discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

explicit residual based a posteriori error estimation for finite element discretizations of the helmholtz equation computation of the constant and new measures of error estimator quality abstract this paper continues the study of an explicit residual based a posteriori error estimator developed for finite element discretizations of the helmholtz equation in the context of time harmonic exterior acoustics problems in previous papers 1 3 the error estimator was derived and subsequently applied to h adaptive computations of model problems in two dimensions in the present paper a methodology is established for computing error estimates these error estimates are then analyzed in detail the error estimates are made possible by computing the scaling constant which exists in error estimators of this type in our case this constant is difficult to obtain an algorithm for computing the constant is described it is noted that the error indicators used for adaptive computations do not require knowledge of this constant several measures are used to analyze the quality of the error estimates providing a complete description of the error estimator in the context of these problems we compute global effectivity indices and utilize the local element wise effectivity index in computing three additional measures the first of these is the robustness index the others are the modified robustness index and the distribution index which are introduced for the first time it is found that the global effectivity index varies with problem parameters in particular the wave number robustness indices also vary due to the local error definition and wave number values of the distribution index are much better however indicating the suitability of the error estimator in guiding efficient adaptive mesh refinement the analysis of the error estimator is carried out on relatively coarse meshes typical of those used in practical engineering computations

a symmetric formulation for computing transient shallow water flows abstract the shallow water equations fall into the category of symmetric advective diffusive systems with source terms these types of equations can be very effectively solved using stabilized methods such as supg and gls a semi discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows special care has been taken in the design of the operators in order to improve the performance for unsteady calculations the solution is advanced in time via a predictor multi corrector algorithm which includes as special cases the second order trapezoidal rule a first order explicit method and a first order implicit method equivalent to the constant in time element of the space time formulation

on the continuous differentiability of inter spike intervals of synaptically connected cortical spiking neurons in a neuronal network we derive conditions for continuous differentiability of inter spike intervals isis of spiking neurons with respect to parameters decision variables of an external stimulating input current that drives a recurrent network of synaptically connected neurons the dynamical behavior of individual neurons is represented by a class of discontinuous single neuron models we report here that isis of neurons in the network are continuously differentiable with respect to decision variables if 1 a continuously differentiable trajectory of the membrane potential exists between consecutive action potentials with respect to time and decision variables and 2 the partial derivative of the membrane potential of spiking neurons with respect to time is not equal to the partial derivative of their firing threshold with respect to time at the time of action potentials our theoretical results are supported by showing fulfillment of these conditions for a class of known bidimensional spiking neuron models

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

importance of the cutoff value in the quadratic adaptive integrate and fire model the quadratic adaptive integrate and fire model izhikevich 2003 2007 is able to reproduce various firing patterns of cortical neurons and is widely used in large scale simulations of neural networks this model describes the dynamics of the membrane potential by a differential equation that is quadratic in the voltage coupled to a second equation for adaptation integration is stopped during the rise phase of a spike at a voltage cutoff value vc or when it blows up subsequently the membrane potential is reset and the adaptation variable is increased by a fixed amount we show in this note that in the absence of a cutoff value not only the voltage but also the adaptation variable diverges in finite time during spike generation in the quadratic model the divergence of the adaptation variable makes the system very sensitive to the cutoff changing vc can dramatically alter the spike patterns furthermore from a computational viewpoint the divergence of the adaptation variable implies that the time steps for numerical simulation need to be small and adaptive however divergence of the adaptation variable does not occur for the quartic model touboul 2008 and the adaptive exponential integrate and fire model brette gerstner 2005 hence these models are robust to changes in the cutoff value

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

random matrix theory and portfolio optimization in moroccan stock exchange abstract in this work we use random matrix theory to analyze eigenvalues and see if there is a presence of pertinent information by using marcenko pastur distribution thus we study cross correlation among stocks of casablanca stock exchange moreover we clean correlation matrix from noisy elements to see if the gap between predicted risk and realized risk would be reduced we also analyze eigenvectors components distributions and their degree of deviations by computing the inverse participation ratio this analysis is a way to understand the correlation structure among stocks of casablanca stock exchange portfolio

a dual petrovgalerkin finite element method for the convectiondiffusion equation we present a minimum residual finite element method based on a dual petrovgalerkin formulation for convectiondiffusion problems in a higher order adaptive continuous galerkin setting the method borrows concepts from both the discontinuous petrovgalerkin dpg method by demkowicz and gopalakrishnan 2011 and the method of variational stabilization by cohen dahmen and welper 2012 and it can also be interpreted as a variational multiscale method in which the fine scales are defined through a dual orthogonality condition a key ingredient in the method is the proper choice of dual norm used to measure the residual and we present two choices which are observed to be robust in both convection and diffusion dominated regimes as well as a proof of stability for quasi uniform meshes and a method for the weak imposition of boundary conditions numerically obtained convergence rates in 2d are reported and benchmark numerical examples are given to illustrate the behavior of the method

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

bifurcation analysis of a general class of nonlinear integrate and fire neurons in this paper we define a class of formal neuron models being computationally efficient and biologically plausible i e able to reproduce a wide range of behaviors observed in in vivo or in vitro recordings of cortical neurons this class includes for instance two models widely used in computational neuroscience the izhikevich and the brette gerstner models these models consist of a 4 parameter dynamical system we provide the full local bifurcation diagram of the members of this class and show that they all present the same bifurcations an andronov hopf bifurcation manifold a saddle node bifurcation manifold a bogdanov takens bifurcation and possibly a bautin bifurcation i e all codimension two local bifurcations in a two dimensional phase space except the cusp among other global bifurcations this system shows a saddle homoclinic bifurcation curve we show how this bifurcation diagram generates the most prominent cortical neuron behaviors this study leads us to introduce a new neuron mode

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

a robust dpg method for convection dominated diffusion problems ii adjoint boundary conditions and mesh dependent test norms we introduce a dpg method for convection dominated diffusion problems the choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter demkowicz and heuer 2011 18 we propose a new inflow boundary condition which regularizes the adjoint problem allowing the use of a stronger test norm the robustness of the method is proven and numerical experiments demonstrate the method s robust behavior

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

a finite element formulation of compressible flows using various sets of independent variables abstract this paper presents a finite element method for the simulation of compressible flows the navier stokes and euler equations are solved in the conservation form using various sets of independent variables a variational formulation is developed based upon a variant of the petrov galerkin method and uses a shock capturing operator an adaptive algorithm based on a particular residual norm is proposed several numerical examples are presented to demonstrate the performances of each set of variables in solving compressible high speed flows

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

an arbitrary lagrangian eulerian finite element method for solving three dimensional free surface flows abstract this paper discusses numerical solution of unsteady three dimensional free surface flows the governing equilibrium equations are written in the framework of the arbitrary lagrangian eulerian kinematic description the corresponding variational formulation is established afterwards since the variational problems are nonlinear with respect to the moving coordinates a second order approximate variational problem is derived after a consistent linearization of the referential motion stability of the discrete formulations is ensured with the help of a new stabilization method a robust preconditioned gmres algorithm is then used to solve the resulting set of nonlinear equations finally the computational algorithms are assessed through numerical studies of various problems a large sloshing flow in a three dimensional reservoir a discharge flow from a reservoir simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom and a three dimensional practical hydraulic problem

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

a quantum anharmonic oscillator model for the stock market abstract a financially interpretable quantum model is proposed to study the probability distributions of the stock price return the dynamics of a quantum particle is considered an analog of the motion of stock price then the probability distributions of price return can be computed from the wave functions that evolve according to schrodinger equation instead of a harmonic oscillator in previous studies a quantum anharmonic oscillator is applied to the stock in liquid market the leptokurtic distributions of price return can be reproduced by our quantum model with the introduction of mixed state and multi potential the trend following dominant market in which the price return follows a bimodal distribution is discussed as a specific case of the illiquid market

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

discontinuous galerkin methods for advection diffusion reaction problems we apply the weighted residual approach recently introduced in f brezzi et al comput methods appl mech engrg 195 2006 pp 3293 3310 to derive discontinuous galerkin formulations for advection diffusion reaction problems we devise the basic ingredients to ensure stability and optimal error estimates in suitable norms and propose two new methods

a generalized linear integrate and fire neural model produces diverse spiking behaviors for simulations of neural networks there is a trade off between the size of the network that can be simulated and the complexity of the model used for individual neurons in this study we describe a generalization of the leaky integrate and fire model that produces a wide variety of spiking behaviors while still being analytically solvable between firings for different parameter values the model produces spiking or bursting tonic phasic or adapting responses depolarizing or hyperpolarizing after potentials and so forth the model consists of a diagonalizable set of linear differential equations describing the time evolution of membrane potential a variable threshold and an arbitrary number of firing induced currents each of these variables is modified by an update rule when the potential reaches threshold the variables used are intuitive and have biological significance the model s rich behavior does not come from the differential equations which are linear but rather from complex update rules this single neuron model can be implemented using algorithms similar to the standard integrate and fire model it is a natural match with event driven algorithms for which the firing times are obtained as a solution of a polynomial equation

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

a space time coupled p version least squares finite element formulation for unsteady fluid dynamics problems this paper presents a p version least squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled the dimensionless form of the differential equations describing the problem are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c element approximation the element properties are derived by utilizing p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach for non linear problems the non linear algebraic equations resulting from the least squares process are solved using newton s method with a line search this procedure results in a symmetric hessian matrix equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the generality success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection diffusion and burgers equations the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements the work presented in this paper provides the basis for the extension of the space time coupled least squares minimization concept to two and three dimensional unsteady fluid flow

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

econophysics financial time series from a statistical physics point of view in recent years physicists have started applying concepts and methods of statistical physics to study economic problems the word econophysics is sometimes used to refer to this work much recent work is focused on understanding the statistical properties of nancial time series one reason for this interest is that nancial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that nancial time series viewed from a dierent perspective might yield new results this article reviews the results of three recent phenomenological studies i the probability distribution of stock price uctuations stock price uctuations occur in all magnitudes in analogy to earthquakes from tiny uctuations to drastic events such as market crashes the distribution of price uctuations decays with a power law tail well outside the levy stable regime and describes uctuations that dier by as much as eight orders of magnitude in addition this distribution preserves its functional form for uctuations on time scales that dier by three orders of magnitude from 1 min up to approximately 10 d ii correlations in nancial time series while price uctuations themselves have rapidly decaying correlations the magnitude of uctuations measured by either the absolute value or the square of the price uctuations has correlations that decay as a power law and persist for several months iii correlations among dierent companies the third result bears on the application of random matrix theory to understand the correlations among price uctuations of any two dierent stocks from a study of the eigenvalue statistics of the cross correlation matrix constructed from price uctuations of the leading 1000 stocks we nd that the largest 1 of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix whereas the rest of the eigenvalues conform to random matrix behavior suggesting that these 1 of the eigenvalues contain system specic information about correlated time evolution of dierent companies c 2000 published by elsevier science b v all rights reserved

a random matrix theory approach to financial cross correlations it is common knowledge that any two rms in the economy are correlated even rms belonging to dierent sectors of an industry may be correlated because of indirect correlations how can we analyze and understand these correlations this article reviews recent results regarding cross correlations between stocks specically we use methods of random matrix theory rmt which originated from the need to understand the interactions between the constituent elements of complex interacting systems to analyze the cross correlation matrix c of returns we analyze 30 min returns of the largest 1000 us stocks for the 2 year period 1994 1995 we nd that the statistics of approximately 20 of the largest eigenvalues 2 show deviations from the predictions of rmt to test that the rest of the eigenvalues are genuinely random we test for universal properties such as eigenvalue spacings and eigenvalue correlations and demonstrate that c shares universal properties with the gaussian orthogonal ensemble of random matrices the statistics of the eigenvectors of c conrm the deviations of the largest few eigenvalues from the rmt prediction we also nd that these deviating eigenvectors are stable in time in addition we quantify the number of rms that participate signicantly to an eigenvector using the concept of inverse participation ratio borrowed from localization theory c 2000 published by elsevier science b v all rights reserved

a fic based stabilized finite element formulation for turbulent flows abstract we present a new stabilized finite element fem formulation for incompressible flows based on the finite increment calculus fic framework onate 1998 in comparison to existing fic approaches for fluids this formulation involves a new term in the momentum equation which introduces non isotropic dissipation in the direction of velocity gradients we also follow a new approach to the derivation of the stabilized mass equation inspired by recent developments for quasi incompressible flows onate et al 2014 the presented fic fem formulation is used to simulate turbulent flows using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation

neural dynamics bifurcations and firing rates in a quadratic integrate and fire model with a recovery variable i deterministic behavior we study the dynamics of a quadratic integrate and fire model of a single compartment neuron with a slow recovery variable as input current and parameters describing timescales recovery variable and postspike reset change analysis of a codimension 2 bifurcation reveals that the domain of attraction of a stable hyperpolarized rest state interacts subtly with reset parameters which reposition the system state after spiking we obtain explicit approximations of instantaneous firing rates for fixed values of the recovery variable and use the averaging theorem to obtain asymptotic firing rates as a function of current and reset parameters along with the different phase plane geometries these computations provide explicit tools for the interpretation of different spiking patterns and guide parameter selection in modeling different cortical cell types

