a fast combined field volume integral equation solution to em scattering by 3 d dielectric objects of arbitrary permittivity and permeability a fast solution to the combined field volume integral equation cfvie for electromagnetic scattering by large three dimensional dielectric bodies of arbitrary permittivity and permeability is presented the cfvie is formulated in the region of the scatterers by expressing the total fields as the sum of the incident wave and the radiated wave due to both the electric and magnetic polarization currents the resultant integral equation is solved using the method of moments mom then the precorrected fast fourier transform p fft method is applied to reduce the memory requirement and accelerate the matrix vector multiplication in the mom solution in the implementation of the p fft method two sets of projection operators are constructed respectively for the projections of the electric sources and magnetic sources in addition two sets of interpolation operators are also applied respectively for the computation of the vector scalar potentials and the curl of the vector potentials in the support of the testing functions the resultant method has a memory requirement of o n and a computational complexity of o nlogn respectively where n denotes the number of unknowns

a fast analysis of scattering from large scale finite periodic microstrip patch arrays arranged on a non orthogonal lattice using sub entire domain basis functions a sub entire domain sed basis function method which was first introduced for modeling large scale finite periodic pec structures in free space has been extended for fast characterization of electromagnetic scattering from an electrically large planar finite periodic microstrip patch array the microstrip array may have a nonrectangular layout and non orthogonal lattice configurations e g hexagons or quadrangles based on the mixed potential integral equation and utilizing the proposed sed basis function algorithm the original large scale finite periodic array of microstrip patches can be efficiently simulated by decomposing it into two problems with matrix equations of small dimensions the first is to construct the sed basis functions for the corresponding microstrip arrays with orthogonal non orthogonal lattices three kinds of the sed basis functions are constructed including those related to the edge patch elements the interior patch elements and the corner patch elements the second is to solve the system equation with significantly reduced problem dimension as compared to the original larger problem based on the obtained sed basis functions the reduced matrix equation of small size can be generated by the galerkin procedure and solved by use of the lu lower upper decomposition based direct solver which results in a fast solution the accuracy and efficiency of the developed algorithms are demonstrated by numerical tests that include the scattering from several large scale finite periodic arrays of microstrip patches with rectangular non orthogonal lattices

a highly accurate fgg fg fft for the combined field integral equation in this paper a novel realization of the integral equation in combination with the fast fourier transform for the cfie is established by fitting both the green s function and its gradient onto the nodes of a uniform cartesian grid the new method has been compared with several existing popular fft based methods including the aim the ie fft and the p fft the accuracy of the proposed method is significantly higher than other fft based methods and the method is not sensitive to both the grid spacing and the expansion order the outstanding merit of the proposed method is that the fitting procedure is independent of the basis functions therefore when the higher order basis functions would be adopted in the method of moments only one fitting procedure for the green s function and its gradient on a basis function support is needed to meet all of basis functions defined on this support some numerical examples are provided in this paper to demonstrate the accuracy and efficiency of the proposed method

electromagnetic analysis of large scale periodic arrays using a two level cbfs method accelerated with fmm fft to realize efficient analysis of large scale planar periodic arrays a two level characteristic basis functions method accelerated with fast multipole method fast fourier transform fmm fft is developed in this paper the characteristic basis functions cbfs for each element are constructed in a two level framework in which the cbfs construction based on plane wave derivation for single element at child level and the cbfs construction based on local interaction between array elements at father level are combined the number of unknowns is reduced significantly by the two level cbfs further fmm fft is applied to expedite the computation of interaction between elements numerical results of large scale planar periodic arrays are given to demonstrate the validity and efficiency of the present method

computing covariant lyapunov vectors oseledets vectors and dichotomy projectors a comparative numerical study abstract covariant lyapunov vectors or oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations nonautonomous differentiable dynamical systems and random dynamical systems these vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles in recent years new computational methods for approximating oseledets vectors have been developed motivated by increasing model complexity and greater demands for accuracy in this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of ginelli et al 2007 36 and wolfe and samelson 2007 37 we compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry spectral separation and dimension we also investigate which methods perform well with limited data

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

dynamical indicators for the prediction of bursting phenomena in high dimensional systems drawing upon the bursting mechanism in slow fast systems we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates the indicators are associated with functionals defined in terms of optimally time dependent otd modes one such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes in contrast to other choices of subspaces the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful we illustrate the application of these indicators on three examples a prototype low dimensional model a body forced turbulent fluid flow and a unidirectional model of nonlinear water waves we use bayesian statistics to quantify the predictive power of the proposed indicators

perturbation theory for the approximation of stability spectra by qr methods for sequences of linear operators on a hilbert space abstract in this paper we develop a perturbation analysis for stability spectra lyapunov exponents and sacker sell spectrum for products of operators on a hilbert space both real and complex based upon the discrete qr technique error bounds are obtained in both the integrally separated and non integrally separated cases that correspond to distinct and multiple eigenvalues respectively for a single linear operator we illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation the time varying analogue of gaps between eigenvalues determines the sensitivity of the stability spectra to perturbation

noncausal inverses for linear systems suppose we are to guide a plant with a square transfer matrix through a series of way points this can be accomplished by output tracking of trajectories which are fitted together in a sufficiently smooth manner each segment of each output is prescribed in terms of a polynomial or sine function or the product thereof our technique is to compute 1 a noncausal particular solution based on the desired trajectory over each segment and the movement from one segment to the next and this particular solution consists of a particular control and a particular state trajectory and 2 a regulator to handle modeling errors disturbances etc we concentrate on the particular solution

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

almost periodic solutions of delay difference systems in the present paper we study the properties of almost periodic solutions for delay difference systems and establish some theorems on the existence of almost periodic solutions for delay difference systems by means of liapunov functionals the obtained results confirm not only the existence but also the uniqueness as well as the uniform asymptotic stability of the almost periodic solution

brief paper on the equivalence between stable inversion for nonminimum phase systems and reciprocal transfer functions defined by the two sided laplace transform in this paper the two sided laplace transform a classical but not very common mathematical tool is revived to express the stable inversion for linear nonminimum phase systems that was recently proposed from the viewpoint of state space representations it is demonstrated that those two different expressions for the stable inversion are mathematically equivalent simple examples are presented to illustrate the two sided laplace transform as a direct and intuitive approach to stable inversion the two sided laplace transform approach is also applied to the development of an iterative learning control for nonminimum phase systems that needs neither a precise inversion model nor fourier transform computations but instead requires only measuring the system response with time reversals

a fast analysis of scattering and radiation of large microstrip antenna arrays an accurate and efficient method that combines the precorrected fast fourier transform fft method and the discrete complex image method dcim is presented to characterize the scattering and radiation properties of arbitrarily shaped microstrip patch antennas in this method the mixed potential integral equation mpie is discretized in the spatial domain by means of the discrete complex image method the resultant system is solved iteratively using the generalized conjugate residual method gcr and the precorrected fft technique is used to speed up the matrix vector multiplication the precorrected fft eliminates the need to generate and store the usual square impedance matrix and thus leads to a significant reduction in memory requirement and computational cost numerical results are presented for arbitrarily shaped microstrip antenna arrays to demonstrate the accuracy and efficiency of this technique

stable inversion of lptv systems with application in position dependent and non equidistantly sampled systems abstractmany control applications including feedforward and learning control involve the inverse of a dynamical system for nonminimum phase systems the response of the inverse system is unbounded for linear time invariant lti nonminimum phase systems a bounded noncausal inverse response can be obtained through an exponential dichotomy for general linear time varying ltv systems such a dichotomy does not exist in general the aim of this paper is to develop an inversion approach for an important class of ltv systems namely linear periodically time varying lptv systems which occur in e g position dependent systems with periodic tasks and non equidistantly sampled systems the proposed methodology exploits the periodicity to determine a bounded inverse for general lptv systems conditions for existence are provided the method is successfully demonstrated in several application cases including position dependent and non equidistantly sampled systems

sparse dynamics for partial differential equations we investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis the restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation by reducing or compressing the information needed to represent the solution at every step only the essential dynamics are represented in many cases there are natural bases derived from the differential equations which promote sparsity we find that our method successfully reduces the dynamics of convection equations diffusion equations weak shocks and vorticity equations with high frequency source terms

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

a more scalable and efficient parallelization of the adaptive integral method part i algorithm a more effective parallelization of the adaptive integral method aim is proposed for solving the 3 d volume electric field integral equation the aim computations are distributed among processes by employing two workload decomposition strategies 1 a novel 2 d pencil decomposition of the auxiliary regular grid is used to parallelize the anterpolation interpolation and 3 d fft accelerated propagation steps and 2 a balanced 3 d block decomposition of the scattering volume is used to parallelize the correction step the scalability of the proposed parallelization method is investigated theoretically to compare it with competing methods the concepts of resource constraints parallel efficiency constraints scalability limits and acceptable parallelization regions are introduced by using n p plots where n and p denote the number of unknowns and processes respectively using these concepts it is shown that the proposed parallelization of aim has better weak and strong scalability compared to the traditional ones i e it enables not only larger problems to be solved but also faster solution of a given problem by increasing p

a fast full wave analysis of scattering and radiation from large finite arrays of microstrip antennas a fast full wave analysis technique that can be used to analyze the scattering and radiation from large finite arrays of microstrip antennas is presented the technique discretizes the mixed potential integral equation mpie in the spatial domain by means of a full wave discrete complex image method the del operators on the green s functions are transferred from the singular kernel to the expansion and testing functions the resultant system of equations is solved using the biconjugate gradient bcg method in which the matrix vector product is evaluated efficiently using the fast fourier transform fft this results in an efficient and accurate computation of the scattering and radiation from finite arrays of microstrip antennas several numerical results are presented demonstrating the accuracy efficiency and capability of this technique

exponential separation exponential dichotomy and almost periodicity of linear difference equations abstract in this paper we study the relation between the concept of exponential separation and those of reducibility exponential dichotomy and almost periodicity of linear difference equations

topological equivalence and structural stability for linear difference equations abstract in this paper first we prove that if two linear difference equations with invertible coefficient matrices are topologically equivalent and one of them has bounded coefficient matrix together with its inverse then the coefficient matrix of the other equation is also bounded together with its inverse we also prove that if a linear difference equation with bounded and invertible coefficient matrix is structurally stable then the inverse of the coefficient matrix is also bounded

a mixed form thin stratified medium fast multipole algorithm for both low and mid frequency problems a mixed form thin stratified medium fast multipole algorithm is proposed for fast simulation of general microstrip structures at both low and mid frequencies the newly developed matrix friendly formula of layered medium green s function is applied in this algorithm for well separated interactions the path deformation technique is implemented to achieve a smoother and exponentially convergent integral the two dimensional addition theorem is then incorporated into the integrand to expedite the matrix vector product in our approach multipole expansion low frequency fast multiple algorithm as well as the plane wave expansion mid frequency fast multipole algorithm of the translational addition theorem are combined into a single multilevel tree to capture quasi static physics and wave physics simultaneously the outgoing wave is represented first in terms of multipole expansion at leafy levels and then switched to plane wave expansion automatically at higher levels this seamless connection makes the algorithm applicable in simulations where subwavelength interaction and wave physics both exist

extended horizon liftings for stable inversion of nonminimum phase systems the lifting of lozano 1989 is generalized and shown to be one in a large class of liftings which enjoy the same zero annihilation properties i e placement of transmission zeros to the origin the new class of liftings allows for longer horizons and leads to plant inverse controllers for nonminimum phase systems having significantly reduced control gains and peak torque requirements

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

some characterizations of domination we show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates then we consider sets in rm gl d mathbb r with the property that any cocycle with values in has a dominated splitting we characterize these sets in terms of existence of invariant multicones thus extending a two dimensional result by avila bochi and yoccoz we give an example showing how these multicones can fail to have convexity properties

stable inversion for nonlinear nonminimum phase time varying systems we extend stable inversion to nonlinear time varying systems and study computational issues the technique is applicable to minimum phase as well as nonminimum phase systems the inversion technique is new even in the linear time varying case and relies on partitioning the dichotomic split of the linearized system dynamics into time varying stable and unstable submanifolds this dichotomic split is used to build time varying filters which are in turn the basis of a contraction used to find a bounded inverse input state trajectory finding the inverse input state trajectory allows the development of exact output tracking controllers the method is local to the time varying trajectory and requires that the internal dynamics vary slowly however the method represents a significant advance relative to presently available tracking controllers present techniques are restricted to time invariant nonlinear systems and in the general case track only asymptotically

