rotation based rls algorithms unified derivations numerical properties and parallel implementations this work presents a unified derivation of four rotation based recursive least squares rls algorithms they solve the adaptive least squares problems of the linear combiner the linear combiner without a desired signal the single channel and the multichannel linear prediction and transversal filtering compared to other approaches the authors derivation is simpler and unified and may be useful to readers for better understanding the algorithms and their relationships moreover it enables improvements of some algorithms in the literature in both the computational and the numerical issues all algorithms derived in this work are based on givens rotations they offer superior numerical properties as shown by computer simulations they are computationally efficient and highly concurrent aspects of parallel implementation and parameter identification are discussed

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

a finite subgroup of the exceptional lie group g2 with a view to further refining the use of the exceptional group g2 in atomic and nuclear spectroscopy it is confirmed that a simple finite subgroup l168 psl2 7 of order 168 of the symmetric group s8 is also a subgroup of g2 it is established by character theoretic and other methods that there are two distinct embeddings of l168 in g2 analogous to the two distinct embeddings of so 3 in g2 relevant branching rules tensor products and symmetrized tensor products are tabulated as a stimulus to further applications the branching rules are given for the restriction from l168 to the octahedral crystallographic point group o

the ultra weak variational formulation for elastic wave problems the ultra weak variational formulation has been used effectively to solve time harmonic acoustic and electromagnetic wave propagation in inhomogeneous media we develop the ultra weak variational formulation for elastic wave propagation in two space dimensions in order to improve the accuracy and stability of the method we find it necessary to approximate the s and p wave components of the solution in a balanced way some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements

on efficient preconditioners for iterative solution of a galerkin boundary element equation for the three dimensional exterior helmholtz problem the paper presents a galerkin numerical method for solving the hyper singular boundary integral equations for the exterior helmholtz problem in three dimensions with a neumann s boundary condition previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and galerkin methods are not available due to the presence of a hypersingular integral operator this paper proposes a high order galerkin method by using singularity subtraction technique to reduce the hyper singular operator to a weakly singular one moreover we show here how to extend the previous work j appl numer math 36 4 2001 475 489 on sparse preconditioners to the galerkin case leading to fast convergence of two iterative solvers the conjugate gradient normal method and the generalised minimal residual method a comparison with the collocation method is also presented for the helmholtz problem with several wavenumbers

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

a collocation method for high frequency scattering by convex polygons we consider the problem of scattering of a time harmonic acoustic incident plane wave by a sound soft convex polygon for standard boundary or finite element methods with a piecewise polynomial approximation space the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave recently chandler wilde and langdon proposed a novel galerkin boundary element method for this problem for which by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency here we propose a related collocation method using the same approximation space for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the galerkin scheme but with a substantially reduced computational cost

conjugacy ofalt5andsl 2 5 subgroups ofe6 c f4 c and a subgroup ofe8 c of typea2e6 we classifyalt5andsl 2 5 subgroups ofe6 c andf4 c up to conjugacy we then use our results to answer conjugacy questions aboutalt5andsl 2 5 subgroups ofe8 c which were not resolved in earlier work

on the duality between the fast transversal and the fast qrd adaptive least squares algorithms the duality between the fast transversal and the fast qrd adaptive least squares algorithms is established it is shown that these two algorithmic families are related via a time varying state transformation the state transformation method is then applied to both the fast kalman as well as the faest ptf algorithm to derive fast qrd counterparts

on the duality between fast qr methods and lattice methods in least squares adaptive filtering the authors show that fast qr methods and lattice methods in least squares adaptive filtering are duals and follow from identical geometric principles whereas the lattice methods compute the residuals of a projection operation via the forward and backward prediction errors the qr methods compute instead the weights used in the projections within this framework the parameter identification problem is solved using fast qr methods by showing that the reflection coefficients and tap parameters of a least squares lattice filter operating in the joint process mode are immediately available as internal variables in the fast qr algorithms this parameter set can be readily exploited in system identification signal analysis and linear predictive coding for example the relations derived also lead to a fast least squares algorithm of minimal complexity that is a hybrid between a qr and a lattice algorithm the algorithm combines the order recursive properties of the lattice approach with the robust numerical behavior of the qr approach

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

analysis of a spectral galerkin approximation to the helmholtz equation in exterior domains an error analysis is presented for the spectral galerkin method to the helmholtz equation in 2 and 3 dimensional exterior domains the problem in unbounded domains is first reduced to a problem on a bounded domain via the dirichlet to neumann operator and then a spectral galerkin method is employed to approximate the reduced problem the error analysis is based on exploring delicate asymptotic behaviors of the hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems explicit error bounds with respect to the wave number are derived and some illustrative numerical examples are also presented

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

fast recursive least squares transversal filters for adaptive filtering fast transversal filter ftf implementations of recursive least squares rls adaptive filtering algorithms are presented in this paper substantial improvements in transient behavior in comparison to stochastic gradient or lms adaptive algorithms are efficiently achieved by the presented algorithms the true not approximate solution of the rls problem is always obtained by the ftf algorithms even during the critical initialization period first n iterations of the adaptive filter this true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic gradient algorithms factor of 1 6 to 3 5 depending upon application additionally the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing fast rls algorithms such as the fast kalman algorithms of morf ljung and falconer 1976 and the fast ladder lattice algorithms of morf and lee 1977 1981 they are further shown to attain steady state unnormalized or improve upon first n initialization steps the very low computational requirements of the efficient rls solutions of carayannis manolakis and kalouptsidis 1983 finally several efficient procedures are presented by which to ensure the numerical stability of the transversal filter algorithms including the incorporation of soft constraints into the performance criteria internal bounding and rescuing procedures and dynamic range increasing square root normalized variations of the transversal filters

high frequency scattering by convex curvilinear polygons we consider the scattering of a time harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with lipschitz boundary for standard boundary or finite element methods with a piecewise polynomial approximation space the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave here we propose a novel galerkin boundary element method with a hybrid approximation space consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes a uniform mesh on illuminated sides and graded meshes refined towards the corners of the polygon on illuminated and shadow sides numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases

the fast adaptive rotor s rls algorithm a fast algorithm for implementation of the qr factorization based recursive least squares rls adaptive filter is discussed this fast adaptive rotors far algorithm can be implemented with a pipelined array of processors called rotors and cisors the rotors compute 2 2 orthogonal givens rotations and the cisors compute the cosines and sines of the angles used in the rotors the algorithm requires 4n rotors and 2n cisors at each iteration to compute the solution to the rls problem the algorithm is numerically stable the far algorithm is derived using a single generic updating formula for orthogonal matrices which is introduced and derived whereas the generic updating formula is reminiscent of previous fast transversal filters and fast lattice algorithms the set of internally propogated adaptive filter quantities is entirely different and constitutes yet another complete characterization of the rls covariance and the forward backward and pinning estimation problems

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

embeddings of altn and its perfect covers for 6 in exceptional complex lie groups abstract we classify alt n and its perfect covers for n 6 up to conjugacy as subgroups of the exceptional complex lie groups with some exceptions where we give lower bounds and discuss the possibilities of lie primitive subgroups

towards a full bandwidth numerical acoustic model prediction models are at the heart of modern acoustic engineering current commercial room acoustic simulation software almost exclusively approximates the propagation of sound geometrically as rays or beams these assumptions yield efficient algorithms but the maximum accuracy they can achieve is limited by how well the geometric assumption represents sound propagation in a given space this comprises their accuracy at low frequencies in particular methods that directly model wave effects are more accurate but they have a computational cost that scales with problem size and frequency effectively limiting them to small or low frequency scenarios this paper will report the results of initial research into a new full bandwidth model which aims to be accurate and efficient for all frequencies the name proposed for this is the wave matching method this builds on the boundary element method with the premise that if an appropriate interpolation scheme is designed then the model will become geometrically dominated at high frequencies other propagation modes may then be removed without significant error yielding an algorithm which is accurate and efficient this paper will present the general concepts of wave matching and the results from some numerical test cases

boundary value problems and hardy spaces associated to the helmholtz equation in lipschitz domains abstract in this paper we introduce and discuss in the clifford algebra framework certain hardy like spaces which are well suited for the study of the helmholtz equation u k 2 u 0 in lipschitz domains of r n 1 in particular in the second part of the paper these results are used in connection with the classical boundary value problems for the helmholtz equation in lipschitz domains in arbitrary space dimensions in this setting existence uniqueness and optimal estimates are obtained by inverting the corresponding layer potential operators on l p for sharp ranges of p s also a detailed discussion of the helmholtz eigenvalues of lipschitz domains is presented

a hybrid numerical asymptotic method for scattering problems we develop a hybrid numerical asymptotic method for the helmholtz equation the method is a galerkin finite element method in which the space of trial solutions is spanned by asymptotically derived basis functions the basis functions are very efficient in representing the solution because each is the product of a smooth amplitude and an oscillatory phase factor like the asymptotic solution the phase is determined a priori by solving the eiconal equation using the ray method while the smooth amplitude is represented by piecewise polynomials the number of unknowns necessary to achieve a given accuracy with this new basis is dramatically smaller than the number necessary with a standard method and it is virtually independent of the wavenumber k we apply the method to the problems of scattering from a parabola and from a circle and compare the results with those of a standard finite element method

coupling of fast multipole method and microlocal discretization for the 3 d helmholtz equation we are concerned with an integral method applied to the solution of the helmholtz equation where the linear system is solved using an iterative method we need to perform matrix vector products whose time and memory requirements increase as a function of the wavenumber k many methods have been developed to speed up the matrix vector product calculation or to reduce the size of the system microlocal discretization methods enable one to consider new systems with reduced size another method the fast multipole method is one of the most efficient and robust methods used to speed up the calculation of matrix vector products in this paper a coupling of these two recently developed methods is presented this coupling enables one to reduce cpu time very efficiently for large wavenumbers satisfactory numerical tests are also presented to confirm the theoretical study within a new integral formulation results are obtained for a sphere with a size of 26 using a resolution based on a mesh with an average edge length of about 2 where is the wavelength results are also given for an industrial test case from dassault aviation the cetaf

a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains