meshes such estimators cannot be analyzed in the common way known from isotropic meshes the first estimator proposed here is a slight modification of a familiar isotropic counterpart by a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator hence the local problem error estimator also yields reliable upper and lower error bounds on anisotropic meshes the local problems are shown to be well conditioned two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here the results reveal the significance of the proper choice of local problem furthermore the analysis covers dirichlet neumann and robin boundary conditions particular attention is paid to the robin boundary conditions since they require a different treatment than the neumann boundary conditions a numerical example supports the error analysis

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

a geometric discretization and a simple implementation for variational mesh generation and adaptation we present a simple direct discretization for functionals used in the variational mesh generation and adaptation meshing functionals are discretized on simplicial meshes and the jacobian matrix of the continuous coordinate transformation is approximated by the jacobian matrices of affine mappings between elements the advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional moreover the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form which allows a simple implementation of variational mesh generation and adaptation on computer since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods the result in this work can be used to develop simple implementations of those methods numerical examples are given

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

minimization of gradient errors of piecewise linear interpolation on simplicial meshes abstract the paper is devoted to the analysis of optimal simplicial meshes which minimize the gradient error of the piecewise linear interpolation over all conformal simplicial meshes with a fixed number of cells nt we present theoretical results on asymptotic dependencies of lp norms of the gradient error on nt for spaces of arbitrary dimension d our analysis is based on a geometric representation of the gradient error of linear interpolation on a simplex and a relaxed saturation assumption we derive a metric field m p such that a m p quasi uniform mesh is quasi optimal for arbitrary d and p 0 quasi optimal meshes provide the same asymptotics of the lp norm of the gradient error as the optimal meshes

numerical meshfree path integration method for non linear dynamic systems abstract we present a new numerical meshfree path integration mpi method for non linear dynamic systems the obtained mpi method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the mpi method and the ordinary differential equations for the first and second order moments on the basis of gaussian closure method the piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions the good performance of the resulting method is finally shown in the numerical examples by using three specific non linear dynamic systems duffing oscillator subjected to both harmonic and stochastic excitations duffing rayleigh oscillator subjected to both harmonic and stochastic excitations and chen system driven by three different gaussian white noises

an anisotropic zienkiewicz zhu type error estimator for 3d applications we extend the anisotropic zienkiewicz zhu a posteriori error estimator of proceedings of the enumath 2009 uppsala sweden 29 june 3 july 2009 to three dimensions like the standard zienkiewicz zhu estimator the proposed estimator is designed to be independent of the problem at hand is cheap to compute and easy to implement in contrast to the standard zienkiewicz zhu estimator the elementwise counterpart of the proposed estimator explicitly takes into account the geometrical properties of the actual tetrahedron thus in a wide variety of applications the estimator is able to detect the anisotropic features exhibited by the solution of the governing equations a metric based optimization procedure rigorously addressed drives the adaptation of the mesh it is shown numerically to yield quasi optimal triangulations dictating the accuracy vs number of elements behaviour despite being heuristic to some extent in practice the overall anisotropic adaptation procedure turns out to be effective copyright 2010 john wiley sons ltd

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

a posteriori error analysis for finite element solution of one dimensional elliptic differential equations using equidistributing meshes the paper is concerned with the adaptive linear finite element solution of linear one dimensional elliptic differential equations using equidistributing meshes a strategy is developed for defining such meshes based on a residual based a posteriori error estimate the mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation i e the mesh equation an iterative algorithm is proposed for solving this system for the mesh and the finite element solution the existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh which is also established in the current work error bounds for the finite element solution are obtained for the equidistributing and quasi equidistributing meshes they show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes the departure from the equidistributing meshes has only a mild effect on the finite element error numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings

anisotropic norm oriented mesh adaptation for a poisson problem we present a novel formulation for the mesh adaptation of the approximation of a partial differential equation pde the discussion is restricted to a poisson problem the proposed norm oriented formulation extends the goal oriented formulation since it is equation based and uses an adjoint at the same time the norm oriented formulation somewhat supersedes the goal oriented one since it is basically a solution convergent method indeed goal oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that as mesh size is increased more degrees of freedom only this output is proven to tend to its continuous analog while the solution field itself may not converge a remarkable quality of goal oriented metric based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization in the well identified set of metrics of a well defined functional in the new proposed formulation we amplify this advantage we search in the same well identified set of metrics the minimum of a norm of the approximation error the norm is prescribed by the user and the method allows addressing the case of multi objective adaptation like for example in aerodynamics adaptating the mesh for drag lift and moment in one shot in this work we consider the basic linear finite element approximation and restrict our study to l 2 norm in order to enjoy second order convergence numerical examples for the poisson problem are computed

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

an l lp mesh adaptive method for computing unsteady bi fluid flows this paper discusses the contribution of mesh adaptation to high order convergence of unsteady multi fluid flow simulations on complex geometries the mesh adaptation relies on a metric based method controlling the lp norm of the interpolation error and on a mesh generation algorithm based on an anisotropic delaunay kernel the mesh adaptive time advancing is achieved thanks to a transient fixed point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure in the time direction we enforce the equidistribution of the error i e the error minimization in l norm this adaptive approach is applied to an incompressible navier stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows copyright 2010 john wiley sons ltd

continuous mesh framework part i well posed continuous interpolation error in the context of mesh adaptation riemannian metric spaces have been used to prescribe orientation density and stretching of anisotropic meshes but such structures are only considered to compute lengths in adaptive mesh generators in this article a riemannian metric space is shown to be more than a way to compute a length it is proven to be a reliable continuous mesh model in particular we demonstrate that the linear interpolation error can be evaluated continuously on a riemannian metric space from one hand this new continuous framework proves that prescribing a riemannian metric field is equivalent to the local control in the mathbf l 1 norm of the interpolation error this proves the consistency of classical metric based mesh adaptation procedures on the other hand powerful mathematical tools are available and are well defined on riemannian metric spaces calculus of variations differentiation optimization dots whereas these tools are not defined on discrete meshes

an anisotropic fully adaptive algorithm for the solution of convection dominated equations with semi lagrangian schemes abstract we present in this paper an anisotropic fully adaptive spatial temporal algorithm for the solution of convection dominated equations in the context of semi lagrangian schemes we devise the algorithm within a finite element framework suitable for higher order finite elements and derive a newly proposed a posteriori error indicator which allows us to control the local or truncation error in the l 2 norm at each time step this a posteriori error is split into temporal and spatial contributions leading us to define an optimal time step size and an optimal triangulation respectively as regards the spatial adaptation anisotropic unstructured triangular meshes are used to capture the distinctive features of the evolving discrete solution of the governing equations for solutions exhibiting strong anisotropies the orientation shape and size of the mesh triangles are provided by a metric tensor valid for linear and quadratic finite elements finally we show the capabilities of the algorithm for linear and quadratic finite elements by a series of two and three dimensional benchmarks taken from the literature involving purely convective as well as convection dominated problems

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

multiscaling and clustering of volatility abstract the dynamics of prices in stock markets has been studied intensively both experimentally data analysis and theoretically models nevertheless while the distribution of returns of the most important indices is known to be a truncated levy the behaviour of volatility correlations is still poorly understood what is well known is that absolute returns have memory on a long time range this phenomenon is known in financial literature as clustering of volatility in this paper we show that volatility correlations are power laws with a non unique scaling exponent this kind of multiscale phenomenology is known to be relevant in fully developed turbulence and in disordered systems and it is pointed out here for the first time for a financial series in our study we consider the new york stock exchange nyse daily index from january 1966 to june 1998 for a total of 8180 working days

local averaged path integration method approach for nonlinear dynamic systems abstract traditional path integration method always has a limited time step length whose minimum is based on the subdivision degree of integral space in order to numerically analyze a smaller time step probability evolution without significant increase in computing time a new local averaged path integration lapi method is investigated the basic idea of this method is giving a local averaged value of transition probability density function tpdf then the numerical integration of chapman kolmogorov ck equation to the local averaged probability density function pdf will be calculated in lower space subdivision degree second order partial derivative of pdf can be used to correct the deviation between local averaged pdf and the real one it is demonstrated that the proposed method can give a more efficient calculation compared with traditional gauss legendre path integration glpi in the case of our method smaller time step value is permitted with the same space subdivision degree and smaller space subdivision degree is needed to keep the accuracy of stationary pdf with the same time step value this merits also exist when analyzing transient pdf which leads to a better capability of capturing the evolution of the pdf

an a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes a new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes all considerations carry over to anisotropic triangular meshes with minor changes only the lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities in order to prove the upper error bound it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration to measure this correspondence a so called matching function is defined and its discussion shows it to be a useful tool with its help anisotropic interpolation estimates and subsequently the upper error bound are proven additionally it is pointed out how to treat robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes a numerical example supports the anisotropic error analysis

continuous mesh framework part ii validations and applications this paper gives a numerical validation of the continuous mesh framework introduced in part i a loseille and f alauzet siam j numer anal 49 2011 pp 38 60 we numerically show that the interpolation error can be evaluated analytically once analytical expressions of a mesh and a function are given in particular the strong duality between discrete and continuous views for the interpolation error is emphasized on two dimensional and three dimensional examples in addition we show the ability of this framework to predict the order of convergence given a specific adaptive strategy defined by a sequence of continuous meshes the continuous mesh concept is then used to devise an adaptive strategy to control the mathbf l p norm of the continuous interpolation error given the mathbf l p norm of the continuous interpolation error we derive the optimal continuous mesh minimizing this error this exemplifies the potential of this framework as we use a calculus of variations that is not defined on the space of discrete meshes anisotropic adaptations on analytical functions correlate the optimal predicted theoretical order of convergence the extension to a solution of nonlinear pdes is also given comparisons with experiments show the efficiency and the accuracy of this approach

inverse statistics in stock markets universality and idiosyncracy abstract investigations of inverse statistics a concept borrowed from turbulence in stock markets exemplified with filtered dow jones industrial average sp physica a 324 2003 338 343 int j mod phys b 17 2003 4003 4012 we have performed extensive analyses based on unfiltered daily indices and stock prices as well as high frequency 5 min records in numerous stock markets all over the world our analysis confirms that the power law distribution of exit times with an exponent of about 1 5 is universal for all the data sets analyzed in addition all data sets show that the power law scaling in the optimal investment horizon holds but with idiosyncratic exponents specifically 1 5 for the daily data in most of the developed stock markets and the 5 min high frequency data while the values for the daily indexes and stock prices in emerging markets are significantly less than 1 5 we show that there is little chance that the discrepancy in is due to the difference in sample sizes of the two kinds of stock markets

relaxation dynamics of aftershocks after large volatility shocks in the ssec index abstract the relaxation dynamics of aftershocks after large volatility shocks are investigated based on two high frequency data sets of the shanghai stock exchange composite ssec index compared with previous relevant work we have defined main financial shocks based on large volatilities rather than large crashes we find that the occurrence rate of aftershocks with the magnitude exceeding a given threshold for both daily volatility constructed using 1 minute data and minutely volatility using intra minute data decays as a power law the power law relaxation exponent increases with the volatility threshold and is significantly greater than 1 taking financial volatility as the counterpart of seismic activity the power law relaxation in financial volatility deviates remarkably from the omori law in geophysics

anisotropic mesh quality measures and adaptation for polygonal meshes abstract anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes three sets of alignment and equidistribution measures are developed based on least squares fitting generalized barycentric mappings and the singular value decomposition respectively numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given anisotropic metrics based on the second set of quality measures and using a moving mesh partial differential equation an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second order elliptic equations numerical examples are presented to demonstrate the effectiveness of the method

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

image segmentation with eigenfunctions of an anisotropic diffusion operator we propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation the diffusion matrix is defined based on the input image the eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image an important property of the model is that for many input images the first few eigenfunctions are close to being piecewise constant which makes them useful as the basis for a variety of applications such as image segmentation and edge detection the eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models the relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods the new continuous model is also different from energy minimization methods such as active contour models in that no initial guess is required for in the current model a numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented it is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes several interesting features of the model are examined in numerical examples and possible applications are discussed

anisotropic adaptive simulation of transient flows using discontinuous galerkin methods an anisotropic adaptive analysis procedure based on a discontinuous galerkin finite element discretization and local mesh modification of simplex elements is presented the procedure is applied to transient two and three dimensional problems governed by euler s equation a smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified the mesh metric field in smooth portions of the domain is controlled by a hessian matrix constructed using a variational procedure to calculate the second derivatives the transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain copyright c 2004 john wiley sons ltd

