unitary beamformer designs for mimo interference broadcast channels interference broadcast channel ibc is a generalized scenario for interference channel ic where each base station bs transmits to multiple users in its own cell while sharing same time frequency resources with other adjacent bss in this case it has been shown that by using interference alignment ia and under the availability of perfect channel state information csi the achievable degrees of freedom dofs can be linearly scaled up with the number of users however in the presence of imperfect csi the sum rate becomes degraded and the achievability of full dof can not be guaranteed in this paper considering a multiple input multiple output mimo ibc with constant channel coefficients we propose ia algorithms that result in unitary beamformers first we consider two improved beamformer designs based on minimizing the weighted leakage interference min wli and maximizing the signal to interference plus noise ratio max sinr second we propose a novel minimum mean square error mmse based ia algorithm both the improved max sinr and the mmse based ia schemes are optimized upon the knowledge of the channel estimation error variance so as to achieve better performance under csi mismatch we then demonstrate that these two schemes outperform min wli and the conventional max sinr ia under perfect and imperfect csi furthermore it is shown that the mmse based ia achieves the same performance trend as the improved max sinr ia does however the former needs less csi to be available and has less computational complexity compared to the latter moreover it is established that the mmse based ia can result in diagonalized decoupled subchannels under perfect csi the superiority of the proposed algorithms over the conventional schemes is corroborated by numerical simulations

dynamically orthogonal field equations for continuous stochastic dynamical systems abstract in this work we derive an exact closed set of evolution equations for general continuous stochastic fields described by a stochastic partial differential equation spde by hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component we derive a system of field equations consisting of a partial differential equation pde for the mean field a family of pdes for the orthonormal basis that describe the stochastic subspace where the stochasticity lives as well as a system of stochastic differential equations that defines how the stochasticity evolves in the time varying stochastic subspace these new evolution equations are derived directly from the original spde using nothing more than a dynamically orthogonal condition on the representation of the solution if additional restrictions are assumed on the form of the representation we recover both the proper orthogonal decomposition equations and the generalized polynomial chaos equations we apply this novel methodology to two cases of two dimensional viscous fluid flows described by the navier stokes equations and we compare our results with monte carlo simulations

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a nine dimensional lorenz system to study high dimensional chaos we examine the dynamics of three dimensional cells with square planform in dissipative rayleigh benard convection by applying a triple fourier series ansatz up to second order we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations depending on two control parameters namely the rayleigh number and the prandtl number the asymptotic behaviour can be stationary periodic quasiperiodic or chaotic a period doubling cascade is identified as a route to chaos hereafter the asymptotic behaviour progressively evolves towards a hyperchaotic attractor for given values of control parameters beyond the accumulation point we observe a low dimensional chaotic attractor as is currently done for dissipative systems although the correlation dimension strongly suggests that this attractor could be embedded in a three dimensional space a topological characterization reveals that a higher dimensional space must be used thus we reconstruct a four dimensional model which is found to be in agreement with the properties of the original dynamics the nine dimensional lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3

optimal galerkin approximations of partial differential equations using principal interaction patterns the standard approach to the numerical study of the dynamical behavior of partial differential equations pdes a priori vested with an infinite number of degrees of freedom consists in their approximation by finite dimensional dynamical systems given the well known fact that the dynamics of pdes are often confined to attractor sets of relatively low dimension the construction of minimal dynamical models capturing the principal properties of the complete system is an interesting task a class of frequently used approximation schemes is formed by the galerkin methods the dynamical field is expanded into a finite set of timeindependent global basis functions projection of the pde onto these basis functions yields a finite dimensional system of ordinary differential equations odes for the time evolution of the expansion coefficients the efficiency of such a dynamical description i e the number of degrees of freedom required to capture the dynamics of the pde depends crucially on a proper choice of the basis functions the most traditional approach lies in using eigenfunctions of a suitably chosen linear differential operator commonly fourier modes as basis functions despite the compact and elegant mathematical framework of these fourier galerkin methods they often provide a system of equations that is much larger than the true dimensionality of the system since fourier modes are completely general and not adapted to the particular system under consideration hence information on the dynamics of the system has to be incorporated into the choice of the basis functions in order to arrive at a dynamical description reflecting more closely the intrinsic dimensionality of the system