modeling of steep layers in singularly perturbed diffusion reaction equation via flexible fine scale basis abstract we present a new stabilized method for the diffusion reaction equation which develops sharp boundary and or internal layers for the reaction dominated case i e singularly perturbed case the method relies on an improved expression for the stabilization parameter that is derived via the fine scale variational formulation facilitated by the variational multiscale vms framework the proposed fine scale basis consists of enrichment functions which may be nonzero at element edges the derived stabilization parameter enjoys spatial variation over element interiors and along inter element boundaries that helps model the rapid variation of the residual of the euler lagrange equations over the domain this feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects the method is able to better satisfy the maximum principle as compared to other existing methods we present a priori mathematical analysis of the stability and convergence of the method for the diffusion reaction equation optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems as well as for problems with steep boundary layers stability and accuracy features of the method for problems with discontinuous forcing function internal layers and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted

an adaptive stabilized conforming finite element method via residual minimization on dual discontinuous galerkin norms abstract we design and analyze a new adaptive stabilized finite element method we construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf sup stability we formulate this residual minimization as a stable saddle point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement numerical results on an advection reaction model problem show competitive error reduction rates when compared to discontinuous galerkin methods on uniformly refined meshes and smooth solutions moreover the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

continuous interior penalty hp finite element methods for advection and advection diffusion equations a continuous interior penalty hp finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced error estimates are obtained for first order and advection dominated transport operators the analysis relies on three technical results that are of independent interest an hp inverse trace inequality a local discontinuous to continuous hp interpolation result and hp error estimates for continuous l2 orthogonal projections

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

collective behavior of stock price movements a random matrix theory approach we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory rmt developed in physics to describe the energy levels ofcomplex systems rmt makes predictions f universal properties that do not depend on the interactions between the elements comprising the system deviations from rmt provide clues regarding system speci c properties we compare the statistics ofthe cross correlation matrix c whose elements cij are the correlation coe cients ofprice uctuations ofstock i and j against a random matrix having the same symmetry properties it is found that rmt methods can distinguish random and non random parts of c the non random part of c which deviates from rmt results provides information regarding genuine collective behavior among stocks c 2001 published by elsevier science b v

random matrix theory filters in portfolio optimisation a stability and risk assessment abstract random matrix theory rmt filters applied to covariance matrices of financial returns have recently been shown to offer improvements to the optimisation of stock portfolios this paper studies the effect of three rmt filters on the realised portfolio risk and on the stability of the filtered covariance matrix using bootstrap analysis and out of sample testing we propose an extension to an existing rmt filter based on krzanowski stability which is observed to reduce risk and increase stability when compared to other rmt filters tested we also study a scheme for filtering the covariance matrix directly as opposed to the standard method of filtering correlation where the latter is found to lower the realised risk on average by up to 6 7 we consider both equally and exponentially weighted covariance matrices in our analysis and observe that the overall best method out of sample was that of the exponentially weighted covariance with our krzanowski stability based filter applied to the correlation matrix we also find that the optimal out of sample decay factors for both filtered and unfiltered forecasts were higher than those suggested by riskmetrics j p morgan reuters riskmetrics technical document technical report 1996 http www riskmetrics com techdoc html with those for the latter approaching a value of 1 in conclusion rmt filtering reduced the realised risk on average and in the majority of cases when tested out of sample but increased the realised risk on a marked number of individual days in some cases more than doubling it

random matrix theory for portfolio optimization a stability approach we apply random matrix theory rmt to an empirically measured financial correlation matrix c and show that this matrix contains a large amount of noise in order to determine the sensitivity of the spectral properties of a random matrix to noise we simulate a set of data and add different volumes of random noise having ascertained that the eigenspectrum is independent of the standard deviation of added noise we use rmt to determine the noise percentage in a correlation matrix based on real data from s p500 eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart finally we attempt to separate the noisy part from the non noisy part of c we apply an existing technique to cleaning c and then discuss its associated problems we propose a technique of filtering c that has many advantages from the stability point of view over the existing method of cleaning

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

on the simulation of nonlinear bidimensional spiking neuron models bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations these models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time coupled to a second equation for adaptation spikes are emitted when the membrane potential blows up or reaches a cutoff i the precise simulation of the spike times and of the adaptation variable is critical for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times we thoroughly study the precision of fixed time step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded as the cutoff value is increased in the evaluation of the two crucial quantities the spike time and the value of the adaptation variable at this time precise evaluation of these quantities therefore involves very small time steps and long simulation times in order to achieve a fixed absolute precision in a reasonable computational time we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane and compare this algorithm with fixed time step euler scheme and other more accurate simulation algorithms