local projection stabilization for the oseen problem and its interpretation as a variational multiscale method we propose to apply the recently introduced local projection stabilization to the numerical computation of the oseen equation at high reynolds number the discretization is done by nested finite element spaces using a priori error estimation techniques we prove the convergence of the method the a priori estimates are independent of the local peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth moreover we show how this method may be cast in the framework of variational multiscale methods we indicate what modeling assumptions must be made to use the method for large eddy simulations

a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

lattice tiling and density conditions for subspace gabor frames abstract the well known density theorem in time frequency analysis establishes the connection between the existence of a gabor frame g g a b e 2 i b m x g x a n m n z d for l 2 r d and the density of the time frequency lattice a z d b z d this is also tightly related to lattice tiling and packing in this paper we investigate the density theorem for gabor systems in l 2 s with s being an a z d periodic subset of r d we characterize the existence of a gabor frame for l 2 s in terms of a condition that involves the haar measure of the group generated by a z d and b t 1 z d this new characterization is used to recover the density theorem and several related known results in the literature additionally we apply this approach to obtain the density theorems for multi windowed and super gabor frames for l 2 s

the balian low theorem for the symplectic form on r2d in this paper we extend the balian low theorem which is a version of the uncertainty principle for gabor weyl heisenberg systems to functions of several variables in particular we first prove the balian low theorem for arbitrary quadratic forms then we generalize further and prove the balian low theorem for differential operators associated with a symplectic basis for the symplectic form on r2d

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

vector valued weak gabor dual frames on discrete periodic sets the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples

on the weyl heisenberg frames generated by simple functions let x n 0 cn e x n with cn n 0 l1 and let a 1 0 a 1 be a weyl heisenberg system e2 imx x na m n z we show that if e 0 1 and some modulo extension of e then a 1 is a frame for each 0 a 1 for certain a respectively if and only if the analytic function h z n 0 cnzn has no zero on the unit circle z z 1 these results extend the case of casazza and kalton 2002 6 that x i 1k 0 1 x ni and a 1 which brought together the frame theory and the function theory on the closed unit disk our techniques of proofs are based on the zak transform and the distribution of fractional parts of na n z

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

balian low phenomenon for subspace gabor frames in this work the balian low theorem is extended to gabor also called weyl heisenberg frames for subspaces and more particularly its relationship with the unique gabor dual property for subspace gabor frames is pointed out to achieve this goal the subspace gabor frames which have a unique gabor dual of type i resp type ii are defined and characterized in terms of the zak transform for the rational parameter case this characterization is then used to prove the balian low theorem for subspace gabor frames along the same line the same characterization is used to prove a duality theorem for the unique gabor dual property which is an analogue of the ron and shen duality theorem

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

gabor super frames with hermite functions we investigate vector valued gabor frames sometimes called gabor superframes based on hermite functions hn let h h0 h1 hn be the vector of the first n 1 hermite functions we give a complete characterization of all lattices lambda subseteq mathbb r 2 such that the gabor system rm e 2 pi i lambda 2 t bf h t lambda 1 lambda lambda 1 lambda 2 in lambda is a frame for l 2 mathbb r mathbb c n 1 as a corollary we obtain sufficient conditions for a single hermite function to generate a gabor frame and a new estimate for the lower frame bound the main tools are growth estimates for the weierstrass function a new type of interpolation problem for entire functions on the bargmann fock space and structural results about vector valued gabor frames

approximation of exponential order of the attractor of a turbulent flow abstract in this work we consider the two dimensional navier stokes equations with periodic boundary conditions describing the evolution of homogeneous turbulent flows we use the method of debussche and temam a debussche and r temam convergent families of approximate inertial manifolds j math pures appl to appear to construct approximate inertial manifolds aims whose order decreases exponentially fast with respect to the dimension of the manifold we recall that an aim is a smooth manifold of solutions m such that the attractor a is contained in a neighborhood of m of thinness is called the order of the manifold the dependence of all the constants with respect to physical parameters especially the grashof number is made explicit