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

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

on the ensemble average in the study of approximate inertial manifolds ii abstract in this paper we present new estimates on the ensemble averages of the fluid dynamical quantities that have significances in the study of approximate inertial manifolds of the 2 d navier stokes equations some immediate implications of these estimates on the physics of 2 d turbulent flows are discussed

data driven non markovian closure models abstract this paper has two interrelated foci i obtaining stable and efficient data driven closure models by using a multivariate time series of partial observations from a large dimensional system and ii comparing these closure models with the optimal closures predicted by the mori zwanzig mz formalism of statistical physics multilayer stochastic models msms are introduced as both a generalization and a time continuous limit of existing multilevel regression based approaches to closure in a data driven setting these approaches include empirical model reduction emr as well as more recent multi layer modeling it is shown that the multilayer structure of msms can provide a natural markov approximation to the generalized langevin equation gle of the mz formalism a simple correlation based stopping criterion for an emr msm model is derived to assess how well it approximates the gle solution sufficient conditions are derived on the structure of the nonlinear cross interactions between the constitutive layers of a given msm to guarantee the existence of a global random attractor this existence ensures that no blow up can occur for a broad class of msm applications a class that includes non polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants the emr msm methodology is first applied to a conceptual nonlinear stochastic climate model of coupled slow and fast variables in which only slow variables are observed it is shown that the resulting closure model with energy conserving nonlinearities efficiently captures the main statistical features of the slow variables even when there is no formal scale separation and the fast variables are quite energetic second an msm is shown to successfully reproduce the statistics of a partially observed generalized lotka volterra model of population dynamics in its chaotic regime the challenges here include the rarity of strange attractors in the model s parameter space and the existence of multiple attractor basins with fractal boundaries the positivity constraint on the solutions components replaces here the quadratic energy preserving constraint of fluid flow problems and it successfully prevents blow up

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

numerical taylor expansions of invariant manifolds in large dynamical systems in this paper we develop a numerical method for computing higher order local approximations of invariant manifolds such as stable unstable or center manifolds near steady states of a dynamical system the underlying system is assumed to be large in the sense that a large sparse jacobian at the equilibrium occurs for which only a linear black box solver and a low dimensional invariant subspace is available but for which methods like the qr algorithm are considered to be too expensive our method is based on an analysis of the multilinear sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions these conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold the final algorithm requires the solution of several large linear systems with a bordered jacobian to these systems we apply a block elimination method recently developed by govaerts and pryce 12 14

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

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

inertial manifolds and finite dimensional reduction for dissipative pdes this paper is devoted to the problem of finite dimensional reduction for parabolic partial differential equations we give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so called mane projection theorems the recent counter examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated as well as a discussion of the most important open problems are also included

numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows the quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions such situations are common in multiscale intermittent and non homogeneous fluid and ocean flows the dynamically orthogonal do field equations provide an adaptive methodology to predict the probability density functions of such flows the present work derives efficient computational schemes for the do methodology applied to unsteady stochastic navier stokes and boussinesq equations and illustrates and studies the numerical aspects of these schemes semi implicit projection methods are developed for the mean and for the do modes and time marching schemes of first to fourth order are used for the stochastic coefficients conservative second order finite volumes are employed in physical space with new advection schemes based on total variation diminishing methods other results include i the definition of pseudo stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic ii symmetric advection schemes for the stochastic velocities iii the use of generalized inversion to deal with singular subspace covariances or deterministic modes and iv schemes to maintain orthonormal modes at the numerical level to verify our implementation and study the properties of our schemes and their variations a set of stochastic flow benchmarks are defined including asymmetric dirac and symmetric lock exchange flows lid driven cavity flows and flows past objects in a confined channel different reynolds number and grashof number regimes are employed to illustrate robustness optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations

broyden s method in hilbert space broyden s method is formulated for the solution of nonlinear operator equations in hilbert spaces the algorithm is proven to be well defined and a linear rate of convergence is shown under an additional assumption on the initial approximation for the derivative we prove the superlinear rate of convergence

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