convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

a mixed formulation for the brinkman problem the brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium where fractures bubbles or channels alternate inside a porous matrix in this work we explore a novel mixed formulation of the brinkman problem based on the hodge decomposition of the vector laplacian introducing the flow s vorticity as an additional unknown this formulation allows for a uniformly stable and conforming discretization by standard finite elements nedelec raviart thomas piecewise discontinuous a priori error estimates for the discretization error in the h rm curl omega h rm div omega l 2 omega norm of the solution which are optimal with respect to the approximation properties of finite element spaces are obtained the theoretical results are illustrated with numerical experiments finally the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for h rm curl and h

coarse stability and bifurcation analysis using time steppers a reaction diffusion example evolutionary pattern forming partial differential equations pdes are often derived as limiting descriptions of microscopic kinetic theory based models of molecular processes e g reaction and diffusion the pde dynamic behavior can be probed through direct simulation time integration or more systematically through stability bifurcation calculations time stepper based approaches like the recursive projection method shroff g m keller h b 1993 siam j numer anal 30 1099 1120 provide an attractive framework for the latter we demonstrate an adaptation of this approach that allows for a direct effective coarse bifurcation analysis of microscopic kinetic based models this is illustrated through a comparative study of the fitzhugh nagumo pde and of a corresponding lattice boltzmann model

uniformly stable discontinuous galerkin discretization and robust iterative solution methods for the brinkman problem we consider robust iterative methods for discontinuous galerkin dg h div omega conforming discretizations of the brinkman equations we describe a simple uzawa iteration for the solution of this problem which requires the solution of a nearly incompressible linear elasticity type equation with mass term on every iteration we prove the uniform stability of the dg discretization for both problems then we analyze variable v cycle and w cycle multigrid methods with nonnested bilinear forms we prove that these algorithms are robust and their convergence rates are independent of the parameters in the brinkman problem and of the mesh size the theoretical analysis is confirmed by numerical results

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

multilevel monte carlo for stochastic differential equations with small noise we consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by brownian motions in the commonly occurring small noise regime we consider i standard monte carlo methods combined with numerical discretization algorithms tailored to the small noise setting and ii a multilevel monte carlo method combined with a standard euler maruyama implementation under the assumptions we make on the underlying model the multilevel method combined with euler maruyama is often found to be the most efficient option moreover under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of o 1 per sample a key step in our analysis is to analyze the variance between two coupled paths directly as opposed to their l 2 distance careful simulations are provided to illustrate the asymptotic results

an answer to an open problem on triangular norms in this paper we shall give an example to show that an open problem of fodor on triangular norm which is collected by klement et al in problems on triangular norms and related operators fuzzy sets and systems 145 2004 471 479 has a negative solution

a staggered discontinuous galerkin method for the stokes system discontinuous galerkin dg methods are a class of efficient tools for solving fluid flow problems there are in the literature many greatly successful dg methods in this paper a new staggered dg

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

ortholinear and paralinear semi copulas abstract a new method to construct semi copulas is introduced these semi copulas are called ortholinear resp paralinear semi copulas and their construction is based on linear interpolation on segments that are perpendicular resp parallel to the diagonal of the unit square the classes of ortholinear and paralinear quasi copulas are characterized as well

analysis of a pseudostress based mixed finite element method for the brinkman model of porous media flow in this paper we introduce and analyze a new mixed finite element method for the two dimensional brinkman model of porous media flow with mixed boundary conditions we use a dual mixed formulation in which the main unknown is given by the pseudostress the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply the babuska brezzi theory to establish sufficient conditions for the well posedness of the resulting continuous and discrete formulations in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k ge 0 k 0 for the pseudostress and continuous piecewise polynomials of degree k 1 k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem suitable auxiliary problems the continuous inf sup conditions satisfied by the bilinear forms involved a discrete helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators are the main tools for proving the reliability then helmholtz s decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are employed to show the efficiency finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithm are provided

a new staggered dg method for the brinkman problem robust in the darcy and stokes limits abstract in this paper we propose a novel staggered discontinuous galerkin method for the brinkman problem on general polygonal meshes the proposed method is robust in the stokes and darcy limits in addition hanging nodes can be automatically incorporated in the construction of the method which are desirable features in practical applications there are three unknowns involved in our formulation namely velocity gradient velocity and pressure unlike the original staggered dg formulation proposed for the stokes equations in kim et al 2013 we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns which is tailored to yield an optimal l 2 error estimates for velocity gradient velocity and pressure independent of the viscosity coefficient moreover by choosing suitable projection superconvergence can be proved for l 2 error of velocity finally several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions the equivalence between the method of centre manifold and the method of multiple scales was examined when they are applied to the weakly nonlinear stability analysis for fluid motions the relationship between the landau equations of the seventh order which were reduced from the navier stokes equations by applying these two methods is clarified how we should define the disturbance amplitude to obtain the complete equivalence between them is discussed

a staggered dg method of minimal dimension for the stokes equations on general meshes abstract in this paper a locally conservative lowest order staggered discontinuous galerkin method is developed for the stokes equations the proposed method allows rough grids and is based on the partition of the domain into arbitrary shapes of quadrilaterals or polygons which makes the method highly desirable for practical applications a priori error analysis covering low regularity is demonstrated a new postprocessing scheme for the velocity earning faster convergence is constructed next adaptive mesh refinement is highly appreciated on quadrilateral and polygonal meshes since hanging nodes are allowed therefore we propose two guaranteed type error estimators in l 2 error of stress and energy error of the postprocessed velocity respectively numerical experiments confirm our theoretical findings and illustrate the flexibility of the proposed method and accuracy of the guaranteed upper bounds

parametric estimation of stationary stochastic processes under indirect observability for many natural turbulent dynamic systems observed high dimensional dynamic data can be approximated at slow time scales by a process xt driven by a systems of stochastic differential equations sdes when one tries to estimate the parameters of this unobservable sdes systems there is a clear mismatch between the available data and the sdes dynamics to be parametrized here we formalize this indirect observability framework as follows

fitting sde models to nonlinear kac zwanzig heat bath models we study a class of particle in a heat bath models which are a generalization of the well known kac zwanzig class of models but where the coupling between the distinguished particle and the n heat bath particles is through nonlinear springs the heat bath particles have random initial data drawn from an equilibrium gibbs density the primary objective is to approximate the forces exerted by the heat bath which we do not want to resolve by a stochastic process by means of the central limit theorem for gaussian processes and heuristics based on linear response theory we demonstrate conditions under which it is natural to expect that the trajectories of the distinguished particle can be weakly approximated as n by the solution of a markovian sde the quality of this approximation is verified by numerical calculations with parameters chosen according to the linear response theory alternatively the parameters of the effective equation can be chosen using time series analysis this is done and agreement with linear response theory is shown to be good

some new characterizations and properties of quasi copulas in this paper we characterize the concept of quasi copula in three different ways as a special subclass of aggregation operators in terms of its associated volume and for the bivariate case in terms of non increasing tracks on 0 1 2 we also provide sufficient conditions and new properties for quasi copulas

mortar formulation for a class of staggered discontinuous galerkin methods abstract a mortar formulation is developed and analyzed for a class of staggered discontinuous galerkin sdg methods applied to second order elliptic problems in two dimensions the computational domain consists of nonoverlapping subdomains and a triangulation is provided for each subdomain which need not conform across subdomain interfaces this feature allows a more flexible design of discrete models for problems with complicated geometries shocks or singular points a mortar matching condition is enforced on the solutions across the subdomain interfaces by introducing a lagrange multiplier space moreover optimal convergence rates in both l 2 and discrete energy norms are proved numerical results are presented to show the performance of the method

solution of an open problem for associative copulas an open problem on triangular norms recently posed by klement et al fuzzy sets and systems 145 2004 471 is discussed in the class of associative copulas

a staggered discontinuous galerkin method of minimal dimension on quadrilateral and polygonal meshes in this paper we first propose and analyze a locally conservative lowest order staggered discontinuous galerkin method of minimal dimension on general quadrilateral polygonal meshes for elliptic

support recovery for the drift coefficient of high dimensional diffusions consider the problem of learning the drift coefficient of a p dimensional stochastic differential equation from a sample path of length t we assume that the drift is parametrized by a high dimensional vector and study the support recovery problem when both p and t can tend to infinity in particular we prove a general lower bound on the sample complexity t by using a characterization of mutual information as a time integral of conditional variance due to kadota zakai and ziv for linear stochastic differential equations the drift coefficient is parametrized by a p times p matrix that describes which degrees of freedom interact under the dynamics in this case we analyze a ell 1 regularized least squares estimator and prove an upper bound on t that nearly matches the lower bound on specific classes of sparse matrices

a stable numerical algorithm for the brinkman equations by weak galerkin finite element methods this paper presents a stable numerical algorithm for the brinkman equations by using weak galerkin wg finite element methods the brinkman equations can be viewed mathematically as a combination of the stokes and darcy equations which model fluid flow in a multi physics environment such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain in such applications the flow is dominated by darcy in some regions and by stokes in others it is well known that the usual stokes stable elements do not work well for darcy flow and vice versa the challenge of this study is on the design of numerical schemes which are stable for both the stokes and the darcy equations this paper shows that the wg finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both darcy and the stokes dominated flows error estimates of optimal order are established for the corresponding wg finite element solutions the paper also presents some numerical experiments that demonstrate the robustness reliability flexibility and accuracy of the wg method for the brinkman equations

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

estimation of space dependent diffusions and potential landscapes from non equilibrium data in scientific topics ranging from protein folding to the thermohaline ocean circulation it is useful to model the effective macroscopic dynamics of complex systems as noise driven motion in a potential landscape in this paper we consider the estimation of such models from a collection of short non equilibrium trajectories between two points in phase space we generalize a recently introduced spectral methodology for the estimation of diffusion processes from timeseries so that it can be used for non equilibrium data this methodology makes use of the spectral properties leading eigenvalue eigenfunction pairs of the fokker planck operator associated with the diffusion process it is well suited to infer stochastic differential equations that give effective coarse grained descriptions of multiscale systems the generalization to the non equilibrium situation is illustrated with numerical examples in which potentials and diffusion coefficients are estimated from ensembles of short trajectories

path space variational inference for non equilibrium coarse grained systems in this paper we discuss information theoretic tools for obtaining optimized coarse grained molecular models for both equilibrium and non equilibrium molecular simulations the latter are ubiquitous in physicochemical and biological applications where they are typically associated with coupling mechanisms multi physics and or boundary conditions in general the non equilibrium steady states are not known explicitly as they do not necessarily have a gibbs structure the presented approach can compare microscopic behavior of molecular systems to parametric and non parametric coarse grained models using the relative entropy between distributions on the path space and setting up a corresponding path space variational inference problem the methods can become entirely data driven when the microscopic dynamics are replaced with corresponding correlated data in the form of time series furthermore we present connections and generalizations of force matching methods in coarse graining with path space information methods we demonstrate the enhanced transferability of information based parameterizations to different observables at a specific thermodynamic point due to information inequalities we discuss methodological connections between information based coarse graining of molecular systems and variational inference methods primarily developed in the machine learning community however we note that the work presented here addresses variational inference for correlated time series due to the focus on dynamics the applicability of the proposed methods is demonstrated on high dimensional stochastic processes given by overdamped and driven langevin dynamics of interacting particles

equal order finite elements with local projection stabilization for the darcy brinkman equations abstract for the darcy brinkman equations which model porous media flow we present an equal order h1 conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique the method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation we prove a priori error estimates in a mesh dependent norm as well as in the l2 norm for velocity and pressure in particular we obtain optimal order of convergence in l2 for the pressure in the darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient numerical results for different values of the coefficients in the darcy brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory

a unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

an optimization principle for deriving nonequilibrium statistical models of hamiltonian dynamics a general method for deriving closed reduced models of hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation given a vector of resolved variables selected to describe the macroscopic state of the system a family of quasi equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities specifically a cost function is constructed to quantify the lack of fit to the microscopic dynamics of any feasible path of densities from the statistical model it is an ensemble averaged weighted squared norm of the residual that results from submitting the path of densities to the liouville equation the path that minimizes the time integral of the cost function determines the best fit evolution of the mean resolved vector the closed reduced equations satisfied by the optimal path are derived by hamilton jacobi theory when expressed in terms of the macroscopic variables these equations have the generic structure of governing equations for nonequilibrium thermodynamics in particular the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes the adjustable closure parameters in the best fit reduced equations depend explicitly on the arbitrary weights that enter into the lack of fit cost function two particular model reductions are outlined to illustrate the general method in each example the set of weights in the optimization principle contracts into a single effective closure parameter

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

pulse dynamics in low reynolds number interfacial hydrodynamics experiments and theory abstract we analyze interaction of nonlinear pulses in active dispersive dissipative nonlinear media a particular example of such media is a viscous thin film coating a vertical fibre experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place in such events larger pulses catch up with smaller ones and annihilate them however for certain flow conditions and after a certain distance from the inlet no more coalescence is observed and the flow is described by quasi equilibrium solitary pulses interacting continuously with each other through attractions and repulsions and eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole this experimental study represents the first evidence of formation of bound states in low reynolds number interfacial hydrodynamics to gain theoretical insight into the interaction of the pulses and formation of bound states we derive a weakly nonlinear model for the flow the generalized kuramoto sivashinsky gks equation that retains the fundamental mechanisms of the wave evolution namely dominant nonlinearity instability stability and dispersion much like in the experiments the spatio temporal evolution of the gks equation is dominated by quasi stationary solitary pulses which continuously interact with each other through coalescence events or attractions repulsions to understand the latter case we utilize a weak interaction theory for the solitary pulses of the gks equation the theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses or alternatively the evolution of the separation distances by analyzing the fixed points of this system we obtain bound states of interacting pulses for two pulses we provide a criterion for the existence of a countable infinite or finite number of bound states depending on the strength of the dispersive term in the equation the interaction theory and resulting bound states are corroborated by computations of the full equation we also find qualitative agreement between the theory and the experiments

problems on triangular norms and related operators abstract a number of open problems on triangular norms and related operators was posed during the 24th linz seminar on fuzzy set theory triangular norms and related operators in many valued logics held in february 2003 they are collected here together with some other open problems in this context and with some problems which were posed earlier and have been solved in the meantime

discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics many physical systems are described by nonlinear differential equations that are too complicated to solve in full a natural way to proceed is to divide the variables into those that are of direct interest and those that are not formulate solvable approximate equations for the variables of greater interest and use data and statistical methods to account for the impact of the other variables in the present paper we consider time dependent problems and introduce a fully discrete solution method which simplifies both the analysis of the data and the numerical algorithms the resulting time series are identified by a narmax nonlinear autoregression moving average with exogenous input representation familiar from engineering practice the connections with the mori zwanzig formalism of statistical physics are discussed as well as an application to the lorenz 96 system

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

additive global noise delays turing bifurcations we apply a stochastic center manifold method to the calculation of noise induced phase transitions in the stochastic swift hohenberg equation this analysis is applied to the reduced mode equations that result from fourier decomposition of the field variable and of the temporal noise the method shows a pitchfork bifurcation at lower perturbation order but reveals a novel additive noise induced postponement of the turing bifurcation at higher order good agreement is found between the theory and the numerics for both the reduced and the full system the results are generalizable to a broad class of nonlinear spatial systems

new stochastic mode reduction strategy for dissipative systems we present a new methodology for studying non hamiltonian nonlinear systems based on an information theoretical extension of a renormalization group technique using a modified maximum entropy principle we obtain a rigorous dimensionally reduced description for such systems the neglected degrees of freedom by this reduction are replaced by a systematically defined stochastic process under a constraint on the second moment this then forms the basis of a computationally efficient method numerical computations for the generalized kuramoto sivashinsky equation support our method and reveal that the long time underlying stochastic process of the fast unresolved modes obeys a universal distribution that does not depend on the initial conditions and which we rigorously derive by the maximum entropy principle

analysis of an sdg method for the incompressible navier stokes equations in this paper we analyze a staggered discontinuous galerkin sdg method for the incompressible navier stokes equations the method is based on a novel splitting of the nonlinear convection term and results in a skew symmetric discretization of it as a result the sdg discretization has a better conservation property and numerical stability property the aim of this paper is to develop a mathematical theory for this method in particular we will show that the sdg method is well posed and has an optimal rate of convergence a superconvergence result will also be shown

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

a center stable manifold theorem for differential equations in banach spaces we prove a center stable manifold theorem for a class of differential equations in infinite dimensional banach spaces

numerical strategies for filtering partially observed stiff stochastic differential equations in this paper we present a fast numerical strategy for filtering stochastic differential equations with multiscale features this method is designed such that it does not violate the practical linear observability condition and more importantly it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach the proposed filtering algorithm comprises of a macro filter that borrows ideas from the heterogeneous multiscale methods and a micro filter that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro filter and macroscopic observations at asynchronous times we will show that the proposed micro filter is equivalent to solving an inverse problem for parameterizing differential equations numerically we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions especially when the microscopic dynamics is not mixing at all

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

adaptive sub sampling for parametric estimation of gaussian diffusions we consider a gaussian diffusion xt ornstein uhlenbeck process with drift coefficient and diffusion coefficient 2 and an approximating process y varepsilon t converging to xt in l2 as e 0 we study estimators hat gamma varepsilon hat sigma 2 varepsilon which are asymptotically equivalent to the maximum likelihood estimators of and 2 respectively we assume that the estimators are based on the available n n e observations extracted by sub sampling only from the approximating process y varepsilon t with time step e we characterize all such adaptive sub sampling schemes for which hat gamma varepsilon hat sigma 2 varepsilon are consistent and asymptotically efficient estimators of and 2 as e 0 the favorable adaptive sub sampling schemes are identified by the conditions e 0 0 e and n which implies that we sample from the process y varepsilon t with a vanishing but coarse time step e e this study highlights the necessity to sub sample at adequate rates when the observations are not generated by the underlying stochastic model whose parameters are being estimated the adequate sub sampling rates we identify seem to retain their validity in much wider contexts such as the additive triad application we briefly outline

a robust finite element method for darcy stokes flow finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed the system is approximately a linear stokes problem when the perturbation parameter is large while it degenerates to a mixed formulation of poisson s equation as the perturbation parameter tends to zero it is established basically by numerical experiments that most of the proposed finite element methods for stokes problem or the mixed poisson s system are not well behaved uniformly in the perturbation parameter this is used as the motivation for introducing a new robust finite element which exhibits this property

the role of additive and multiplicative noise in filtering complex dynamical systems covariance inflation is an ad hoc treatment that is widely used in practical real time data assimilation algorithms to mitigate covariance underestimation owing to model errors nonlinearity or an

low rank solvers for unsteady stokes brinkman optimal control problem with random data abstract we consider the numerical simulation of an optimal control problem constrained by the unsteady stokes brinkman equation involving random data more precisely we treat the state the control the target or the desired state as well as the viscosity as analytic functions depending on uncertain parameters this allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic galerkin finite element method discretization of the model the discrete problem yields a prohibitively high dimensional saddle point system with kronecker product structure we develop a new alternating iterative tensor method for an efficient reduction of this system by the low rank tensor train representation besides we propose and analyze a robust schur complement based preconditioner for the solution of the saddle point system the performance of our approach is illustrated with extensive numerical experiments based on two and three dimensional examples where the full problem size exceeds one billion degrees of freedom the developed tensor train scheme reduces the solution storage by two three orders of magnitude depending on discretization parameters

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

convexity conditions on t norms and their additive generators abstract let t be an archimedean continuous t norm and 0 1 2 fodor and ovchinnikov have studied a relation between the inequality t max x 0 min x 1 t x x and the convexity of the additive generator of t here we clarify this relation and answer open problems from previous studies

a hybrid high order discretisation of the brinkman problem robust in the darcy and stokes limits abstract in this work we develop and analyse a novel hybrid high order discretisation of the brinkman problem the method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures based on the discrete unknowns we reconstruct inside each element a stokes velocity one degree higher than face unknowns and a darcy velocity in the raviart thomas nedelec space these reconstructed velocities are respectively used to formulate the discrete versions of the stokes and darcy terms in the momentum equation along with suitably designed penalty contributions the proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local stokes or darcy dominated regimes as identified by a dimensionless number which can be interpreted as a friction coefficient the singular limit corresponding to the darcy equation is also fully supported by the method numerical examples corroborate the theoretical results this paper also contains two contributions whose interest goes beyond the specific method and application treated in this work an investigation of the dependence of the constant in the second korn inequality on star shaped domains and its application to the study of the approximation properties of the strain projector in general sobolev seminorms

a new framework for extracting coarse grained models from time series with multiscale structure in many applications it is desirable to infer coarse grained models from observational data the observed process often corresponds only to a few selected degrees of freedom of a high dimensional dynamical system with multiple time scales in this work we consider the inference problem of identifying an appropriate coarse grained model from a single time series of a multiscale system it is known that estimators such as the maximum likelihood estimator or the quadratic variation of the path estimator can be strongly biased in this setting here we present a novel parametric inference methodology for problems with linear parameter dependency that does not suffer from this drawback furthermore we demonstrate through a wide spectrum of examples that our methodology can be used to derive appropriate coarse grained models from time series of partial observations of a multiscale system in an effective and systematic fashion

a weakening of schur concavity for copulas the class of copulas satisfying a new property called weak schur concavity is introduced and studied and the relationships of this property with other known notions of concavity are investigated

a tale of two time scales it is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns however market microstructure poses challenges to this estimation approach as evidenced by recent empirical studies in finance the present work attempts to lay out theoretical grounds that reconcile continuous time modeling and discrete time samples we propose an estimation approach that takes advantage of the rich sources in tick by tick data while preserving the continuous time assumption on the underlying returns under our framework it becomes clear why and where the usual volatility estimator fails when the returns are sampled at the highest frequencies if the noise is asymptotically small our work provides a way of finding the optimal sampling frequency a better approach the two scales estimator works for any size of the noise

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

a priori tests of a stochastic mode reduction strategy several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors proc natl acad sci 96 1999 14687 commun pure appl math 54 2001 891 are developed here in this procedure reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables while the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big typically the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1 2 the examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated burgers hopf equations as a nonlinear heat bath systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables in an important illustrative example presented here the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises even for large value of the correlation time ratio of the order of 1 2 the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non gaussian statistics of the original nonlinear systems with 102 modes extremely well furthermore it is shown here that the standard regression fitting of the second order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example 2002 elsevier science b v all rights reserved

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

comparisons and bounds for expected lifetimes of reliability systems sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non identically distributed lifetimes are expressed in terms of expected lifetimes of components similar evaluations are concluded for the respective mean residual lifetimes in the iid case improved inequalities dependent on a concentration parameter connected to the gini dispersion index are obtained the results can be used to compare systems with component lifetimes ordered in the convex ordering in the inid case some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes and the expected lifetime of single unit for the equivalent systems with iid components the latter can be further simplified in the case of weak schur concavity and schur convexity of the system generalized domination polynomial

an ensemble kalman filter for statistical estimation of physics constrained nonlinear regression models a central issue in contemporary science is the development of nonlinear data driven statistical dynamical models for time series of noisy partial observations from nature or a complex model it has been established recently that ad hoc quadratic multi level regression models can have finite time blow up of statistical solutions and or pathological behavior of their invariant measure recently a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior here a new finite ensemble kalman filtering algorithm is developed for estimating the state the linear and nonlinear model coefficients the model and the observation noise covariances from available partial noisy observations of the state several stringent tests and applications of the method are developed here in the most complex application the perfect model has 57 degrees of freedom involving a zonal east west jet two topographic rossby waves and 54 nonlinearly interacting rossby waves the perfect model has significant non gaussian statistics in the zonal jet with blocked and unblocked regimes and a non gaussian skewed distribution due to interaction with the other 56 modes we only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation numerically we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion including the skew non gaussian distribution and autocorrelation decay on the other hand a similar stochastic model with zero memory levels fails to capture the crucial non gaussian behavior of the zonal jet from the perfect 57 mode model