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

explicit small scale velocity simulation for high re turbulent flows a new approach two level simulation tls is developed based on the explicit reconstruction of the small scale velocity by solving the small scale governing equations on the domain with reduced dimension representing a collection of one dimensional lines embedded in the three dimensional flow domain a coupled system of equations that is not based on an eddy viscosity hypothesis is derived based on the decomposition of flow variables into the large scale and the small scale components without introducing the concept of filtering simplified treatment of the small scale equations is proposed based on neglecting the small scale advective and dissipative terms in the line orthogonal directions fully coupled simulation of the forced isotropic turbulence at re l 65 114 demonstrates potential benefits and challenges of the tls approach for high re turbulent flows using coarse grids

on the fock space of metaanalytic functions abstract we develop the theory on the fock space of metaanalytic functions a generalization of some recent results on the fock space of polyanalytic functions we show that the metaanalytic bargmann transform is a unitary mapping between vector valued hilbert spaces and metaanalytic fock spaces a reproducing kernel of the metaanalytic fock space is derived in an explicit form furthermore we establish a complete characterization of all lattice sampling and interpolating sequence for the fock space of metaanalytic functions

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

spectrum efficient localized orthogonal waveforms closing the gap with the balian low theorem the balian low theorem blt states the fundamental impossibility to design waveforms for rm l 2 mathbb r which 1 form an orthogonal set 2 are time frequency localized and 3 attain a critical waveform density such that they form an orthogonal basis this article closes the gap between existing waveform designs and the blt the main contribution is the design of orthogonal time frequency localized spectrum efficient waveforms for hexagonal lattices the waveform design is adaptive by a single design parameter which tradesoff time frequency localization with the waveform density as the orthogonalization procedure is based on employing the minimum number of most time frequency localized waveforms hermite functions it is argued that the results may be optimal in terms of combined spectrum efficiency and time frequency localization an example is provided for waveforms for a hexagonal lattice which are quasi orthogonal time frequency localized and up to 99 of the critical waveform density although the designed waveforms are not strictly orthogonal their cross correlation can be made arbitrarily small the robustness in doubly dispersive channels and the efficiency for multiuser scenarios are discussed and compared to conventional orthogonal frequency division multiplexing ofdm

characterization of coorbit spaces with phase space covers abstract we show that coorbit spaces can be characterized in terms of arbitrary phase space covers which are families of phase space multipliers associated with partitions of unity this generalizes previously known results for time frequency analysis to include time scale decompositions as a by product we extend the existing results for time frequency analysis to an irregular setting

sequential optimization and mixed uncertainty analysis method for reliability based optimization a reliability based optimization method under both aleatory and epistemic uncertainties is studied the mixed uncertainties are analyzed by combined probability and evidence theory if the mixed uncertainty analysis is directly embedded in reliability based optimization to quantify the uncertain features of each search point it would be computationally prohibitive to address this problem a sequential optimization and mixed uncertainty analysis method is proposed to decompose the reliability based optimization problem into separate deterministic optimization and mixed uncertainty analysis subproblems which are solved sequentially and alternately until convergence is achieved the research focus is how to transform the reliability based optimization problem into its quasi equivalent deterministic formulation according to the information obtained in the uncertainty analysis it is proposed to first decompose the total reliability target into each focal element of the epistemic uncertainties so as to sim

gabor schauder bases and the balian low theorem the balian low theorem is a strong form of the uncertainty principle for gabor systems that form orthonormal or riesz bases for l2 r in this paper we investigate the balian low theorem in the setting of schauder bases we prove that new weak versions of the balian low theorem hold for gabor schauder bases but we constructively demonstrate that several variants of the blt can fail for gabor schauder bases that are not riesz bases we characterize a class of gabor schauder bases in terms of the zak transform and product a2 weights the riesz bases correspond to the special case of weights that are bounded away from zero and infinity

vector valued nonstationary gabor frames abstract as an extension of gabor frames nonstationary gabor nsg frames were recently introduced in adaptive signal analysis they allow for efficient reconstruction with flexible sampling and varying window functions in this paper we generalize the notion of nsg frames from l 2 r to the vector valued hilbert space l 2 r c l and investigate the resulting vector valued nsg frames we derive a walnut s representation of the mixed frame operator and provide some necessary sufficient conditions for a vector valued nsg system to be a frame for l 2 r c l furthermore we show the existence of painless vector valued nsg frames and of vector valued nsg frames with fast decaying window functions

a multiscale v p discretization for flow problems this paper gives a comprehensive numerical analysis of a multiscale method for equilibrium navier stokes equations the method includes pressure regularization and eddy viscosity stabilizations both acting only on the finest scales this method allows for equal order velocity pressure spaces as well as the linear constant pair and the usual pk pk 1 pair we show the method is optimal in a natural energy norm for all of these pairs of spaces and provide guidance in choosing the regularization parameters

multilevel methods in space and time for the navier stokes equations we investigate the discretization in time of numerical schemes based on multilevel spatial splittings for the two dimensional periodic navier stokes equations the approximate solution is computed as the sum of a low frequency component and a high frequency one these two terms are advanced in time using different stepsizes we show improved stability conditions with respect to the classical galerkin method we derive error estimates that indicate that the high frequency term can be integrated less often we address implementation issues and show that the method should yield a significant gain in computing time

a multilevel algorithm for large eddy simulation of turbulent compressible flows abstract a multilevel method for large eddy simulation of turbulent compressible flows is proposed the method relies on the splitting of the turbulent flowfield into several frequency bands in space and time each band being associated to a specific computational grid in physical space this allows to take into account in a deterministic way the information contained on finer grids a subgrid model adapted to such a decomposition based on a generalization of the germano s identity to multilevel decomposition is also introduced the approach is validated by several multilevel simulations in a subsonic plane channel flow configuration for a low and a high value of the reynolds number while reductions of the cpu times up to 80 are obtained

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

rational time frequency super gabor frames and their duals abstract this paper addresses general super gabor systems with rational time frequency product lattices a zak transform matrix method is developed for such super gabor systems we characterize complete super gabor systems and super gabor frames given a super gabor frame we obtain an explicit expression of its canonical dual and a parametrization of all its super gabor duals and prove that the canonical dual is the norm minimal one among all super gabor duals we also prove that a super gabor frame has a unique super gabor dual if and only if it is a riesz basis

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

the existence of tight gabor duals for gabor frames and subspace gabor frames abstract let k and l be two full rank lattices in r d we give a complete characterization for all the gabor frames that admit tight dual of the same type the characterization is given in terms of the center valued trace of the von neumann algebra generated by the left regular projective unitary representations associated with the time frequency lattice k l two applications of this characterization were obtained i we are able to prove that every gabor frame has a tight dual if and only if the volume of k l is less than or equal to 1 2 ii we are able to obtain sufficient or necessary conditions for the existence of tight gabor pseudo duals for subspace gabor frames in various cases in particular we prove that every subspace gabor frame has a tight gabor pseudo dual if either the volume v k l 1 2 or v k l 2 moreover if k z d l z d with 1 then a subspace gabor frame g g l k has a tight gabor pseudo dual only when g g l k itself is already tight

density theorems for sampling and interpolation in the bargmann fock space we give a complete description of sampling and interpolation in the bargmann fock space based on a density concept of beurling roughly speaking a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von neumann lattice and similarly a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von neumann lattice

reliability based design optimization with dependent interval variables summary reliability based design optimization rbdo ensures design requirements are satisfied at required reliability levels in the presence of uncertainty the recent rbdo methodologies can accommodate two types of uncertainty aleatory type represented by random variables and epistemic type represented by interval variables however only independent interval variables can be handled the objective of this research is to introduce dependent interval variables into rbdo the dependency of interval variables comes from equality and inequality constraints with the intervals the reliability is also an interval and its lower bound is considered to preserve conservativeness an efficient optimization method is developed to solve rbdo problems the method is an extension of the existing framework of sequential optimization and reliability analysis with the inclusion of inequality constraints of interval variables the new method is efficient because it decouples the optimization process from the process of reliability and interval analyses the new method is applied to the rbdo of a slider mechanism with joint clearances which are known as dependent interval variables copyright 2012 john wiley sons ltd

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

sampling theorems on locally compact groups from oscillation estimates we present a general approach to derive sampling theorems on locally compact groups from oscillation estimates we focus on the l2 stability of the sampling operator by using notions from frame theory this approach yields particularly simple and transparent reconstruction procedures we then apply these methods to the discretization of discrete series representations and to paley wiener spaces on stratified lie groups

a p q version of bourgain s theorem let 1 p q satisfy 1 p 1 q 1 we construct an orthonormal basis b n for l 2 r such that p b n and q bn are both uniformly bounded in n here f inf r x a f x 2 dx 1 2 this generalizes a theorem of bourgain and is closely related to recent results on the balian low theorem

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

dilation of the weyl symbol and balian low theorem the key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their weyl symbols this observation has two implications in time frequency analysis first it implies the stability of general gabor frames under small dilations of the time frequency set previously known only for the case that the time frequency set is a lattice secondly it allows us to derive a new balian low theorem blt for gabor systems with window in the standard window class and with general time frequency families in contrast to the classical versions of blt the new blt does not only exclude orthonormal bases and riesz bases at critical density but indeed it even excludes irregular gabor frames at critical density

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

stability analysis of the nonlinear galerkin method our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations and to study the stability of the schemes spatial discretization can be performed by either spectral or pseudospectral methods finite elements or finite differences time discretization is done by two level schemes partly or fully explicit the algorithms that we present stem from the study of the evolution equations from the dynamical systems point of view they are based on a differentiated treatment of the small and large wave lengths and they are particularly adapted to the integration of such equations on large intervals of time introduction the long time integration of dissipative evolution equations has emerged as a new type of problem in numerical analysis and its appearance is due to the considerable increase in computing power that we have seen during the last years much effort has been devoted in the past to the approximation of evolution equations on finite intervals of time or in the dynamically simple case where the solution converges to a steady stationary state as time goes to infinity however in many physically relevant situations the solutions to a dissipative evolution equation do not converge to a stationary solution they rather remain time dependent and they converge to a compact attractor that encompasses the nonlinear dynamics hence and whether this is addressed implicitly or explicitly the numerical integration of the evolution equation is then closely related to the approximation of the attractor that may be a complicated set even a fractal inertial algorithms are new integration algorithms that are based on the approximation of the attractor by smooth simple manifolds namely the approximate inertial manifolds these algorithms will be recalled below but at this point let us point out that one of their main features is a differentiated treatment of small and large wave lengths an inertial manifold is an exact law of dependence of small wave lengths in terms of the large ones the small wave received april 3 1990 1980 mathematics subject classification 1985 revision primary 65f10 65f35 65n20 35a40 1991 american mathematical society 0025 5718 91 1 00 25 perpage

vector valued gabor frames associated with periodic subsets of the real line the notion of vector valued frame also called superframe was first introduced by balan in the context of multiplexing it has significant applications in mobile communication satellite communication and computer area network for vector valued gabor analysis existent literatures mostly focus on l 2 r c l instead of its subspace let a 0 and s be an a z periodic measurable set in r i e s a z s this paper addresses gabor frames in l 2 s c l with rational time frequency product they can model vector valued signals to appear periodically but intermittently and the projections of gabor frames in l 2 r c l onto l 2 s c l cannot cover all gabor frames in l 2 s c l if s r by introducing a suitable zak transform matrix we characterize completeness and frame condition of gabor systems obtain a necessary and sufficient condition on gabor duals of type i resp ii for a general gabor frame and establish a parametrization expression of gabor duals of type i resp ii all our conclusions are closely related to corresponding zak transform matrices this allows us to easily realize these conclusions by designing the corresponding matrix valued functions an example theorem is also presented to illustrate the efficiency of our method

reliability analysis for multidisciplinary systems with random and interval variables tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems most of the approaches are only capable of dealing with random variables modeled by probability distributions both random and interval variables however may exist in multidisciplinary systems their propagation through coupled subsystems make s reliability analysis computationally expensive in this work a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems the framework is an extension of an existing unified uncertainty analysis framework for single disciplinary problems the new framework involves probabilistic analysis and interval analysis both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially the first order reliability method is used for probabilistic analysis three supporting algorithms are developed the effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application

residual based variational multiscale turbulence models for unstructured tetrahedral meshes abstract this paper presents three level residual based turbulence models for the incompressible navier stokes equations employing the variational multiscale vms framework the velocity and pressure fields are decomposed into two overlapping hierarchical scales thereby leading to a system of coupled mixed field problems the mixed problem at the fine scales is stabilized via a subsequent vms application that results in a further decomposition of the fine scale velocity field into level i and level ii scales the level ii scales are modeled using higher order bubble functions that are then variationally embedded in the level i formulation to stabilize it the level i problem is modeled via a second set of bubble functions that are linearly independent of the bubbles employed at level ii finally the resulting level i fine scales are variationally embedded in the coarse scale formulation this yields a residual based turbulence model for the larger or coarser scales a significant feature of the proposed method is that it results in a concurrent top down and bottom up two way nesting of the scales in addition the resulting turbulence model does not possess any embedded tunable parameters another attribute of the formulation is that the fine scales at every level are driven by the residuals of the euler lagrange equations of the coarser scales at the preceding levels thereby resulting in a method that is variationally consistent various algorithmic generalizations of the method are presented that lead to computationally economic residual based turbulence models the proposed telescopic depth in scales approach helps make these models accurate for low order tetrahedral and hexahedral elements a feature that is facilitated by the higher order bubble functions over element interiors and it results in an enhanced representation of the fine scale terms modeling the fine viscous effects from a computational perspective this method results in easy to implement equal order pressure velocity elements and possesses the desirable p refinement feature numerical performance of the method is assessed on turbulent channel flow problems at re 395 and re 590 also presented are the results for turbulent sd 7003 airfoil at re 60 000 and comparison is made with the published experimental data and numerical results

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

the duals of gabor frames on discrete periodic sets due to its good potential for digital signal processing analysis the discrete version of gabor analysis has interested many mathematicians in this paper we focus on gabor analysis on discrete periodic sets which can model a signal to appear periodically but intermittently for a gabor frame on a discrete periodic set three types of dual frames with gabor structure are introduced and investigated in terms of discrete zak transform given a gabor frame on a discrete periodic set a characterization of a gabor system being separately its dual of types i iii is obtained a method to construct these three types of duals is presented an explicit expression of its canonical dual is established the uniqueness of these three types of duals is characterized some examples are also provided to illustrate the generality of the theory

density and duality theorems for regular gabor frames abstract we investigate gabor frames on locally compact abelian groups with time frequency shifts along non separable closed subgroups of the phase space density theorems in gabor analysis state necessary conditions for a gabor system to be a frame or a riesz basis formulated only in terms of the index subgroup in the classical results the subgroup is assumed to be discrete we prove density theorems for general closed subgroups of the phase space where the necessary conditions are given in terms of the size of the subgroup from these density results we are able to extend the classical wexler raz biorthogonal relations and the duality principle in gabor analysis to gabor systems with time frequency shifts along non separable closed subgroups of the phase space even in the euclidean setting our results are new

the structure of shift modulation invariant spaces the rational case in this paper we study structural properties of shift modulation invariant smi spaces also called gabor subspaces or weyl heisenberg subspaces in the case when shift and modulation lattices are rationally dependent we prove the characterization of smi spaces in terms of range functions analogous to the well known description of shift invariant spaces c de boor r devore a ron the structure of finitely generated shift invariant spaces in l2 r d j funct anal 119 1994 37 78 m bownik the structure of shift invariant subspaces of l2 r n j funct anal 177 2000 282 309 h helson lectures on invariant subspaces academic press new york london 1964 we also give a simple characterization of frames and riesz sequences in terms on their behavior of the fibers of the range function next we prove several orthogonal decomposition results of smi spaces into simpler blocks called principal smi spaces then this is used to characterize operators invariant under both shifts and modulations in terms of families of linear maps acting on the fibers of the range function we also introduce the fundamental concept of the dimension function for smi spaces as a result this leads to the classification of unitarily equivalent smi spaces in terms of their dimension functions finally we show several results illustrating our fiberization techniques to characterize dual gabor frames

universal grey system theory for analysis of uncertain structural systems the uncertainty present in many structural systems is commonly modeled using probabilistic fuzzy or interval approaches depending on the nature of uncertainty present the probabilistic approach

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

a variational multiscale method for incompressible turbulent flows bubble functions and fine scale fields this paper presents a residual based turbulence model for the incompressible navier stokes equations the method is derived employing the variational multiscale vms framework a multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse scale and the fine scale sub problems the fine scale velocity field is assumed to be nonlinear and time dependent and is modeled via the bubble functions approach applied directly to the fine scale sub problem a significant contribution in this paper is a systematic and consistent derivation of the fine scale variational operator commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits and variationally projects the fine scale solution onto the coarse scale space a direct treatment of the fine scale problem via bubble functions offers several fine scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems numerical accuracy of the proposed method is shown on a forced isotropic turbulence problem statistically stationary turbulent channel flow problems at ret 395 and 590 and non equilibrium turbulent flow around a cylinder at re 3 900