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

a local anisotropic adaptive algorithm for the solution of low mach transient combustion problems a novel numerical algorithm for the simulation of transient combustion problems at low mach and moderately high reynolds numbers is presented these problems are often characterized by the existence of a large disparity of length and time scales resulting in the development of directional flow features such as slender jets boundary layers mixing layers or flame fronts this makes local anisotropic adaptive techniques quite advantageous computationally in this work we propose a local anisotropic refinement algorithm using for the spatial discretization unstructured triangular elements in a finite element framework for the time integration the problem is formulated in the context of semi lagrangian schemes introducing the semi lagrange galerkin slg technique as a better alternative to the classical semi lagrangian sl interpolation the good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem the flame vortex interaction first a premixed methane air flame vortex interaction with simplified transport and chemistry description test i is considered results are found to be in excellent agreement with those in the literature proving the superior performance of the slg scheme when compared with the classical sl technique and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement as a more realistic example we then conduct simulations of non premixed hydrogen air flame vortex interactions test ii using a more complex combustion model which involves state of the art transport and chemical kinetics in addition to the analysis of the numerical features this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature

numerical simulation for a duffing oscillator driven by colored noise using nonstandard difference scheme abstract the present paper is devoted to the numerical solution of the fokker planck fp equation associated to the duffing oscillator driven by colored noise we propose an improved discretization of the standard fp operator splitting method which renders the scheme unconditionally stable this scheme is successfully applied to explore the influence of the colored noise on the probability density

anisotropic mesh adaptation for finite element solution of anisotropic porous medium equation abstract anisotropic porous medium equation apme is developed as an extension of the porous medium equation pme for anisotropic porous media a special analytical solution is derived for apme for time independent diffusion anisotropic mesh adaptation for linear finite element solution of apme is discussed and numerical results for two dimensional examples are presented the solution errors using anisotropic adaptive meshes show second order convergence

the econometrics of ultra high frequency data ultra high frequency data is defined to be a full record of transactions and their associated characteristics the transaction arrival times and accompanying measures can be analyzed as marked point processes the acd point process developed by engle and russell 1998 is applied to ibm transactions arrival times to develop semiparametric hazard estimates and conditional intensities combining these intensities with a garch model of prices produces ultra high frequency measures of volatility both returns and variances are found to be negatively influenced by long durations as suggested by asymmetric information models of market micro structure

a new functional for variational mesh generation and adaptation based on equidistribution and alignment conditions abstract a new functional is presented for variational mesh generation and adaptation it is formulated based on combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter the functional is shown to be coercive but not convex a solution procedure using a discrete moving mesh partial differential equation is employed it is shown that the element volumes and altitudes of a mesh trajectory of the mesh equation associated with the new functional are bounded away from zero and the mesh trajectory stays nonsingular if it is so initially numerical examples demonstrate that the new functional performs comparably as an existing one that is also based on the equidistribution and alignment conditions and known to work well but contains an additional parameter

path integral solutions of the governing equation of sdes excited by levy white noise abstract in this paper the probability density functions pdfs of scalar stochastic differential equations sdes subject to stable levy white noise are investigated the path integral pi method is extended to solve one dimensional space fractional fokker planck kolmogorov fpk equations which are the governing equations corresponded to scalar sdes excited by stable levy white noise first we derive a short time solution of the one dimensional space fractional fpk equation which is used in the chapman kolmogorov smoluchowski cks equation to obtain the pi solution then the accuracy of the pi solution is analyzed theoretically in terms of its characteristic function our results demonstrate that the pi method has a higher accuracy than the first order finite difference method for one step iteration in time finally several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the pi method for solving one dimensional space fractional fpk equations we find that the pi solution agrees well with the exact solution or the monte carlo one

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

how a nonconvergent recovered hessian works in mesh adaptation hessian recovery has been commonly used in mesh adaptation for obtaining the required magnitude and direction information of the solution error unfortunately a recovered hessian from a linear finite element approximation is nonconvergent in general as the mesh is refined it has been observed numerically that adaptive meshes based on such a nonconvergent recovered hessian can nevertheless lead to an optimal error in the finite element approximation this also explains why hessian recovery is still widely used despite its nonconvergence in this paper we develop an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered hessian to the exact one numerical results show that this closeness assumption is satisfied by the recovered hessian obtained with commonly used hessian recovery methods moreover it is shown that the finite element error changes gradually with the closeness of the recovered hessian this provides a

grid adaption based on modified anisotropic diffusion equations formulated in the parametric domain a new grid adaption algorithm for problems in computational fluid dynamics is presented the basic equations are derived from a variational problem formulated in the parametric domain of the mapping that defines the existing grid modification of the basic equations provides desirable properties in boundary layers the resulting modified anisotropic diffusion equations are solved for the computational coordinates as functions of the parametric coordinates and these functions are numerically inverted numerical examples show that the algorithm is robust that shocks and boundary layers are well resolved on the adapted grid and that the flow solution becomes a globally smooth function of the computational coordinates

on the stabilized finite element method for steady convection dominated problems with anisotropic mesh adaptation in this work we combine the use of the streamline upwind petrov galerkin supg method with anisotropic mesh adaptation to obtain accurate solutions for steady convection dominated problems the anisotropic mesh adaptation framework is introduced in the context of a local mesh generation method based on a mesh topology modification and a minimal volume principle a new route to get a metric field directly at the node of the mesh is highlighted using the length distribution tensor and an edge based error analysis an a posteriori error estimation is applied to the stabilized finite element solution detecting automatically all sharp gradients inner and boundary layers the numerical examples show that the use of the anisotropic mesh adaptation algorithm allows the recovery of the global convergence order of the numerical schemes while producing accurate and oscillation free numerical solutions

adaptive anisotropic meshing for steady convection dominated problems obtaining accurate solutions for convection diffusion equations is challenging due to the presence of layers when convection dominates the diffusion to solve this problem we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution three main ingredients are used first the streamline upwind petrov galerkin method is used to produce a stabilized solution second an adapted metric tensor is computed from the approximate solution third optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal voronoi tessellation algorithm our algorithm is tested on a variety of two dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non physical oscillations in the numerical approximation

probability of large movements in financial markets abstract based on empirical financial time series we show that the silence breaking probability follows a super universal power law the probability of observing a large movement is inversely proportional to the length of the on going low variability period such a scaling law has been previously predicted theoretically r kitt j kalda physica a 353 2005 480 assuming that the length distribution of the low variability periods follows a multi scaling power law

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

on the shape of tetrahedra from bisection we present a procedure for bisecting a tetrahedron t successively into an infinite sequence of tetrahedral meshes j 0 91 g2 which has the following properties 1 each mesh sfn is conforming 2 there are a finite number of classes of similar tetrahedra in all the 3n n 0 3 for any tetrahedron t7 in s7n jj t cl1 t where ti is a tetrahedron shape measure and cl is a constant 4 3 j c2 1 2 ni3j t where d t denotes the diameter of tetrahedron t1 and c2 is a constant estimates of cl and c2 are provided properties 2 and 3 extend similar results of stynes and adler and of rosenberg and stenger respectively for the 2 d case the diameter bound in property 4 is better than one given by kearfott

a new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates a new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented it generates anisotropic adaptive meshes as quasi uniform ones in some metric space with the metric tensor being computed based on hierarchical a posteriori error estimates a global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution instead of solving the global error problem exactly which is costly in general we solve it iteratively using the symmetric gaus seidel method numerical results show that a few gs iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation the new method is compared with several strategies using local error estimators or recovered hessians numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

memory effects in the statistics of interoccurrence times between large returns in financial records we study the statistics of the interoccurrence times between events above some threshold q in two kinds of multifractal data sets multiplicative random cascades and multifractal random walks with vanishing linear correlations we show that in both data sets the relevant quantities probability density functions and the autocorrelation function of the interoccurrence times as well as the conditional return period are governed by power laws with exponents that depend explicitly on the considered threshold by studying a large number of representative financial records market indices stock prices exchange rates and commodities we show explicitly that the interoccurrence times between large daily returns follow the same behavior in a nearly quantitative manner we conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner