the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

direct singular integral equation methods in scattering and propagation in strip or slot loaded structures problems of three dimensional 3 d scattering hybrid wave propagation for strip or slot loaded structures are often formulated in terms of systems of singular integral integrodifferential equations sie side of the first kind proper handling of the singular part of the kernels constitutes a major difficulty in carrying out method of moments mom three powerful techniques explored in the present paper provide efficient solutions by direct recourse to the theory of singular integral equations in contrast to low frequency methods wherein similar concepts are utilized for electrically narrow strips slots the proposed procedures are applicable uniformly to the whole range of widths from very narrow to very wide scatterers with remarkable accuracy numerical results are presented to validate and compare to one another the various numerical codes

a faffa mlfma algorithm for electromagnetic scattering based on the multilevel fast multipole algorithm mlfma an efficient method is proposed to accelerate the solution of the combined field integral equation in electromagnetic scattering and radiation where the fast far field approximation faffa is combined with mlfma the translation between groups in mlfma is expensive because spherical hankel functions and legendre polynomials are involved and the translator is defined on an eward sphere with many k spl circ directions when two groups are in the far field region however the translation can be greatly simplified by faffa where only a single k spl circ direction is involved in the translator the condition for using faffa and the way to efficiently incorporate faffa with mlfma are discussed complexity analysis illustrates that the computational cost in faffa mlfma can be asymptotically cut by half compared to the conventional mlfma numerical results are given to verify the efficiency of the algorithm

spheroidal vector wave eigenfunction expansion of dyadic green s functions for a dielectric spheroid the dyadic green s functions for defining the electromagnetic em fields for the inner and outer regions of a dielectric spheroid are formulated the dyadic green s function for an unbounded medium is expanded in terms of the spheroidal vector wave functions and the singularity at source points is extracted the principle of scattering superposition is then applied into the analysis to obtain the scattering spheroidal dyadic green s functions due to the existing interface coupled equation systems satisfied by scattering i e reflection and transmission coefficients of the dyadic green s functions are obtained so that these coefficients can be uniquely solved for the characteristics of the spheroidal dyadic green s functions as compared with the spherical and cylindrical green s dyadics are described and the improper developments of the spheroidal dyadic green s function for the outer region of a conducting spheroid in the existing work are pointed out

analyticity of the stokes semigroup in spaces of bounded functions the analyticity of the stokes semigroup with the dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the helmholtz decomposition the proof is based on a blow up argument this is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years

effect of small rank modification on the condition number of a matrix the condition number of a matrix plays an important role in numerical matrix computations in this paper we investigate how much the small rank modification method can reduce the condition number of a matrix

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

on l3 stability of strong solutions of the navier stokes equations with the navier type boundary conditions abstract we show that a strong solution u of the navier stokes initial boundary value problem on the time interval 0 t where 0 t is stable in the l 3 norm provided u l 2 0 t w 2 2 l 0 t w 1 2 the problem is considered with the so called navier type boundary conditions

exact image theory for the calculation of fields transmitted through a planar interface of two media exact image theory recently developed for the calculation of fields reflected from a planar interface between two homogenous media is extended for transmitted field calculation the image of a point source is found to be a line source located in complex space for proper convergence unlike for the reflection image the transmission image line is seen to be curved in complex space for efficient computation expressions are derived for the transmission image of the most general three dimensional electric and magnetic sources and for media with both epsilon and mu discontinuous across the interface the new theory is verified with several asymptotic tests and properties of the transmission image function are discussed as an application radiation from a resonant horizontal loop antenna into the ground is calculated and results are seen to agree with those given in the literature

on the asymptotic limit of the navier stokes system on domains with rough boundaries abstract we study the asymptotic behavior of solutions to the incompressible navier stokes system considered on a sequence of spatial domains whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non degenerate meaning not oriented in a single direction

excitation of complex modes of periodic structures using inhomogeneous plane wave scattering in fast and slow wave regions periodic metamaterial unit cells are analyzed by using inhomogeneous plane wave scattering the complex modes of the unit cell are derived using either the poles or the zeroes of the complex reflection coefficient of the fundamental floquet mode the required reflection coefficients are computed by a doubly periodic finite element boundary integral technique the search for the eigenvalues is accelerated by the nonlinear nelder mead method usually pairs of eigenvalues are observed which can be both real or conjugate complex also the eigenvalues correspond either to proper or improper modes by selecting the proper or the improper modes during field computation the other type of modes is removed from the observed response and instead zeroes are found in the reflection coefficient this can explain the absorption behavior of single sided open periodic leaky structures by investigating their reflection coefficient with the proposed inhomogeneous plane wave excitation approach the obtained results for various single sided open periodic unit cells are in good agreement with the results of other methods where in the presented excitation based method there is no need to solve any dispersion equation in the complex plane

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

additive preconditioning eigenspaces and the inverse iteration we incorporate our recent preconditioning techniques into the classical inverse power rayleigh quotient iteration for computing matrix eigenvectors every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations due to our modification we solve a well conditioned linear system instead we prove that this modification preserves local quadratic convergence show experimentally that fast global convergence is preserved as well and yield similar results for higher order inverse iteration covering the cases of multiple and clustered eigenvalues

discretization of continuous spectra based on perfectly matched layers as a tool of analysis in physics wavefields are often expanded in a set of eigensolutions obtained from a sturm liouville problem for singular sturm liouville problems subject to radiation boundary conditions i e problems defined on an infinite domain this set of eigensolutions has continuous parts in this paper we will show that it is possible to approximate this continuous set of eigensolutions by a discrete set of eigensolutions of the same sturm liouville operator but subject to dirichlet boundary conditions in complex space the idea of dirichlet boundary conditions in complex space stems from the perfectly matched layer pml absorbing boundary condition the pml was introduced in 1994 j p berenger j comput phys 114 1994 pp 185 200 as an absorbing termination of a finite difference time domain grid these complex space dirichlet boundary conditions have been used recently to close open electromagnetic waveguide structures in the present paper we aim at developing a mathemati

an efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures an efficient multilevel fast multipole algorithm mlfma formalism to model radiation and scattering by from large planar microwave structures is presented the technique relies on an electric field integral equation efie formulation and a series expansion for the green dyadic based on the use of perfectly matched layers pml in this way a new pml mlfma is developed to efficiently evaluate matrix vector multiplications arising in the iterative solution of the scattering problem the computational complexity of the new algorithm scales down to o n for electrically large structures the theory is validated by means of several illustrative numerical examples

random multipliers numerically stabilize gaussian and block gaussian elimination proofs and an extension to low rank approximation abstract we study two applications of standard gaussian random multipliers at first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize gaussian elimination with no pivoting as well as block gaussian elimination then by extending our analysis we prove that such a multiplier is also expected to support low rank approximation of a matrix without customary oversampling our test results are in good accordance with this formal study the results remain similar when we replace gaussian multipliers with random circulant or toeplitz multipliers which involve fewer random parameters and enable faster multiplication we formally support the observed efficiency of random structured multipliers applied to approximation but we still continue our research in the case of elimination we specify a narrow class of unitary inputs for which gaussian elimination with no pivoting is numerically unstable and then prove that with a probability close to 1 a gaussian random circulant multiplier does not fix numerical stability problems for such inputs we also prove that the power of the random circulant preprocessing increases if we also include random permutations

general solutions of the narrow strip and slot integral equations analytic solutions of the integral equations for the current induced on a narrow conducting strip are obtained by two methods one method follows from an expansion of the strip current in a series of chebyshev polynomials while the other is based on a special power series expansion in the former fourier type coefficients depending on integrals of the excitation are derived while in the latter the coefficients depend upon derivatives of the excitation from knowledge of the strip current simple expressions for the far zone scattered field are derived explicit results are given for plane wave excitation the solution methods apply to the equations for the narrow slot in a planar conducting screen

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

series representation of green dyadics for layered media using pmls the green dyadics for closed layered media i e layered media bounded by a perfectly conducting plate at the bottom and top of the structure can be expanded in a discrete surface wave series for open layered media with semi infinite layers at the top and or bottom of the structure the discrete series needs to be complemented by a branch cut integral of space waves in this paper we present a technique to circumvent this branch cut integral by truncating the semi infinite layers with a perfectly matched layer pml that is backed by a perfect electric conductor pec it is demonstrated that in this way it is possible to obtain an accurate series or closed form representation for the green dyadic of the open layered medium the series allows a very efficient calculation and storage of the green dyadic if it is needed for multiple observation and or excitation points very close to the source the series loses efficiency it is shown that the determination of the surface waves in the pml truncated layered medium has the same complexity as the determination of the surface waves in a pec truncated layered medium without a pml

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

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

an accurate derivation of spatial green s function for finite dielectric structures using characteristic green s function perfectly matched layer method a closed form spatial green s function for a truncated dielectric slab is derived by using a combination of the characteristic green s function cgf and perfectly matched layer pml method the original structure is terminated by pml that is backed by perfect electric conductor pec in semi infinite layer at the top and or bottom the eigenmodes of the closed structure by pml construct the continuous spectrum contribution of the original structure efficiently derived green s function is exact for separable finite structure while it is approximate for nonseparable one especially in the corners generalized scattering matrix gsm of truncating surface which contains possible conversions between all modes is computed with mode matching method it is demonstrated that by importing gsm in cgf formulation more exact results for nonseparable structures can be obtained the source and observation points dependence is analytically available in derived expression and allows very efficient computation and storage of the spatial green s function very close to the source where the large number of modes must be considered shank s transform is used to preserve the efficiency of the method the main advantage of the method lies in its accuracy and low dependency of the result to the pml parameters excellent agreements with the rigorous method of moments mom are shown in several examples

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

a homotopy method for locating all the poles of a parallel plate waveguide with the pml in this paper a homotopy method for locating all the poles of a parallel plate waveguide with the perfectly matched layer is proposed as is well known the green s functions for an open multilayered medium can be approximated by those for a parallel plate waveguide with the perfectly matched layer while the latter green s functions can be theoretically expressed into series expansions of their own eigenmodes without involving any numerical integrations the homotopy method here is applied for finding the desired mode poles in this method the newton raphson iteration is employed to solve every node equation except the start node equation while the dichotomy method is used to solve the start node equation the roots of the end node equation are just the desired mode poles with the proposed method the green s functions for a lossy or lossless multilayered medium can be efficiently calculated by the corresponding series expansions some numerical examples are provided to examine the robustness and accuracy of the proposed method

current induced on a conducting strip which resides on the planar interface between two semi infinite half spaces an analysis is described for determining the current induced by a known excitation on a conducting strip which resides on the planar interface between two semi infinite homogeneous half spaces of different electromagnetic properties the perfectly conducting flat strip is of infinite extent and the excitation is transverse magnetic to the strip axis an integral equation for the induced current is formulated and it is shown that its kernel which is in general a sommerfeld type integral can be expressed in closed form when the permeabilities of the two media are the same under this practical condition the integral equation is solved numerically and data are presented for cases of interest for the electrically narrow strip the integral equation is approximated and this approximate equation is solved analytically

error control in the perfectly matched layer based multilevel fast multipole algorithm the development of efficient algorithms to analyze complex electromagnetic structures is of topical interest application of these algorithms in commercial solvers requires rigorous error controllability in this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm pml mlfma a dedicated technique constructed to efficiently analyze large planar structures more specifically the crux of the algorithm viz the pertinent layered medium green functions is under investigation therefore particular attention is paid to the plane wave decomposition for 2 d homogeneous space green functions in very lossy media as needed in the pml mlfma the result of the investigations is twofold first upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived these formulas can be used in 2 d homogeneous lossy media mlfmas second a more heuristic approach to control the error of the pml mlfma s green functions is presented the theory is verified by means of several numerical experiments

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

comments on spheroidal vector wave eigenfunction expansion of dyadic green s functions for a dielectric spheroid with reply dudley and johnson see ibid vol 50 p xx xx nov 2002 comments that li leong kooi and yeo see ibid vol 49 p 645 659 april 2001 formulate in detail the dyadic green s functions for the dielectric spheroid stating that one result of their derivations the singularity of the electric green s dyadics is extracted they also comment that there is nothing formally wrong with extracting the delta function they object strongly however to the authors stating that they have extracted the singularity of the electric green s dyadics they have not done so li et al reply that they are grateful for the comments of dudley et al clarifying the singular behavior of the electric and magnetic field green s dyadics they intended to state irrotational part of the electric dyadic green s function which is highly singular but misused the term singularily of the dyadic green s function

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

current induced on a conducting cylinder located near the planar interface between two semi infinite half spaces an analysis is described for determining the current induced by a known excitation on a conducting cylinder located near the planar interface between two semi infinite homogeneous half spaces of different electromagnetic properties the perfectly conducting cylinder of general cross section is of infinite extent and the excitation is transverse magnetic to the cylinder axis an integral equations for the induced current is derived and a numerical method for solving it is developed the kernel of the integral equation contains a term corresponding to the usual open space green s function plus a term proportional to a sommerfeld type integral in two dimensions various forms of the sommerfeld type integral are given and the choices of form amenable to efficient evaluation are discussed for a flat strip a circular cylinder and a rectangular cylinder data are presented and discussed for selected parameters data are presented for cylinders above and below the interface as well as for a cylinder resting on the interface

boundary behavior of viscous fluids influence of wall roughness and friction driven boundary conditions we consider a family of solutions to the evolutionary navier stokes system supplemented with the complete slip boundary conditions on domains with rough boundaries we give a complete description of the asymptotic limit by means of convergence arguments and identify a general class of boundary conditions

matrix computations and polynomial root finding with preprocessing solution of homogeneous linear systems of equations is a basic operation of matrix computations the customary algorithms rely on pivoting orthogonalization and svd but we employ randomized preprocessing instead this enables us to accelerate the solution dramatically both in terms of the estimated arithmetic cost and the observed cpu time the approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations matrix eigen solving the solution of polynomial and secular equations and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure e g of the toeplitz or hankel type our analysis and extensive experiments show the power of the presented algorithms

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

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

viscous boundary layers for the navier stokes equations with the navier slip conditions we tackle the issue of the inviscid limit of the incompressible navier stokes equations when the navier slip with friction conditions are prescribed on impermeable boundaries we justify an asymptotic expansion which involves a weak amplitude boundary layer with the same thickness as in prandtl s theory and a linear behavior this analysis holds for general regular domains in both dimensions two and three

on the completeness of eigenmodes in a parallel plate waveguide with a perfectly matched layer termination an explicit proof of the completeness of the eigenmodes of a grounded parallel plate waveguide with a perfectly matched layer pml termination is given the proof is based on a general theorem governing the completeness of sets of complex exponentials the green s function of the structure is then obtained by means of a biorthogonalization procedure a comparison with the green s function of a grounded halfspace indicates that a pml termination simulates the open halfspace efficiently index terms completeness eigenmodes green s function parallel plate waveguide perfectly matched layer pml

a fast multipole method for layered media based on the application of perfectly matched layers the 2 d case an efficient fast multipole method fmm formalism to model scattering from two dimensional 2 d microstrip structures is presented the technique relies on a mixed potential integral equation mpie formulation and a series expression for the green functions based on the use of perfectly matched layers pml in this way a new fmm algorithm is developed to evaluate matrix vector multiplications arising in the iterative solution of the scattering problem novel iteration schemes have been implemented and a computational complexity of order o n is achieved the theory is validated by means of several illustrative numerical examples this paper aims at elucidating the pml fmm mpie concept and can be seen as a first step toward a pml based multilevel fast multipole algorithm mlfma for 3 d microstrip structures embedded in layered media

very weak solutions and large uniqueness classes of stationary navier stokes equations in bounded domains of r2 extending the notion of very weak solutions developed recently in the three dimensional case to bounded domains r2 we obtain a new class of unique solutions u in lq q 2 to the stationary navier stokes system u u u p f divu k u g with data f k g of low regularity as a main consequence we obtain a new uniqueness class also for classical weak or strong solutions indeed such a solution is unique if its lq norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

root finding by expansion with independent constraints elimination methods are highly effective for the solution of linear and nonlinear systems of equations but reversal of the elimination principle can be beneficial as well competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms that is their convergence from the start a well known example is the dual linear and nonlinear programming which enhances the power of optimization algorithms we believe that this is just an ad hoc application of general principle of expansion with independent constraints it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization at the end of this paper we comment on other applications and extensions of this principle presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros such as weierstrass durand kerner s root finder which reduces its root finding task to viete s vieta s system of n polynomial equations with n unknowns we adjust this root finder to the approximation of a single zero of p preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear together with computing a zero of a polynomial p the algorithm deflates this polynomial as by product and then could be reapplied to the quotient to approximate the next zero of p alternatively by using m processors that exchange no data one can concurrently approximate up to m zeros of p our tests confirm the efficiency of the proposed algorithms technically our root finding boils down to computations with structured matrices polynomials and partial fraction decompositions our study of these links can be of independent interest e g as by product we express the inverse of a sylvester matrix via its last column thus extending the celebrated result of gohberg and sementsul 1972 22 from toeplitz to sylvester matrix inverses

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

additive preconditioning for matrix computations abstract our randomized additive preconditioners are readily available and regularly facilitate the solution of linear systems of equations and eigen solving for a very large class of input matrices we study the generation of such preconditioners and their impact on the rank and the condition number of a matrix we also propose some techniques for their refinement and two alternative versions of randomized preprocessing our analysis and experiments show the power of our approach even where we employ weak randomization that is generate sparse and structured preconditioners defined by a small number of random parameters

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

a nondirective plane wave mlfma stable at low frequencies a novel method called the nondirective stable plane wave multilevel fast multipole algorithm nspwmlfma is presented to evaluate the low frequency lf interactions that cannot be handled by the multilevel fast multipole algorithm mlfma it is well known that the mlfma cannot be used for lf interactions since it suffers from numerical instability contrary to current techniques the proposed technique is not based on the spectral representation of the green function instead the addition theorem of the mlfma is manipulated into a form that allows numerically stable translations along the z axis the translation operator for these translations is derived in closed form a qr based method is devised to allow stable translations in all the other directions interpolations and anterpolations are also provided allowing a full multilevel algorithm since the nspwmlfma is based on the same mathematical foundations as the mlfma it requires limited adaptations to existing mlfma codes the fact that a qr is needed limits this algorithm to lf interactions however a coupling with the mlfma is straightforward allowing the easy construction of a broadband algorithm the dc limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems finally it is shown that the error introduced in the different steps of the algorithm is controllable and a single level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity

scattering of h polarized waves from conducting strips in the presence of an electrically uniaxial half space a singular integrodifferential equation approach electromagnetic scattering from perfectly conducting infinitely long strips in the presence of an electrically uniaxial half space is treated analytically a uniform h polarized plane wave is incident from the isotropic region as a prerequisite to this approach the h mode scalar green s function of the structure is developed its use in connection with the reciprocity theorem leads to a singular integrodifferential equation for the current density induced on the scatterer surface the equation is solved for strips parallel or perpendicular to the interface located above or embedded in the anisotropic region following a special technique based on the theory of singular integral equations that has been developed in connection with strips and slots and e or h polarized incident waves numerical results for the induced current density and for the scattered far field in a variety of cases are presented in graphical form whenever possible they are compared with existing results based on purely numerical procedures

scattering and radiation from by 1 d periodic metallizations residing in layered media an efficient technique is proposed to compute the scattering or radiation from by 1 d periodic structures residing in a layered background medium the technique is based on a mixed potential integral equation mpie combined with the method of moments mom solving for the unknown current density flowing within a unit cell of the periodic structure the formalism requires the knowledge of the pertinent layered medium green s functions with 1 d periodicity here these green s functions are derived in closed form by invoking the perfectly matched layer pml paradigm the stationary phase method is applied to determine the far field of the infinite periodic structure leading to a series of floquet modes in addition approximating this series leads to an efficient technique to estimate the scattering or radiation from by large but finite periodic structures with a 1 d periodic character the theory is illustrated and validated by means of various examples stemming from scattering and radiation applications from by antenna arrays residing on microstrip substrates the efficiency of the method is also demonstrated

lp theory for stokes and navier stokes equations with navier boundary condition abstract this work was intended as an attempt at studying stationary stokes and navier stokes problem with navier boundary conditions 1 3 we wish to investigate some results of existence uniqueness and regularity of solutions in hilbert case and in l p theory

spectral discretization of the vorticity velocity and pressure formulation of the stokes problem we consider the stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure next we propose a discretization by spectral methods which relies on this formulation and since it leads to an inf sup condition on the pressure in a natural way we prove optimal error estimates for the three unknowns we present numerical experiments which are in perfect coherence with the analysis

uniform regularity for the navier stokes equation with navier boundary condition we prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the navier stokes equation with the navier boundary condition has a strong solution this solution is uniformly bounded in a conormal sobolev space and has only one normal derivative bounded in l this allows us to obtain the vanishing viscosity limit to the incompressible euler system from a strong compactness argument

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

a high performance upgrade of the perfectly matched layer multilevel fast multipole algorithm for large planar microwave structures an improvement of the perfectly matched layer multilevel fast multipole algorithm pml mlfma for simulating large planar microwave structures is presented by exploiting the low rank property of the pml mlfma multimodal aggregation and disaggregation matrices considerable reductions in memory usage and computation time are obtained the method has been extensively validated demonstrating complete error controllability when simulating large planar microwave structures reductions in memory requirements and cpu time of more than 60 and 40 have been achieved

relevance of the slip condition for fluid flows near an irregular boundary we consider the navier stokes equation in a domain with a rough boundary the roughness is modeled by a small amplitude and small wavelength oscillation with typical scale varepsilon ll 1 for periodic oscillation it is well known that the best homogenized that is regular in varepsilon boundary condition is of navier type such result still holds for random stationary irregularities as shown recently by the first author 5 15 we study here arbitrary irregularity patterns

from the boltzmann equation to the stokes fourier system in a bounded domain we prove that the renormalized solutions of the boltzmann equation considered in a bounded domain with different types of kinetic boundary conditions converge to the stokes fourier system with different types of fluid boundary conditions when the main free path goes to zero this extends the work of f golse and d levermore 9 to the case of a bounded domain 2003 wiley periodicals inc

estimating extremal eigenvalues and condition numbers of matrices the paper describes probabilistic algorithms which may be used to make rough estimates of the largest and smallest eigenvalues of a positive definite matrix and the condition number of a nonsingular matrix in the 2 norm given varepsilon 0 and a prescribed relative error the algorithms compute estimates which with probability at least 1 varepsilon have relative errors less than that prescribed in particular the method gives a reliable way to estimate the condition number of a matrix of large degree

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

randomized preprocessing of homogeneous linear systems of equations abstract our randomized preprocessing enables pivoting free and orthogonalization free solution of homogeneous linear systems of equations in the case of toeplitz inputs we decrease the estimated solution time from quadratic to nearly linear and our tests show dramatic decrease of the cpu time as well we prove numerical stability of our approach and extend it to solving nonsingular linear systems inversion and generalized moore penrose inversion of general and structured matrices by means of newton s iteration approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank matrix eigen solving and root finding for polynomial and secular equations and for polynomial systems of equations some by products and extensions of our study can be of independent technical intersest e g our extensions of the sherman morrison woodbury formula for matrix inversion our estimates for the condition number of randomized matrix products and preprocessing via augmentation

surface electric fields and impedance matrix elements of stratified media for the various geometrical configurations of waves in stratified media we consider the important case when both source and field points are located on the same interface separating two different dielectric media we denote this configuration as surface electric field case in this paper the electric fields are calculated numerically without using potentials for the surface electric field case the integrand of the electric field grows with k sub spl rho sup 3 2 for large spl kappa sub spl rho making the sommerfeld integral singular to calculate the surface electric fields in the spatial domain we previously applied a technique of higher order asymptotic extraction in the higher order asymptotic extraction the higher order asymptotic parts were calculated analytically the remainder which has an integrand decays as spl kappa sub spl rho sup 3 2 was calculated numerically along the sommerfeld contour path of integration in this paper we use a different extraction technique the half space extraction after the half space extraction the integrand of the sommerfeld integral of stratified media decays exponentially and the integral is calculated along the sommerfeld integration path the half space extraction part is calculated by numerical integration along the vertical branch cuts the surface electric fields for stratified media using half space extraction and higher order asymptotic extraction are in good agreement to validate the accuracy of the solution we also compute the impedance matrix elements using surface electric fields testing and basis functions all in the spatial domain the results are then compared with the results of the spectral domain method the comparisons of the complex impedance matrix elements are tabulated and show that the difference is less than 2

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

solving linear systems of equations with randomization augmentation and aggregation abstract seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization augmentation and aggregation to reduce our task to computations with well conditioned matrices of full rank our algorithms avoid pivoting and orthogonalization preserve matrix structure and sparseness and in the case of an ill conditioned input perform only a small part of the computations with high accuracy we extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned our study can be of independent technical interest we extend the known results on conditioning of random matrices to randomized preconditioning estimate the condition numbers of randomly augmented matrices and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations

numerically safe gaussian elimination with no pivoting abstract gaussian elimination with no pivoting and block gaussian elimination are attractive alternatives to the customary but communication intensive gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely that is run into neither division by 0 nor numerical problems empirically safety and numerical safety of genp have been consistently observed in a number of papers where an input matrix was pre processed with various structured multipliers chosen ad hoc our present paper provides missing formal support for this empirical observation and explains why it was elusive so far namely we prove that genp is numerically unsafe for a specific class of input matrices in spite of its pre processing with some well known and well tested structured multipliers but we also prove that genp and bge are safe and numerically safe for the average input matrix pre processed with any nonsingular and well conditioned multiplier this should embolden search for sparse and structured multipliers and we list and test some new classes of them we also seek randomized pre processing that universally that is for all input matrices supports i safe genp and bge with probability 1 and or ii numerically safe genp and bge with a probability close to 1 we achieve goal i with a gaussian structured multiplier and goal ii with a gaussian unstructured multiplier and alternatively with gaussian structured augmentation we consistently confirm all these formal results with our tests of genp for benchmark inputs we have extended our approach to other fundamental matrix computations and keep working on further extensions

scattering of arbitrarily polarized plane waves obliquely incident on infinite slots or strips in a planar stratified medium an exact analysis of three dimensional scattering by a slot or by a strip right on an interface between two adjacent layers of a planar stratified isotropic medium is presented the most general case of arbitrarily polarized obliquely incident plane waves is considered the formulation is based on systems of singular integral integrodifferential equations sie side these systems are discretized with the help of two independent algorithms developed by tsalamengas see ibid vol 46 p 1560 70 1998 filling up the matrix elements involves rapidly convergent spectral integrals the expression of the far radiated field is derived and numerical results are presented for several cases

effective matrix free preconditioning for the augmented immersed interface method we present effective and efficient matrix free preconditioning techniques for the augmented immersed interface method aiim aiim has been developed recently and is shown to be very effective for interface problems and problems on irregular domains gmres is often used to solve for the augmented variable s associated with a schur complement a in aiim that is defined along the interface or the irregular boundary the efficiency of aiim relies on how quickly the system for a can be solved for some applications there are substantial difficulties involved such as the slow convergence of gmres particularly for free boundary and moving interface problems and the inconvenience in finding a preconditioner due to the situation that only the products of a and vectors are available here we propose matrix free structured preconditioning techniques for aiim via adaptive randomized sampling using only the products of a and vectors to construct a hierarchically semiseparable matrix approximation to a several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability the significance of the preconditioners includes 1 they do not require the entries of a or the multiplication of a t with vectors 2 constructing the preconditioners needs only o log n matrix vector products and o n storage where n is the size of a 3 applying the preconditioners needs only o n flops 4 they are very flexible and do not require any a priori knowledge of the structure of a the preconditioners are observed to significantly accelerate the convergence of gmres with heuristical justifications of the effectiveness comprehensive tests on several important applications are provided such as navier stokes equations on irregular domains with traction boundary conditions interface problems in incompressible flows mixed boundary problems and free boundary problems the preconditioning techniques are also useful for several other problems and methods

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

te scattering by conducting strips right on the planar interface of a three layered medium combining the fourier transform with powerful singular integral equation methods the authors investigate the scattering properties of a variety of two dimensional three layered planar structures loaded by a strip or a couple of coplanar strips right on an interface the primary excitation is the lowest order incident tm to strip axis surface wave mode or an incident e polarized plane wave in the practical case where the entire space is magnetically homogeneous the kernel of the integral equations is most conveniently decomposed into a singular closed form term added to an exponentially convergent inverse fourier integral an efficient accurate and stable numerical code is obtained and used to determine the strip induced current densities as well as any quantity of practical interest plotted results reveal that by a proper choice of certain geometrical parameters the structures may become very efficient radiators which can be designed to exhibit either very wideband and high directivity or considerable frequency scanning