a hybrid finite element boundary integral method for the analysis of cavity backed antennas of arbitrary shape an edge based hybrid finite element boundary integral fe bi formulation using tetrahedral elements is described for scattering and radiation analysis of arbitrarily shaped cavity backed patch antennas by virtue of the finite element method fem the cavity irregularities the dielectric super substrate inhomogeneities and the diverse excitation schemes inside the cavity may be readily modeled when tetrahedral elements are used to discretize the cavity on the aperture the volume mesh reduces to a triangular grid allowing the modeling of nonrectangular patches without special handling of the boundary integral system this formulation is typically applicable to cavity backed antenna systems with moderate aperture size to retain an o n memory requirement storage of the full matrix due to the boundary integral equation is avoided by resorting to a structured triangular aperture grid and taking advantage of the integral s convolutional property if necessary this is achieved by overlaying a structured triangular grid on the unstructured triangular grid and relating the edge field coefficients between the two grids via two narrow banded transformation matrices the combined linear system of equations is solved via the biconjugate gradient bicg method and the fft algorithm is incorporated to compute the matrix vector product efficiently with minimal storage requirements

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

null field preconditioner with selected far field contribution for 3 d full wave efie in this paper a stable and effective preconditioner for the method of moments based electric field integral equation matrix based on the null field method is presented to expedite the convergence of krylov subspace based fast iterative solution the preconditioner is computed from all near field blocks and selected far field blocks in accordance with the null field procedure the final form of the preconditioner is block diagonal and therefore generates no additional fill ins in its inverse and is also amenable to parallelization a complexity analysis is presented to show the near linear cost of preconditioner construction and usage in terms of computation time and memory the numerical experiments with a sequential implementation demonstrate on an average 1 5 3 times speed up in the iterative solution time over threshold dependent incomplete lu based preconditioners

method of moments solution for a printed patch slot antenna on a thin finite dielectric substrate using the volume integral equation in this paper a volume integral equation vie based modeling method suitable for a patch or slot antenna on a thin finite dielectric substrate is developed and tested two new key features of the method are the use of proper dielectric basis functions and proper vie conditioning close to the metal surface where the surface boundary condition of the zero tangential e component must be extended into adjacent tetrahedra the extended boundary condition is the exact result for the piecewise constant dielectric basis functions the latter operation allows one to achieve a good accuracy with one layer of tetrahedra for a thin dielectric substrate and thereby greatly reduces computational cost the use of low order basis functions also implies the use of low order integration schemes and faster filling of the impedance matrix for some common patch slot antennas the vie based modeling approach is found to give an error of about 1 or less in the resonant frequency for one layer tetrahedral meshes with a relatively small number of unknowns this error is obtained by comparison with fine finite element method fem simulations or with measurements or with the analytical mode matching approach hence it is competitive with both the method of moments surface integral equation approach and with the fem approach for the printed antennas on thin dielectric substrates

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

almost periodic transformable difference systems abstract we analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so called transformable groups we introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non almost periodic solutions of considered systems

on the zero freeness of tall multirate linear systems in this technical note tall discrete time linear systems with multirate outputs are studied in particular we focus on their zeros in systems and control literature zeros of multirate systems are defined as those of their corresponding time invariant systems obtained through blocking of the original multirate systems we assume that blocked systems are tall i e have more outputs than inputs it is demonstrated that for generic choice of the parameter matrices linear systems with multirate outputs generically have no finite nonzero zeros however they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input state and output dimensions

compressive projection principal component analysis principal component analysis pca is often central to dimensionality reduction and compression in many applications yet its data dependent nature as a transform computed via expensive eigendecomposition often hinders its use in severely resource constrained settings such as satellite borne sensors a process is presented that effectively shifts the computational burden of pca from the resource constrained encoder to a presumably more capable base station decoder the proposed approach compressive projection pca cppca is driven by projections at the sensor onto lower dimensional subspaces chosen at random while the cppca decoder given only these random projections recovers not only the coefficients associated with the pca transform but also an approximation to the pca transform basis itself an analysis is presented that extends existing rayleigh ritz theory to the special case of highly eccentric distributions this analysis in turn motivates a reconstruction process at the cppca decoder that consists of a novel eigenvector reconstruction based on a convex set optimization driven by ritz vectors within the projected subspaces as such cppca constitutes a fundamental departure from traditional pca in that it permits its excellent dimensionality reduction and compression performance to be realized in an light encoder heavy decoder system architecture in experimental results cppca outperforms a multiple vector variant of compressed sensing for the reconstruction of hyperspectral data

multilayered media green s functions in integral equation formulations a compact representation is given of the electric and magnetic type dyadic green s functions for plane stratified multilayered uniaxial media based on the transmission line network analog along the aids normal to the stratification furthermore mixed potential integral equations are derived within the framework of this transmission line formalism for arbitrarily shaped conducting or penetrable objects embedded in the multilayered medium the development emphasizes laterally unbounded environments but an extension to the case of a medium enclosed by a rectangular shield is also included

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

convolution profiles for right inversion of multivariable non minimum phase discrete time systems the problem of the non causal inversion of linear multivariable discrete time systems is analyzed in the geometric approach framework and is solved through the computation of convolution profiles which guarantee perfect tracking under the assumption of infinite length preaction and postaction time intervals it is shown how the shape of the convolution profiles is related to both the relative degree and the invariant zeros of the plant a computational setting for the convolution profiles is derived by means of the standard geometric approach tools feasibility constraints are also taken into account a possible implementation scheme based on a finite impulse response system acting on a stabilized control loop is provided

dichotomy spectra and morse decompositions of linear nonautonomous differential equations abstract recently the existence of morse decompositions for nonautonomous dynamical systems was shown for three different time domains the past the future and in the linear case the entire time in this article notions of exponential dichotomy are discussed with respect to the three time domains it is shown that an exponential dichotomy gives rise to an attractor repeller pair in the projective space which is a building block of a morse decomposition moreover based on the notions of exponential dichotomy dichotomy spectra are introduced and it is proved that the corresponding spectral manifolds lead to morse decompositions in the projective space

efficient solution of efie via low rank compression of multilevel predetermined interactions this paper describes the predetermined interaction list oct tree pilot algorithm and its application in expediting the solution of full wave electric field integral equation efie based scattering problems for three dimensional arbitrarily shaped conductors pilot combines features of the fast multipole method fmm and qr decomposition based matrix compression techniques to optimize setup times solve times and memory requirements the method is kernel independent and stable for electrically small structures unlike traditional fmm the novel features of the algorithm namely the mixed potential compression scheme and the hierarchical multilevel predetermined matrix structure are explained in detail a complexity estimate is presented to demonstrate the scaling in time and memory requirements examples exhibiting the accuracy and the time and memory performances are also presented finally a quantitative study is included to address the expected but gradual degradation of qr based compression techniques for electrically large structures

fast fourier transform based iterative method for electromagnetic scattering from 1d flat surfaces an efficient iterative method is proposed for computing the electromagnetic fields scattered from a one dimensional 1d flat surface the new iterative method is based on a similar implementation to the conjugate gradient fast fourier transform cg fft where acceleration of the matrix vector multiplications is achieved using fast fourier transforms fft however the iterative method proposed is not based on krylov subspace expansions and is shown to converge faster than gmres fft and cgne fft while maintaining the computational complexity and memory usage of those methods analysis is presented deriving an explicit convergence criterion

rcs computation for a large array of waveguide slots with finite wall thickness using the mom accelerated by p fft algorithm this correspondence presents an accurate and efficient method of moments mom for analyzing electromagnetic scattering by a large finite array of waveguide slots with finite thickness in this method the mixed potential integral equation mpie is utilized onto both upper and lower surfaces of the slots and the subdomain mom procedure is implemented to obtain the equivalent magnetic current distributions the precorrected fast fourier transform p fft method is employed to accelerate the entire computational process to reduce significantly the memory requirements for large arrays during the mom procedure the rao wilton glisson rwg functions are used as the basis and testing functions with both z and x directional magnetic currents considered this approach extends applicability of the present method to solve the mpie for characterizing waveguide slots of arbitrary shape and distribution and the capability of the integral equation based fast algorithm numerical results are obtained and compared with the experimental results to verify the accuracy and efficiency of this technique

continuity properties of the lower spectral radius 2014 london mathematical society the lower spectral radius or joint spectral subradius of a set of real d times d matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set the lower spectral radius arises naturally in connection with a number of topics including combinatorics on words the stability of linear inclusions in control theory and the study of random cantor sets in this article we apply some ideas originating in the study of dominated splittings of linear cocycles over a dynamical system to characterize the points of continuity of the lower spectral radius on the set of all compact sets of invertible d times d matrices as an application we exhibit open sets of pairs of 2 times 2 matrices within which the analogue of the lagarias wang finiteness property for the lower spectral radius fails on a residual set and discuss some implications of this result for the computation of the lower spectral radius

fast directional multilevel algorithm for analyzing wave scattering a new method named fast directional multilevel algorithm is proposed for analyzing wave scattering similar to a fast multipole method an oct tree structure is used in the new method unlike the conventional mlfma the kernel is evaluated by a new method instead of approximating the green s function by multipole expansion the fast directional multilevel algorithm is first used to solve the wave scattering problem via combined field integral equation cfie using rao wilton glisson rwg basis functions the low rank representations are extended to apply in the low frequency regime which is proved efficient and stable the numerical results demonstrate that the computational complexity of this new multilevel algorithm can achieve the complexity of o n log n this algorithm is robust and performs well on other oscillatory kernels because of its not depending on the explicit form of kernels

limit periodic homogeneous linear difference systems we study limit periodic homogeneous linear difference systems where the coefficient matrices belong to a bounded group we find groups of matrices with the property that the systems which do not possess any non zero asymptotically almost periodic solution form a dense subset in the space of all considered systems analogously we analyse almost periodic systems as well

a combined full wave cg fft method for rigorous analysis of large microstrip antenna arrays an accurate and efficient technique is presented for the analysis of large microstrip antenna arrays the technique consists of an amalgamation of a spatially discrete scheme consisting of the cg fft method and the complex discrete image cdi technique the unique feature of this approach is the use of the spatially discrete cg fft for analyzing microstrip structures the aliasing and truncation errors are thoroughly eliminated in this approach in addition the grad div operators are transformed from singular green s functions to differentiable expansion and testing functions by using galerkin s procedure thereby improving the accuracy and the rate of convergence to show the accuracy and efficiency of this technique a number of microstrip arrays including a large microstrip reflectarray have been studied it is found that the simulations carried out using this technique are in very good agreement with measurements

nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries we demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters first we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low rank coherent structures here libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points the selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics stability and bifurcations of nonlinear dynamical systems the use of empirical interpolation points and sparse representation facilitates a family of local reduced order models for each physical regime rather than a higher order global model which has the benefit of physical interpretability of energy transfer between coherent structures the method advocated also allows for orders of magnitude improvement in computational speed and memory requirements to illustrate the method the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex ginzburg landau equation it is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present

separation of horizontal and vertical dependencies in a surface volume integral equation approach to model quasi 3 d structures in multilayered media a new and efficient integral equation approach is presented to model heterogeneous dielectric volumes in a multilayered environment the dielectric volumes can be vertically large they can cross dielectric interfaces and they should be quasi 3 d which means that the contour faces should be placed either horizontally or vertically in the multilayered environment in this way a perpendicular prismatic mesh can be fitted on the volumes and the 3 d displacement currents can be expanded in generalized rooftop functions with separated horizontal x y and vertical z dependencies this makes it possible to evaluate all z z reaction integrals fully analytically in the spectral domain and ensures an efficient implementation the formulation of the source field relations is adapted to the quasi 3 d geometries as a hybrid dyadic mixed potential form additionally the electromagnetic coupling between dielectric volumes and metal sheets is included using a coupled volume surface formulation this turns the implementation into a complete full wave solver for planar antennas containing both finite and infinite dielectric regions in order to illustrate and validate the presented approach two patch antennas with a local volumetric inhomogeneity under the patch are numerically analyzed in a multilayered environment eventually a matrix fill time comparison is used to demonstrate the improvement in computation efficiency

modified prufer angle and conditional oscillation of perturbed linear and half linear differential equations abstract the research and results described in this paper belong to the qualitative theory of differential equations more precisely the partial differential equations with the one dimensional p laplacian using a method whose core is formed by the prufer technique we identify a borderline case between oscillatory and non oscillatory equations moreover we are able to decide whether the studied equations are oscillatory or not even in the so called critical i e the borderline case the advantage of our approach is the fact that we obtain new and strong results for linear and half linear equations i e the equations with the one dimensional p laplacian at the same time in addition we are able to work with equations whose coefficients are non constant and non periodic the novelty of our results is documented by examples and corollaries

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

precorrected fft solution of the volume integral equation for 3 d inhomogeneous dielectric objects this work presents a fast solution to the volume integral equation for electromagnetic scattering from three dimensional inhomogeneous dielectric bodies by using the precorrected fast fourier transform fft method the object is modeled using tetrahedral volume elements and the basis functions proposed by schaubert et al are employed to expand the unknown electric flux density the basis functions are then projected onto a fictitious uniform grid surrounding the nonuniform mesh enabling the fft to be used to speed up the matrix vector multiplies in the iterative solution of the matrix equation the resultant method greatly reduces the memory requirement to o n and the computational complexity to o nlogn where n is the number of unknowns as a result this method is capable of computing electromagnetic scattering from large complex dielectric objects

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

exponential dichotomy integral separation and diagonalizability of linear systems of ordinary differential equations abstract this paper is concerned with linear systems of ordinary differential equations a criterion for integral separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for integral separation and the new result that an upper triangular system on a half line is integrally separated if and only if the system corresponding to its diagonal is we then show that a diagonal system on a half line is integrally separated if and only if a certain perturbed system is diagonalizable using this result we are able to deduce that the interior of the set of diagonalizable systems on a half line is the set of systems with integral separation and that the interior for the whole line is the set of systems which are integrally separated on both half lines

the morse spectrum of linear flows on vector bundles for a linear flow on a vector bundle e s a spectrum can be defined in the following way for a chain recurrent component m on the projective bundle pe consider the exponential growth rates associated with finite time e t chains in m and define the morse spectrum mo m over m as the limits of these growth rates as e 0 and t the morse spectrum mo of is then the union over all componentsm pe this spectrum is a synthesis of the topological approach of selgrade and salamon zehnder with the spectral concepts based on exponential growth rates such as the oseledec spectrum or the dichotomy spectrum of sacker sell it turns out that mo contains all lyapunov exponents of for arbitrary initial values and the mo m are closed intervals whose boundary points are actually lyapunov exponents using the fact that is cohomologous to a subflow of a smooth linear flow on a trivial bundle one can prove integral representations of all morse and all lyapunov exponents via smooth ergodic theory a comparison with other spectral concepts shows that in general the morse spectrum is contained in the topological spectrum and the dichotomy spectrum but the spectral sets agree if the induced flow on the base space is chain recurrent however even in this case the associated subbundle decompositions of e may be finer for the morse spectrum than for the dynamical spectrum if one can show that the closure of the floquet spectrum i e the lyapunov spectrum based on periodic trajectories in pe agrees with the morse spectrum then one obtains equality for the floquet the entire oseledec the lyapunov and the morse spectrum we present an example flows induced by c vector fields with hyperbolic chain recurrent components on the projective bundle where this fact can be shown using a version of bowen s shadowing lemma

volume integral equation with higher order hierarchical basis functions for analysis of dielectric electromagnetic scattering an efficient method of moments mom solution of volume electric field integral equation v efie with a kind of higher order basis functions is presented to model scattering from a dielectric object with arbitrary shapes and inhomogenuity this method is based on both higher order geometrical modeling and higher order current modeling which is realized using a family of higher order hierarchical vector hhov basis functions to expand the equivalent volume electric currents within curvilinear volume elements this higher order scheme of mom results in a significant reduction in meshing density and number of unknowns leading to a remarkable reduction of memory and cpu time in comparison with conventional low order mom while the accuracy of the solution remains at a comparable level numerical results agree with analytical solutions or with simulation results obtained by commercial software demonstrating the accuracy convergence efficiency flexibility and adaptability of the new approach

the optimal hard threshold for singular values is 4 sqrt 3 we consider recovery of low rank matrices from noisy data by hard thresholding of singular values in which empirical singular values below a threshold are set to 0 we study the asymptotic mean squared error amse in a framework where the matrix size is large compared with the rank of the matrix to be recovered and the signal to noise ratio of the low rank piece stays constant the amse optimal choice of hard threshold in the case of n by n matrix in white noise of level is simply 4 3 n 2 309 n when is known or simply 2 858 y med when is unknown where y med is the median empirical singular value for nonsquare m by n matrices with m n the thresholding coefficients 4 3 and 2 858 are replaced with different provided constants that depend on m n asymptotically this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner it is always better than hard thresholding at any other value and is always better than ideal truncated singular value decomposition tsvd which truncates at the true rank of the low rank matrix we are trying to recover hard thresholding at the recommended value to recover an n by n matrix of rank r guarantees an amse at most 3 nr 2 in comparison the guarantees provided by tsvd optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nr 2 6 nr 2 and 2 nr 2 respectively the recommended value for hard threshold also offers among hard thresholds the best possible amse guarantees for recovering matrices with bounded nuclear norm empirical evidence suggests that performance improvement over tsvd and other popular shrinkage rules can be substantial for different noise distributions even in relatively small n

tracking sensitivity and achievable h performance in preview control preview control has been explored in the context of linear and nonlinear inverse tracking controllers preview of the reference trajectory has been shown to give significant performance advantages particularly for plants that do not possess a stable inverse without preview it has been known that lack of a stable inverse imposes inherent limitations on the achievable tracking performance of a control loop this has been explored in the context of time and frequency domain integral constraints in this paper we explore the extension of these performance limitations to the preview control case in particular we consider the infimal achievable h performance and poisson sensitivity integral results we also briefly discuss tracking sensitivity properties of preview control

on the error in qr integration an important change of variables for a linear time varying system dot x a t x t ge 0 is that induced by the qr factorization of the underlying fundamental matrix solution x qr with q orthogonal and r upper triangular with positive diagonal to find this change of variable one needs to solve a nonlinear matrix differential equation for q practically this means finding a numerical approximation to q by using some appropriate discretization scheme whereby one attempts to control the local error during the integration our contribution in this work is to obtain global error bounds for the numerically computed q these bounds depend on the local error tolerance used to integrate for q and on structural properties of the problem itself but not on the length of the interval over which we integrate this is particularly important since in principle q may need to be found on the half line t ge 0

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

learning based fast iterative convergence of 3 d mom via eigen agmres method three dimensional 3 d full wave electromagnetic simulation using method of moments mom under the framework of fast solver algorithms like fast multipole method fmm is often bottlenecked by the speed of convergence of the krylov subspace based iterative process this is primarily because the electric field integral equation efie matrix even with cutting edge preconditioning techniques often exhibits bad spectral properties arising from frequency or geometry based ill conditioning which render iterative solvers slow to converge or stagnate occasionally in this communication a novel technique to expedite the convergence of mom matrix solution at a specific frequency is proposed by extracting and applying eigen vectors from a previously solved neighboring frequency in an augmented generalized minimum residual agmres iterative framework this technique can be applied in unison with any preconditioner numerical results demonstrate up to 40 speed up in convergence using the proposed eigen agmres method

a characterization of exponential separation in terms of reducibility for linear difference equations abstract the main aim of this paper is to show that a linear difference equation is exponentially separated if and only if all neighboring equations are reducible

on the use of the singular value decomposition in the fast multipole method the authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method fmm the method is based on the application of the singular value decomposition svd to the plane wave fmm aggregation matrices these matrices are low ranked which is exploited to provide alternative sets of orthonormal singular basis functions obtained as linear combinations of the original basis by choosing only the most relevant singular functions a much more compact representation is obtained to accurately handle the interactions between the fmm groups the new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements with a moderate impact on the accuracy of the solution

exponential separation exponential dichotomy and spectral theory for linear systems of ordinary differential equations abstract this paper is concerned with linear nonautonomous systems of ordinary differential equations a criterion for exponential separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for exponential separation and the new result that an upper triangular system on a half line is exponentially separated if and only if the system corresponding to its diagonal is a minimal decomposition into exponentially separated subspaces is defined it turns out that it is in general finer than the sacker sell spectral decomposition but that the two decompositions coincide for almost periodic systems

structural stability via the density of a class of linear discrete systems abstract in this paper we prove that the set of multiplicatively separated linear discrete systems is dense in the set of linear systems and we find a relationship between the set of structurally stable systems and the set of the systems which have an exponential dichotomy

exponential dichotomy on the real line svd and qr methods abstract in this work we show when and how techniques based on the singular value decomposition svd and the qr decomposition of a fundamental matrix solution can be used to infer if a system enjoys or not exponential dichotomy on the whole real line

globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients abstract a new approach will be used to obtained the existence and uniqueness of positive almost periodic solution x n of the following nonlinear functional difference equation x n a n x n f n x n n also some sufficient conditions are established for global attractivity of x n for the proof of existence and uniqueness of x n the method used here is better than contraction mapping principle meanwhile when our main results applied to some famous discrete models lasota wazewska model and hematopoiesis model some new statements will be obtained and these results complement existing ones

conditions for kinematical similarity of a linear difference equation to a block upper triangular equation abstract in this paper we consider a linear difference equation in which the coefficient matrix is variable and depends on a positive parameter under some conditions we prove that the above equation is kinematically similar to a block upper triangular equation

pseudo almost periodic solutions for equation with piecewise constant argument abstract by using the roughness theory of exponential dichotomies and the contraction mapping some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument d x d t a t x t j 0 r a j t x t j g t x t x t x t r