viscosity versus vorticity stretching global well posedness for a family of navier stokes alpha like models abstract we study global well posedness and regularity of solutions for a family of incompressible three dimensional navier stokes alpha like models that employ fractional laplacian operators this family of equations depends on two parameters 1 and 2 which affect the strength of non linearity vorticity stretching and the degree of viscous smoothing varying 1 and 2 interpolates between the incompressible navier stokes equations and the incompressible lagrangian averaged navier stokes model our main result which contains previously established results of j l lions and others provides a relationship between 1 and 2 that is sufficient to guarantee global existence uniqueness and regularity of solutions

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

sequential estimation of velocity fields using episodic proper orthogonal decomposition abstract in this paper the problem of approximating velocity fields at future and past times based on information available at the current time is addressed a novel method called episodic pod is described and developed that enables us to achieve our objective application of episodic pod to an ensemble of flow data results in a set of spatio temporal eigenfunctions and a set of coefficients associated with the eigenfunctions from these eigenfunctions we develop two models called the forward model and inverse model that enable us to approximate the velocity fields at future and past times based on information provided at the current time a second set of models the forward and inverse sequential models are also developed that enable the dynamic update of approximated velocity fields when new information is made available making these models more adept at on line estimation the various properties associated with these models are described in detail and four different examples are used to validate the models and show the different properties associated with episodic pod it is shown through numerical validation that the episodic pod model has a form that is dynamically consistent with the original system it is also shown that episodic pod outperforms linear kalman filters in the presence of noise

fourier spectral embedded boundary solution of the poisson s and laplace equations with dirichlet boundary conditions a fourier spectral embedded boundary method for solution of the poisson s equation with dirichlet boundary conditions and arbitrary forcing functions including zero forcing function is presented in this paper this iterative method begins by transformation of the dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points which are called the numerical boundaries using a second order interpolation method then the transformed boundary conditions and the forcing function are extended to a square smoothly and periodically via multiplying them by some suitable error functions instead of direct solution of the resulting extended poisson s problem it is suggested to define and solve an equivalent transient diffusion problem on the regular domain until achievement of the steady solution which is considered as the solution of the original problem without need of any numerical time integration method time advancement of the solution is obtained directly from the exact solution of the transient problem in the fourier space consequently timestep sizes can be chosen without stability limitations which it means higher rates of convergence in comparison with the classical relaxation methods the method is presented in details for one and two dimensional problems and a new emerged phenomenon which is called the saturation state is illustrated both in the physical and spectral spaces the numerical experiments have been performed on the one and two dimensional irregular domains to show the accuracy of the method and its superiority from the rate of convergence viewpoint to the other classical relaxation methods capability of the method in dealing with complex geometries and in presence of discontinuity at the boundaries has been shown via some numerical experiments on a four leaf shape geometry

nonlinear transfer and spectral distribution of energy in α turbulence abstract two dimensional turbulence governed by the so called turbulence equations which include the surface quasi geostrophic equation 1 the navier stokes system 2 and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating 3 is studied here in both the unbounded and doubly periodic domains this family of equations conserves two inviscid invariants energy and enstrophy in the navier stokes case the dynamics of which are believed to undergo a dual cascade it is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse cascading quantity approaches its own injection rate constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived for navier stokes turbulence with moderate reynolds numbers the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations

on the growth and stability of real analytic functions this paper addresses the issue of whether integrals of real analytic functions remain finite under small deformations an approach based on uniform estimates for certain classes of one dimensional integrals is introduced it is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of karpushkin as well as produce new results in higher dimensions in dimension three the new stability results are sharp as shown by the well known example of varchenko

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

α regularization of the pod galerkin dynamical systems of the kuramoto sivashinsky equation abstract in order to improve the dynamics and stability of the pod galerkin models of strongly stiff systems an like regularization is suggested and assessed in the present article in this method the pod eigenmodes of the non linear terms are replaced by their helmholtz filtered counterparts while the other terms are remained unchanged as an example the method is applied to the pod galerkin models of the one dimensional kuramoto sivashinsky ks equation in a full chaotic regime and the fidelity of the original and regularized models to the direct numerical simulations dns are investigated moreover the effects of regularization on the dynamics of various terms and whole of the systems are analyzed via eigenvalue analysis of each term separately and the total dynamical system as a whole the numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability by minimum number of free parameters

preserving symmetries in the proper orthogonal decomposition the proper orthogonal decomposition pod also called karhunen loeve expansion has been recently used in turbulence to derive optimally fast converging bases of spatial functions leading to efficient finite truncations whether a finite number of these modes can be used in numerical simulations to derive an accurate finite set of ordinary differential equations over a certain range of bifurcation parameter values still remains an open question it is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite dimensional system in most cases this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric pod basis the kuramoto sivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view however the conclusion is general and can be applied to other equations such as the navier stokes equations the complex g

an fio calculus for marine seismic imaging ii sobolev estimates we establish sharp l 2 sobolev estimates for classes of pseudodifferential operators with singular symbols guillemin and uhlmann duke math j 48 251 267 1981 melrose and uhlmann commun pure appl math 32 483 519 1979 whose non pseudodifferential fourier integral operator parts exhibit two sided fold singularities the operators considered include both singular integral operators along curves in mathbb r 2 with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics felea comm pde 30 1717 1740 2005 felea and greenleaf comm pde 33 45 77 2008 nolan siam j appl math 61 659 672 2000

uniform estimates for oscillatory integral operators abstract in this paper we prove the uniform l p boundedness of oscillatory singular integral operators with smooth phase functions which satisfy a certain finite type condition these phase functions include the class of real analytic functions the uniform boundedness does not hold if the finite type condition is removed we also obtained estimates for related operators including an optimal l 2 estimate for operators with singular kernels whose phase functions have non singular hessians

fourier embedded domain methods periodic and c extension of a function defined on an irregular region to a rectangle via convolution with gaussian kernels abstract one possible way to solve a partial differential equation in an irregular region is the use of the so called domain embedding methods in where the domain of interest is embedded within a rectangle in order to apply a fourier spectral method on the rectangle the inhomogeneous term f x y has to be extended to a new function g x y that is periodic and infinitely differentiable and equal to f x y everywhere within some authors have given explicit methods to compute extensions with infinite order convergence for the cases in where the boundary of can be defined as the zero isoline of a function x y for the cases in where this is not possible we suggest a new method to build these extensions via convolution with gaussian kernels

near peak scattering and the corrected kirchhoff approximation for a convex obstacle abstract the uniform asympotic behavior of the scattering amplitude near the forward peak in the case of classical scattering of waves by a convex obstacle is derived a microlocal model is obtained for the scattering operator this is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of fourier integral operators those with folding canonical relations a crucial ingredient consists of putting a fourier integral operator with folding canonical relation into a normal form the analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary thus providing a corrected version of the kirchhoff approximation

fourier integral operators with cusp singularities we study the boundedness properties on lebesgue and sobolev spaces of fourier integral operators associated with canonical relations such that at least one of the projections is a simple whitney cusp in the process we obtain decay estimates for oscillatory integral operators whose symplectic relations have the same singular structure such singularities occur generically for averages over lines and curves in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on l 2 we show that the operators lose 1 3 derivative to obtain sharp results off of l 2 we need to impose an additional transversality condition satisfied by many geometric averaging operators which leads to the notion of a strong cusp these estimates can be further improved if we impose curvature conditions on the cusp surface one application is the l 2 comp l 2 1 6 loc and l 12 7 comp l 2 loc boundedness of restrictions of the x ray transform on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i to four dimensional families of lines satisfying a natural curvature and torsion condition

on the l p continuity of singular fourier integral operators we derive l p continuity of fourier integral operators with one sided fold singularities the argument is based on interpolation of asymptotics of l 2 estimates and h 1 l 1 estimates we derive the latter estimates elaborating arguments of seeger sogge and stein s 1991 paper we apply our results to the study of the l p regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points

coherent structures in random media and wavelets abstract we construct low dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows these models are derived from the wavelet based galerkin projection of the pde describing the flow and account for the local interaction of a small number of coherent structures we show that under rather general assumptions the wavelet projection is close to the proper orthogonal decomposition in the average energy sense in the specific case of the 1d kuramoto sivashinsky equation we show numerically that the dynamics of our model agrees qualitatively with that of the original ks equation

proper orthogonal decomposition and its applications part i theory abstract in view of the increasing popularity of the application of proper orthogonal decomposition pod methods in engineering fields and the loose description of connections among the pod methods the purpose of this paper is to give a summary of the pod methods and to show the connections among these methods firstly the derivation and the performance of the three pod methods karhunen loeve decomposition kld principal component analysis pca and singular value decomposition svd are summarized then the equivalence problem is discussed via a theoretical comparison among the three methods the equivalence of the matrices for processing the objective functions the optimal basis vectors the mean square errors and the asymptotic connections of the three methods are demonstrated and proved when the methods are used to handle the pod of discrete random vectors

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

a classification algorithm based on geometric and statistical information classification is a long standing problem in computing there are two broad kinds of classifiers frequency based and geometry based frequency based classifiers often ignore the geometry underlying the data conversely geometry based classifiers take into account the frequency only indirectly this paper presents a classification algorithm which considers explicitly geometric and statistical characteristics of the data and combines them into a class representation reported here are initial experiments with this algorithm using two well known data sets both with and without noise the results show that the proposed algorithm is less sensitive to the training data set than other classifiers