one step finite difference time domain algorithm to solve the maxwell equations we present a one step algorithm to solve the time dependent maxwell equations for systems with spatially varying permittivity and permeability we compare the results of this algorithm with those obtained from the yee algorithm and from unconditionally stable algorithms we demonstrate that for a range of applications the one step algorithm may be orders of magnitude more efficient than multiple time step finite difference time domain algorithms we discuss both the virtues and limitations of this one step approach

analysis of multiple scattering iterations for high frequency scattering problems ii the three dimensional scalar case in this paper we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology based on integral equations for the numerical solution of scattering problems at high frequencies in more detail here we extend the two dimensional results in part i of this work to fully three dimensional geometries as in the former case our concern here is the determination of the rate of convergence of the multiple scattering iterations for a collection of three dimensional convex obstacles that are inherent in the aforementioned high frequency schemes to this end we follow a similar strategy to that we devised in part i first we recast the iterated neumann multiple scattering series in the form of a sum of periodic orbits of increasing period corresponding to multiple reflections that periodically bounce off a series of scattering sub structures then we proceed to derive a high frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically and finally we analyze this recurrence to provide an explicit rate of convergence associated with each orbit while the procedure is analogous to its two dimensional counterpart the actual analysis is significantly more involved and perhaps more interestingly it uncovers new phenomena that cannot be distinguished in two dimensional configurations e g the further dependence of the convergence rate on the relative orientation of interacting structures as in the two dimensional case and beyond their intrinsic interest we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple scattering series and thus to provide significant savings in computational times

an efficient high order nystrom scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface this text proposes a fast rapidly convergent nystrom method for the solution of the lippmann schwinger integral equation that mathematically models the scattering of time harmonic acoustic waves by inhomogeneous obstacles while allowing the material properties to jump across the interface the method works with overlapping coordinate charts as a description of the given scatterer in particular it employs partitions of unity to simplify the implementation of high order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations to deal with the discontinuous material interface in a high order manner a specialized quadrature is used in the boundary region the approach further utilizes an fft based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of o n log n for an n point discretization a detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance are presented in this paper

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

construction and analysis of fourth order finite difference schemes for the acoustic wave equation in nonhomogeneous media in this article we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients these schemes are fourth order accurate in space and time in the case of smooth media and are designed to remain stable and optimal for reflection transmission phenomena in the case of discontinuous coefficients together with a detailed mathematical study various numerical results are presented

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

local conditional lyapunov exponent characterization of consistency of dynamical response of the driven lorenz system local conditional lyapunov exponents are introduced and plotted in dynamical phase space to visualize the local susceptibility of the lorenz system driven by chaos and noise signals when the system produces consistent response outputs the dependence of the distribution of the local conditional lyapunov exponent on changes in the drive signal is investigated we consider the existence of multiple basins of consistent response for the same drive signal and the dependence of the basins on the drive signal

applications of ramsey s theorem to decision tree complexity combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented problems that are defined by simple inequalities between inputs called order invariant problems are considered a decision tree is called k bounded if each query depends on at most k variables no further assumptions on the type of queries are made it is proved that one can replace the queries of any k bounded decision tree that solves an order invariant problem over a large enough input domain with k bounded queries whose outcome depends only on the relative order of the inputs as a consequence all existing lower bounds for comparison based algorithms are valid for general k bounded decision trees where k is a constant an o n log n lower bound for the element uniqueness problem and several other problems for any k bounded decision tree such that k o n c and c n 1 2 bounded decision trees of complexity o n that solve the element uniqueness problem all the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well

hp adaptive finite elements in electromagnetics a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for curvilinear elements is presented the proposed method is illustrated and verified by a series of 2d experiments which include elements with curved boundaries and nonhomogeneous media

the complexity of problems on probabilistic nondeterministic and alternating decision trees this work generalizes decision trees in order to study lower bounds on the running times of algorithms that allow probabilistic nondeterministic or alternating control it is shown that decision trees that are allowed internal randomization at the expense of introducing a small probability of error run no faster asymptotically than ordinary decision trees for a collection of natural problems two geometric techniques from the literature for proving lower bounds on the time required by ordinary decision trees are shown to be special cases of one unified technique that in fact applies to nondeterministic decision trees as well finally it is shown that any lower bound on alternating decision tree time also applies to alternating turing machine time

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

a high order mixed vector finite element method for solving the time dependent maxwell equations on unstructured grids we present a mixed vector finite element method for solving the time dependent coupled ampere and faraday laws of maxwell s equations on unstructured hexahedral grids that employs high order discretization in both space and time the method is of arbitrary order accuracy in space and up to 4th order accurate in time making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods in addition the method correctly models both the jump discontinuities and the divergence free properties of the electric and magnetic fields is charge and energy conserving conditionally stable and free of spurious modes several computational experiments are performed to demonstrate the accuracy efficiency and benefits of the method

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

pml enhanced with a self adaptive goal oriented hp finite element method simulation of through casing borehole resistivity measurements we describe the application of a perfectly matched layer pml combined with a self adaptive goal oriented hp finite element fe method to the simulation of borehole resistivity measurements the adaptive refinements and fast convergence of the self adaptive hp fe method enhance the performance of the pml thereby enabling the accurate and efficient truncation of the computational domain in open domain problems we apply this method to the simulation of axisymmetric through casing resistivity measurements acquired in a borehole environment that are typically used for the assessment of rock formation properties our numerical results confirm the accuracy and efficiency of our method and provide evidence of highly accurate and reliable simulations of borehole logging measurements in the presence of a conductive steel casing and material contrast of fourteen orders of magnitude in electrical conductivity moreover the combination of adaptivity and pml enables us to significantly reduce the size of the computational domain with substantial savings in computer time and memory

a k space method for coupled first order acoustic propagation equations a k space method for large scale simulation of ultrasonic pulse propagation is presented the present method which solves the coupled first order differential equations for wave propagation in inhomogeneous media is derived in a simple form analogous to previous finite difference methods with staggered spatial and temporal grids like k space methods based on second order wave equations the present method is exact for homogeneous media unconditionally stable for slow c r c0 media and highly accurate for general weakly scattering media in addition unlike previous k space methods the form of the method allows straightforward inclusion of relaxation absorption and perfectly matched layer pml nonreflecting boundary conditions numerical examples illustrate the capabilities of the present k space method for weakly inhomogeneous media accurate results are obtained using coarser temporal and spatial steps than possible with comparable finite difference and pseudospectral methods the low dispers

acoustical wave propagator in this paper an explicit acoustical wave propagator awp is introduced to described the time domain evolution of acoustical waves to implement its operation on an initial state of wave motion the acoustical wave propagator is approximated as a chebyshev polynomial expansion which converges to machine accuracy the spatial gradient in each polynomial term is evaluated by a fourier transformation scheme analysis and numerical examples demonstrated that this chebyshev fourier scheme is highly accurate and computational effective in predicting time domain acoustical wave propagation and scattering

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

solving the maxwell equations by the chebyshev method a one step finite difference time domain algorithm we present a one step algorithm that solves the maxwell equations for systems with spatially varying permittivity and permeability by the chebyshev method we demonstrate that this algorithm may be orders of magnitude more efficient than current finite difference time domain fdtd algorithms

an efficient high order algorithm for acoustic scattering from penetrable thin structures in three dimensions this paper presents a high order accelerated algorithm for the solution of the integral equation formulation of volumetric scattering problems the scheme is particularly well suited to the analysis of thin structures as they arise in certain applications e g material coatings in addition it is also designed to be used in conjunction with existing low order fft based codes to upgrade their order of accuracy through a suitable treatment of material interfaces the high order convergence of the new procedure is attained through a combination of changes of parametric variables to resolve the singularities of the green function and partitions of unity to allow for a simple implementation of spectrally accurate quadratures away from singular points accelerated evaluations of the interaction between degrees of freedom on the other hand are accomplished by incorporating two face equivalent source approximations on cartesian grids a detailed account of the main algorithmic components of the scheme are presented together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

maxwell eigenvalues and discrete compactness in two dimensions abstract we present an elementary proof of the discrete compactness result for a general class of hp finite elements introduced in 1 2 we discuss h convergence of 2d elements only and in this context the results are not new as the analysis of h curl conforming elements for maxwell s equations can be reduced to the long known results for raviart thomas elements 3 the work is based on the result of kikuchi 4 5 for nedelec s edge triangular elements of the lowest order and presents an alternative to techniques presented in 3 6 in particular the present version does not use an inverse inequality argument and therefore is valid for h adaptive meshes we conclude the presentation with a number of 2d computational experiments including nonconvex domains

hp vector finite element method for eigenmode analysis of waveguides a new stable variational formulation for full wave analysis of waveguides is presented numerical experiments show that the method remains stable at very low frequencies theoretical results apply to a wide class of vector finite elements including two families of nedelec numer math 35 1980 315 numer math 50 1986 57 and their generalization hp edge elements comput methods appl mech engrg 152 1998 103 comput methods appl mech engrg 169 1999 331

characterizing the dynamics of stochastic bistable systems by measures of complexity the dynamics of noisy bistable systems is analyzed by means of lyapunov exponents and measures of complexity we consider both the classical kramers problem with additive white noise and the case when the barrier fluctuates due to additional external colored noise in the case of additive noise we calculate the lyapunov exponents and all measures of complexity analytically as functions of the noise intensity or the mean escape time respectively for the problem of a fluctuating barrier the usual description of the dynamics with the mean escape time is not sufficient the application of the concept of measures of complexity allows us to describe the structures of motion in more detail most complexity measures indicate the value of the correlation time at which the phenomenon of resonant activation occurs with an extremum

a two dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities part i energy norm based automatic hp adaptivity abstract the accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering in some cases the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized in other cases the discontinuities are deliberately introduced into the waveguide to perform a certain electric function this is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp adaptive finite element method in this paper a fully automatic energy norm based hp adaptive finite element fe strategy applied to a number of relevant waveguide structures is presented the methodology produces exponential convergence rates in terms of the energy norm error of the solution against the problem size number of degrees of freedom extensive numerical results demonstrate the suitability of the hp method for solving different rectangular waveguide discontinuities these results illustrate the flexibility reliability and high accuracy of the method the self adaptive hp fem provides similar sometimes more accurate results than those obtained with semi analytical techniques such as the mode matching method for problems where semi analytical methods can be applied at the same time the hp fem provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics metallic screws round corners etc which cannot be easily considered when using semi analytical techniques in the second part the work is extended by presenting a fully automatic goal oriented hp adaptive fe in terms of the scattering or s parameters which are widely used in microwave engineering for the characterization of microwave devices

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

mathematical analysis and finite element simulation of a magnetized ferrite model microwave scattering by obstacles made of ferrite materials has been studied for a long time because of its important applications in radar satellite optical imaging system signal processing atmospheric science and even cloaking device design in this paper we present for the first time the rigorous mathematical analysis of a magnetized ferrite model widely used by engineers the well posedness of the model is established first then we develop a finite element time domain method for solving this model numerical stability similar to the continuous stability is proved for our scheme and the optimal error estimate in the l 2 norm is obtained numerical results are consistent with our theoretical analysis and show how wave propagates in the ferrite material

the regular fourier matrices and nonuniform fast fourier transforms for any triple of positive integers m n q the matrix f m n q called the m n q regular fourier matrix is defined the regular fourier matrices f m n q are then applied to set up new algorithms for nonuniform fast fourier transforms numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity the algorithms require o n cdot log 2n arithmetic operations where n is the number of data points

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

acoustical wave propagator for time domain flexural waves in thin plates in this paper an explicit acoustical wave propagator technique is introduced to describe the time domain evolution of acoustical waves in two dimensional plates a combined scheme with chebyshev polynomial expansion and fast fourier transformation is used to implement the operation of the acoustical wave propagator through this operation the initial wave packet at t 0 is mapped into the wave packet at any instant t 0 by comparison of the results of the exact analytical solution and the euler numerical method we find that this new chebyshev fourier scheme is highly accurate and computationally effective in predicting the acoustical wave propagation in thin plates this method offers an opportunity for future study of dynamic stress concentration and time domain energy flow in coupled structures

an infinite element for maxwell s equations a new infinite element ie approximation for steady state maxwell s equations is proposed the ie is compatible with a new class of hp finite element fe discretizations presented in 13 32 following 12 a convergence analysis for the proposed coupled fe ie approximations is presented

an hp adaptive finite element method for electromagnetics part 1 data structure and constrained approximation abstract this is the first of papers describing an implementation of the hp adaptive mixed finite element fe method for the solution of steady state maxwell s equations proposed in l demkowicz and l vardapetyan modeling of electromagnetic absoption scattering problems using hp adaptive finite elements computer methods in applied mechanics and engineering 152 1998 103 124 the discretization is defined on a hybrid grid consisting of both triangles and quads and allows for both h and p refinements of the mesh the paper focuses on the data structure and constrained approximation issues and provides a number of illustrative examples

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

on the gibbs phenomenon v recovering exponential accuracy from collocation point values of a piecewise analytic function this paper presents a method to recover exponential accuracy at all points including at the discontinuities themselves from the knowledge of an approximation to the interpolation polynomial or trigonometrical polynomial we show that if we are given the collocation point values or a highly accurate approximation at the gauss or gauss lobatto points we can reconstruct an uniform exponentially convergent approximation to the function f x in any sub interval of analyticity the proof covers the cases of fourier chebyshev legendre and more general gegenbauer collocation methods a numerical example is also provided

an hp adaptive finite element method for electromagnetics part 3 a three dimensional infinite element for maxwell s equations this paper is a continuation of reference 26 cecot demkowicz and rachowicz computer methods in applied mechanics and engineering 2000 188 625 643 and describes an implementation of the infinite element for three dimensional time harmonic maxwell s equations proposed in reference 15 demkowicz and pal computer methods in applied mechanics and engineering 1998 164 77 94 the element is compatible with the hp finite element discretizations for maxwell s equations in bounded domains reported in references 16 18 computer methods in applied mechanics and engineering 1998 152 103 124 1999 169 331 344 2000 187 307 337 copyright 2003 john wiley sons ltd

a domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change a domain decomposition technique is proposed for the computation of the acoustic wave equation in which the bulk modulus and density fields are allowed to be discontinuous at the interfaces inside each subdomain the method presented coincides with the second order finite difference schemes traditionally used in geophysical modeling however the possibility of assigning to each subdomain its own space step makes numerical simulations much less expensive another interest of the method lies in the fact that its hybrid variational formulation naturally leads to exact equations for gridpoints on the interfaces transposing babuska brezzi s formalism on mixed and hybrid finite elements provides a suitable functional framework for this domain decomposition formulation and shows that the inf sup condition remains the basic requirement for convergence to occur

a two dimensional infinite element for maxwell s equations abstract the paper describes an implementation of the infinite element for two dimensional steady state maxwell s equations proposed by l demkowicz and m pal comput meth appl mech engrg in press the element is compatible with the hp fe element discretizations for maxwell s equations in bounded domains reported in l demkowicz and l vardapetyan comput meth appl mech engrg 152 1 2 1996 103 124 and w rachowicz and l demkowicz comput meth appl mech engrg in press

the k space formulation of the scattering problem in the time domain an improved single propagator formulation the storage allocation requirement and solution speed of the k space formulation of the scattering problem in the time domain developed by this author n bojarski j acoust soc am 72 570 584 1982 were improved by a factor of 2 and 4 respectively by the replacement of the earlier two temporal propagators by a single more accurate propagator and the utilization of the special case real to complex fft algorithm it is shown that for problems of the size of about one million spatially discretized cells this method is about five times slower than an ideal solution and requires about five times as much in core storage allocation as an ideal solution it is shown that for a modern array processor this solution is i o bound it is pointed out that this method is ideally suited for parallel array processor implementation resulting in even greater speed

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

a self adaptive goal oriented hp finite element method with electromagnetic applications part ii electrodynamics abstract we present the formulation implementation and applications of a self adaptive goal oriented hp finite element fe method for electromagnetic em problems the algorithm delivers without any user interaction a sequence of optimal hp grids this sequence of grids minimizes the error in a prescribed quantity of interest with respect to the problem size and it converges exponentially in terms of the relative error in a user prescribed quantity of interest against the cpu time including problems involving high material contrasts boundary layers and or several singularities the goal oriented refinement strategy is an extension of a fully automatic energy norm based hp adaptive algorithm we illustrate the efficiency of the method with 2d numerical simulations of maxwell s equations using both h 1 conforming continuous elements and h curl conforming nedelec edge elements applications include alternate current ac resistivity logging instruments in a borehole environment with steel casing for the assessment of rock formation properties behind casing logging instruments steel casing and rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole for the presented challenging class of problems the self adaptive goal oriented hp fem delivers results with 5 7 digits of accuracy in the quantity of interest

adaptive finite element techniques for the maxwell equations using implicit a posteriori error estimates we consider an implicit a posteriori error estimation technique for the adaptive solution of the maxwell equations with nedelec edge finite element methods on three dimensional domains on each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis the discrete bilinear form of the local problems is shown to satisfy an inf sup condition which ensures the well posedness of the error equations an adaptive solution algorithm is developed based on the obtained error estimates the performance of the method is tested on various problems including non convex domains with non smooth boundaries the numerical results show that the estimated error computed by the implicit a posteriori error estimation technique correlates well with the actual error on the meshes generated adaptively with the help of the error estimator the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom moreover the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

inhomogeneous lossy waveguide mode analysis abstract this paper discusses electromagnetic numerical mode analysis in waveguides with materially inhomogeneous cross sections and material dissipation a full wave formulation of maxwell s homogeneous equations including gauss electric law stable at vanishing propagation constant is implemented and verified in terms of the hp adaptive version of the finite element method it provides the possibility to use high order polynomial enrichments combined with strongly graded meshes it is considered most efficient in resolving the loss of solution regularity at material interfaces with large contrast numerical examples including materially lossless homogeneous and inhomogeneous cross sections with and without losses are analysed to corroborate the implementation the efficiency of using higher order polynomial enrichments is shown the approach is anticipated to have a broad application from modern on chip interconnect and antenna technologies to the design of low observable aerial vehicles

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

a fast higher order solver for scattering by penetrable bodies in three dimensions in this paper we introduce a new fast higher order solver for scattering by inhomogeneous media in three dimensions as in previously existing methods the low complexity of our integral equation method o nlogn operations for an n point discretization is obtained through extensive use of the fast fourier transform fft for the evaluation of convolutions however the present approach obtains significantly higher order accuracy than these previous approaches yielding at worst third order far field accuracy or substantially better for smooth scatterers even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the increased order of convergence of our method results from i a partition of unity decomposition of the green s function into a smooth part with unbounded support and a singular part with compact support and ii replacement of the possibly discontinuous scatterer by an appropriate fourier smoothed scatterer the resulting convolutions can then be computed with higher order accuracy by means of o nlogn ffts we present a parallel implementation of our approach and demonstrate the method s efficiency and accuracy through a variety of computational examples for a very large scatterer considered earlier in the literature with a volume of 3648 l 3 where l is the wavelength using the same number of points per wavelength and in computing times comparable to those required by the previous approach the present algorithm produces far field values whose errors are two orders of magnitude smaller than those reported previously

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

time domain modeling of acoustic propagation with acoustic wave propagator and absorbing boundary conditions the acoustic wave propagator awp is the application of the time evolution operator on the acoustic wave equation for stationary systems in a polynomial expansion of chebychev polynomials it allows to increase the time step by more than one order of magnitude compared to finite difference time domain fdtd codes in contrast to other implementations of the awp the spatial differentiation is carried out with finite difference techniques because this allows the use of the perfectly matched layer formulation as absorbing boundary conditions the formulation includes the direct implementation of acoustic sources with sinusoidal time evolution other sources can be synthesized by their fourier components for the calculation of large areas the explicit formulation of a large system matrix can be avoided by calculating the propagation equations for each time step at row and column level repeatedly which reduces memory requirements notably this procedure and the suitability of the finite difference approach

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

an eigenfunction expansion method to efficiently evaluate spatial derivatives for media with discontinuous properties pseudo spectral methods are often used as an alternative to the finite difference time domain fdtd method to model wave propagation in heterogeneous moving media the fdtd method is robust and accurate but is numerically expensive pseudo spectral methods make use of the wavelike nature of the solution to obtain more efficient time domain algorithms the most straightforward of the pseudo spectral methods is the fourier method in which a spatial fourier transform is used to evaluate the spatial derivatives in the wave equation whereas this method is accurate for a weakly heterogeneous moving medium it degenerates for media with discontinuous properties the eigenfunction expansion method presented here is a way to accurately and efficiently evaluate spatial derivatives in media with interfaces as in the fourier method transforms may be calculated using fft s and spatial sampling is limited only by the nyquist condition the performance of the method is shown in a time domain implementation for media

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency

higher order fourier approximation in scattering by two dimensional inhomogeneous media this paper provides a theoretical analysis of a higher order fft based integral equation method introduced recently ieee trans antennas and propagation 48 2000 pp 1862 1864 for the evaluation of transverse electric polarized electromagnetic scattering from a bounded penetrable inhomogeneity in two dimensional space roughly speaking this method is based on fourier smoothing of the integral operator and the refractive index n x here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher order accuracy even when n x is a discontinuous function as suggested by the numerical experiments contained in the paper mentioned above in detail we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer and we demonstrate with a few numerical examples that the predicted convergence rates are achieved in practice

computation of resonant modes for axisymmetric maxwell cavities using hp version edge finite elements the computation of the resonant frequencies for closed cavities is not a trivial task multi materials and sharp corners all give rise to highly singular eigenfunctions however an approach using hp finite elements is well suited to such problems and with the correct combination of h and p refinements it yields the theoretically predicated exponential rates of convergence in this paper we present a novel approach to the solution of axisymmetric cavity problems which uses a hierarchic h1 and h curl conforming finite element basis a selection of numerical examples is included and these demonstrate that the exponential rates of convergence are achieved in practice copyright 2005 john wiley sons ltd

an efficient preconditioned high order solver for scattering by two dimensional inhomogeneous media we consider the problem of evaluating the scattering of te polarized electromagnetic waves by two dimensional penetrable inhomogeneities building upon previous work ieee trans antennas propag 48 2000 1862 we present a practical and general fast integral equation algorithm for this problem the contributions introduced in this text include 1 a preconditioner that significantly reduces the number of iterations required by the algorithm in the treatment of electrically large scatterers 2 a new radial integration scheme based on chebyshev polynomial approximation which gives rise to increased accuracy efficiency and stability and 3 an efficient and stable method for the evaluation of scaled high order bessel functions which extends the capabilities of the method to arbitrarily high frequencies these enhancements give rise to an algorithm that is much more accurate and efficient than its previous counterpart and that allows for treatment of much larger problems than permitted by the previous approach in one test case for example the present algorithm results in far field errors of 8 9 10 13 in a 2 12s calculation on a 1 7 ghz pc whereas the original algorithm gave rise to far field errors of 1 1 10 8 in 88 91s on a 400 mhz pc in another example the present algorithm evaluates accurately the scattering by a cylinder of acoustical size r 256 which is of the order of 20 times larger 400 times larger in square wavelengths than the largest scatterers that could be treated by the previous approach yielding at worst third order far field accuracy or substantially better for smooth scatterers in fast computing times o n log n operations for an n point mesh even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the proposed approach is the highest order o n log n solver in existence for the problem under consideration

an exact and fast computation of discrete fourier transform for polar and spherical grid numerous applied problems of two dimensional 2 d and 3 d imaging are formulated in continuous domain they place great emphasis on obtaining and manipulating the fourier transform in polar and spherical coordinates however the translation of continuum ideas with the discrete sampled data on a cartesian grid is problematic there exists no exact and fast solution to the problem of obtaining discrete fourier transform for polar and spherical grids in the literature in this paper we develop exact algorithms to the above problem for 2 d and 3 d which involve only 1 d equispaced fast fourier transform with no interpolation or approximation at any stage the result of the proposed approach leads to a fast solution with very high accuracy we describe the computational procedure to obtain the solution in both 2 d and 3 d which includes fast forward and inverse transforms we find the nested multilevel matrix structure of the inverse process and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

a probabilistic distributed algorithm for set intersection and its analysis a probabilistic algorithm for checking set disjointness and performing set intersection of two sets stored at different machines is presented the algorithm is intended to minimize the amount of communication between the machines if n is the total number of elements and k is the number of bits required to represent each of the elements then it is shown that the expected running time of the set disjointness algorithm is o log log n rounds each round consisting of exchanging one message with o n k bits and performing o n steps of local computation all the constants are small the analysis of the algorithm involves approximating markov chains by deterministic models

minimum memory for generating rare events we classify the rare events of structured memoryful stochastic processes and use this to analyze sequential and parallel generators for these events given a stochastic process we introduce a method to construct a process whose typical realizations are a given process rare events this leads to an expression for the minimum memory required to generate rare events we then show that the recently discovered classical quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations