energetic bem fem coupling for wave propagation in 3d multidomains summary starting from a recently developed energetic space time weak formulation of boundary integral equations for wave propagation problems a coupling algorithm is presented which allows a flexible use of fems and bems as local discretization techniques emphasis is given to theoretical and experimental analysis of the stability of the proposed method several numerical results on model problems are presented and discussed showing that both bounded and unbounded three dimensional domains can be efficiently addressed copyright 2013 john wiley sons ltd

a hamiltonian jacobi algorithm the nonsymmetric jacobi iteration is adapted to the special structure of hamiltonian matrices this hamiltonian jacobi algorithm uses symplectic unitary similarity transformations to solve algebraic riccati equations through the hamiltonian schur form it preserves hamiltonian structure without using a condensed form although it converges too slowly for use on conventional serial computers it may be attractive for some highly parallel architectures

a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

elastodynamic direct boundary element methods with enhanced numerical stability properties evidence of numerical instabilities in two dimensional time domain direct boundary element methods is presented the e ects of numerical versus analytical integration of spatial integrals on stability are shown and two new time stepping algorithms are introduced and compared to existing formulations the so called new direct half step scheme and the epsilon scheme are shown to improve the numerical stability of direct boundary element methods copyright 1999 john wiley sons ltd

robust estimation via minimum distance methods given a fixed parametric family p it is desired to estimate however due to contaminations of various sorts the data actually collected by the statistician follow a distribution that is close to but possibly distinct from thep s it is proved that under these conditions and in an appropriate asymptotic framework the minimum distance estimators of the cramer von mises type are robust specification of a cramer von mises weight functionh defines a notion of distance each such choice ofh then delineates the kind of contamination possible and leads to an estimator which defends optimally against it when the theory is specialized to location models various choices ofh lead to estimators asymptotically equivalent to such familiar ones as trimmed mean median hodges lehmann estimator and so forth the framework developed herein provides some guidelines for choosing among the possible estimators and suggests that the standard cramer von mises estimator of location is probably as good a robust estimator as any

theoretical aspects of the application of convolution quadrature to scattering of acoustic waves in this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non homogeneous obstacles using convolution quadrature cq techniques for the time variable and coupled boundary element method finite element method for the space variable the applicability of cq to waves requires polynomial type bounds for operators related to the operator s 2 in the right half complex plane we propose a new systematic way of dealing with this problem both at the continuous and semidiscrete in space cases we apply the technique to three different situations scattering by a group of sound soft and hard obstacles by homogeneous and non homogeneous obstacles

some remarks on the complex j symmetric eigenproblem abstract the eigenproblem for complex j symmetric matrices h c a c d a t a c c t d d t c n n is considered a proof of the existence of a transformation to the complex j symmetric schur form proposed in 21 is given the complex symplectic unitary qr decomposition and the complex symplectic sr decomposition are discussed it is shown that a qr like method based on the complex symplectic unitary qr decomposition is not feasible here a complex symplectic sr algorithm is presented which can be implemented such that one step of the sr algorithm can be carried out in o n arithmetic operations based on this a complex symplectic lanczos method can be derived moreover it is discussed how the 2 n 2 n complex j symmetric matrix h c can be embedded in a 4 n 4 n real hamiltonian matrix

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

time domain boundary elements for dynamic contact problems abstract this article considers a unilateral contact problem for the wave equation the problem is reduced to a variational inequality for the dirichlet to neumann operator for the wave equation on the boundary which is solved in a saddle point formulation using boundary elements in the time domain as a model problem also a variational inequality for the single layer operator is considered a priori estimates are obtained for galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area where the existence of solutions to the continuous problem is known numerical experiments demonstrate the performance of the proposed mixed method they indicate the stability and convergence beyond flat geometries

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

minimum distance estimators for location and goodness of fit abstract a weighted cramer von mises distance between the empirical distribution function and the assumed model f0 x is minimized to produce estimators n that are asymptotically normal if the weight function is taken proportional to ln f 0 f 0 then n is asymptotically efficient and the minimized distance has the appropriate loss of one degree of freedom special attention is focused on the limiting distribution of this latter goodness of fit statistic in both null and alternative situations

the automatic robustness of minimum distance functionals on montre que les estimateurs de distance minimale forment une classe d estimateurs automatiquement robustes dans le meme sens que l estimateur du maximum de vraisemblance

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

toda lattices with indefinite metric ii topology of the iso spectral manifolds abstract we consider the iso spectral real manifolds of tridiagonal hessenberg matrices with distinct real eigenvalues the manifolds are described by the iso spectral flows of indefinite toda lattice equations introduced by the authors physica d 91 1996 321 339 these toda lattices consist of 2n 1 different systems with hamiltonians h 1 2 k 1 n y k 2 k 1 n 1 s k s k 1 exp x k x k 1 where sk 1 which blow up in finite time except the case with all sksk 1 1 we compactify the manifolds by adding infinities according to the toda flows the resulting manifolds are shown to be nonorientable for n 2 and the symmetry group is the semi direct product of z 2 n 1 and the permutation group s n these properties identify themselves with small covers introduced by davis and januszkiewicz duke math j 62 1991 417 451 as a corollary of our construction we give a formula for the total number of zeros for a system of exponential polynomials generated as hankel determinants

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

minimum distance estimators of scale with censored data abstract this article examines how the level of censoring affects the performance of estimators of scale that minimize a weighted cramervon mises distance between the kaplan meier product limit estimator of the survival distribution function and an assumed model the article shows that under certain conditions these estimators are asymptotically normal if the survival function is a member of the assumed scale family weights that minimize the asymptotic variance under these conditions also are found under stronger conditions the minimum distance estimators mde s are asymptotically normal even if the survival function is not a member of the assumed scale family asymptotic coverage probabilities of confidence intervals are found for percentiles constructed from exponential mde s of scale under departures from the exponential distribution and what happens in the limiting cases of no censoring and complete censoring is examined using the asymptotic results and monte carlo experiments several exponenti

toda versus pfaff lattice and related polynomials the pfaff lattice was introduced by us in the context of a lie algebra splitting of gl infinity into sp infinity and an algebra of lower triangular matrices the pfaff lattice is equivalent to a set of bilinear identities for the wave functions which yield the existence of a sequence of tau functions the latter satisfy their own set of bilinear identities which moreover characterize them in the semi infinite case the tau functions are pfaffians in the same way that for the toda lattice the tau functions are hankel determinants interesting examples occur in the theory of random matrices where one considers symmetric and symplectic matrix integrals for the pfaff lattice and hermitian matrix integrals for the toda lattice there is a striking parallel between the pfaff lattice and the toda lattice and even more striking there is a map from one to the other mapping skew orthogonal to orthogonal polynomials in particular we exhibit two maps dual to each other mapping hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals

a symplectic acceleration method for the solution of the algebraic riccati equation on a parallel computer abstract we give a cubic acceleration method for improving the current symplectic jacobi like algorithm for computing the hamiltonian schur decomposition of a hamiltonian matrix and finding the positive semidefinite solution of the riccati equation the acceleration method can speed up the rate of convergence at the end of the symplectic jacobi like process when the norm of the current strictly j lower triangle has become sufficiently small it has high parallelism and takes o n computational time when implemented on a mesh connected n n array processor system a quantitative analysis of convergence and numerical comparisons of one jacobi sweep versus one correction step are presented

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

coupling discontinuous galerkin methods and retarded potentials for transient wave propagation on unbounded domains this work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities to do so we propose to combine a discretization based on a discontinuous galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain the coupling formula enforces a discrete energy identity ensuring the stability under the usual cfl condition in the interior moreover the scheme allows to use a smaller time step in the interior domain yielding to quasi optimal discretization parameters for both methods the aliasing phenomena introduced by the local time stepping are reduced by a post processing by averaging in time obtaining a stable and second order consistent in time coupling algorithm we compute the numerical rate of convergence of the method for an academic problem the numerical results show the feasibility of the whole discretization procedure

a note on the stable decomposition of skew symmetric matrices computationally stable decompositions for skew symmetric matrices which take advantage of the skew symmetry in order to halve the work and storage are presented for solving linear systems of equations

orthosymmetric block reflectors we develop a general theory of reflectors and block reflectors in a class of non euclidean scalar product spaces generated by orthosymmetric scalar product matrices j these j reflectors are generalizations of ordinary householder transformations and we show that they can always be expressed in a householder like representation reflection and mapping properties of j reflectors are completely described block j reflectors can be used for block annihilation in qr like factorizations where q is j unitary and r is block upper triangular

convex analysis and global optimization of joint actuator location and control problems it is shown that the optimal value of the continuous time linear quadratic problem regarded as a function of the system model and index parameters exhibits properties convexity concavity and monoticity specially suitable for optimization purposes based on this fact a procedure for the global solution determination of eventually nonconvex problems involving the above mentioned function is proposed such problems embody some known designs such as filtering under noise uncertainty or precision constraints and optimal actuator sensor location the last problem is deeply analyzed and two practical applications namely satellite attitude control and large flexible system control are included

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

asymptotic optimality of the product limit estimator the product limit estimator due to kaplan and meier 1958 is well known to be the nonparametric maximum likelihood estimator of a distribution function based on censored data it is shown here that the product limit estimator is an asymptotically optimal estimator in two senses in the sense of a hajek beran type representation theorem for regular estimators and in an asymptotic minimax sense similar to the classical result for the uncensored case due to dvoretzky kiefer and wolfowitz 1956 the proofs rely on the methods of beran 1977 and millar 1979

a galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions we consider retarded boundary integral formulations of the three dimensional wave equation in unbounded domains our goal is to apply a galerkin method in space and time in order to solve these problems numerically in this approach the computation of the system matrix entries is the major bottleneck we will propose new types of finite dimensional spaces for the time discretization they allow variable time stepping variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially the reason is that the basis functions of these spaces are globally smooth and compactly supported in order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere in this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments

a class of hamiltonian symplectic methods for solving the algebraic riccati equation abstract the algebraic riccati equation can be solved by finding a certain invariant subspace of a related hamiltonian matrix g ammar and v mehrmann devised a method for calculating this invariant subspace that exploits the hamiltonian structure of the matrix by using unitary symplectic similarity transformations this paper discusses a class of methods that find the invariant subspace by using symplectic similarity transformations that are not necessarily unitary the method of ammar and mehrmann is a special case

a stable boundary element method for modeling transient acoustic radiation transient acoustic radiation from a closed axisymmetric three dimensional object is modeled using the time domain boundary element method the widely reported instability problems are overcome by reformulating the integral equation to obtain a burton and miller type equation in the time domain the stability of such an approach is mathematically justified and supported by subsequent numerical results the hypersingular integrals which arise are evaluated using a method valid for any surface discretization numerical results for the radiation of a spherical wave are presented and compared with an exact solution the accuracy and stability of the results are verified for several geometrically different radiating objects

the moser veselov equation abstract we study the orthogonal solutions of the matrix equation xj jx t m where j is symmetric positive definite and m is skew symmetric this equation arises in the discrete version of the dynamics of a rigid body investigated by moser and veselov commun math phys 139 1991 217 we show connections between orthogonal solutions of this equation and solutions of a certain algebraic riccati equation this will bring out the symplectic geometry of the moser veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain hamiltonian matrix necessary and sufficient conditions for the existence of orthogonal solutions and methods to compute them are presented our method is contrasted with the moser veselov approach commun math phys 139 1991 217 we also exhibit explicit solutions of a particular case of the moser veselov equation which appears associated with the continuous version of the dynamics of a rigid body

partial skew orthogonal polynomials and related integrable lattices with pfaffian tau functions skew orthogonal polynomials sops arise in the study of the n point distribution function for orthogonal and symplectic random matrix ensembles motivated by the average of characteristic polynomials of the bures random matrix ensemble studied in forrester and kieburg commun math phys 342 1 151 187 2016 we propose the concept of partial skew orthogonal polynomials psops as a modification of the sops and then the psops with a variety of special skew symmetric kernels and weight functions are addressed by considering appropriate deformations of the weight functions we derive nine integrable lattices in different dimensions as a consequence the tau functions for these systems are shown to be expressed in terms of pfaffians and the wave vectors psops in fact the tau functions also admit the multiple integral representations among these integrable lattices some of them are known while the others are novel to the best of our knowledge in particular one integrable lattice is related to the partition function of the bures ensemble besides we derive a discrete integrable lattice which can be used to compute certain vector pade approximants this yields the first example regarding the connection between integrable lattices and generalised inverse vector valued pade approximants about which hietarinta joshi and nijhoff pointed out that this field remains largely to be explored in the recent monograph hietarinta et al in discrete systems and integrability vol 54 cambridge university press cambridge 2016 section 4 4

stability properties of the time domain electric field integral equation using a separable approximation for the convolution with the retarded potential the state of art of time domain integral equation tdie solvers has grown by leaps and bounds over the past decade during this time advances have been made in i the development of accelerators that can be retrofitted with these solvers and ii understanding the stability properties of the electric field integral equation as is well known time domain electric field integral equation solvers have been notoriously difficult to stabilize research into methods for understanding and prescribing remedies have been on the uptick the most recent of these efforts are i lubich quadrature and ii exact integration in this paper we re examine the solution to this equation using i the undifferentiated form of the time domain electric field integral equation tdefie and ii a separable approximation to the spatio temporal convolution the proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable as several numerical results will demonstrate the proposed scheme yields stable results for long simulation times and a variety of targets both of which have proven extremely challenging in the past

a galerkin bem for transient acoustic scattering by an absorbing obstacle this paper deals with the numerical resolution of the so called time domain boundary integral equations the scattering problem by an absorbing obstacle serves as the model for this discussion a new system of retarded potential boundary integral equations rpbie is set up to solve the problem using an energy identity we are able to prove the unconditional stability for standard conforming galerkin approximation of the rpbie the actual space time boundary elements are described in some detail and their implementation has confirmed that stability in a wide range of situations and for large run times never reached before

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

some numerical considerations and newton s method revisited for solving algebraic riccati equations analyzes some of the numerical aspects of solving the algebraic riccati equation are this analysis applies to both the symmetric and unsymmetric cases the author reconsiders the numerically relevant problems of balancing the are and the conditioning properties of the are and shows how these can be exploited by a solution algorithm he proposes an estimator for the condition number of the sylvester equation ax xb c based on iterative refinement also he interprets newton s method as a sequence of similarity transformations on the underlying system matrix this closes the gap between so called global and iterative methods for solving the are and also suggests an altogether revised implementation of newton s method one of the advantages of this revised implementation is that in the case where newton s method converges to a solution different from the desired solution enough information emerges to allow a switch to the desired solution the author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

transient elastic wave analysis of 3 d large scale cavities by fast multipole bem using implicit runge kutta convolution quadrature abstract existing boundary element methods bems have been proven to be efficient numerical techniques particularly in modeling wave propagation problems but still remain limitations in solving large scale problems the main disadvantages may be caused by several problems including high cost and large amount of computer memory for the computation a time domain bem in which a collocation method is used for the time discretization also shows numerical instability for small cfl courant friedrichs lewy number in this work we propose a new three dimensional 3 d approach for solving transient elastic wave propagation problems of large scale spherical and spheroidal cavities by the convolution quadrature bem accelerated by fast multipole method fmm using an implicit runge kutta scheme in abbreviation we name it as irk based cq fmbem two special techniques pertaining to the fmm including the scaling of modified spherical bessel functions and the truncation method are used to enhance the efficiency and stability of our proposed method this approach is found to be particularly suitable for 3 d large scale problems as its high accuracy stability of time marching process and computational efficiency these properties are subsequently illustrated through numerical examples dealing with large scale wave problems of 3 d single spherical and spheroidal cavities and multiple equally and irregularly spherical cavities the accuracy of the present formulation is verified by comparing the obtained results with available reference solutions in the literature some numerical aspects of time increments sizes and topological shapes of cavities and their influences on the deformations and the waveforms are investigated also detailed investigation of the computational efficiency of our proposed method such as cpu time required memory etc is presented all the implementation tasks are carried out using the tokyo tech supercomputer which is called tsubame 2 5 42

fast convolution quadrature for the wave equation in three dimensions this work addresses the numerical solution of time domain boundary integral equations arising from acoustic and electromagnetic scattering in three dimensions the semidiscretization of the time domain boundary integral equations by runge kutta convolution quadrature leads to a lower triangular toeplitz system of size n this system can be solved recursively in an almost linear time o n log 2 n but requires the construction of o n dense spatial discretizations of the single layer boundary operator for the helmholtz equation this work introduces an improvement of this algorithm that allows to solve the scattering problem in an almost linear time the new approach is based on two main ingredients the near field reuse and the application of data sparse techniques exponential decay of runge kutta convolution weights w n h d outside of a neighborhood of d n h where h is a time step allows to avoid constructing the near field i e singular and near singular integrals for most of the discretizations of the single layer boundary operators near field reuse the far field of these matrices is compressed with the help of data sparse techniques namely h matrices and the high frequency fast multipole method numerical experiments indicate the efficiency of the proposed approach compared to the conventional runge kutta convolution quadrature algorithm

toda hierarchy with indefinite metric abstract we consider a generalization of the full symmetric toda hierarchy where the matrix l of the lax pair is given by l ls with a full symmetric matrix l and a nondegenerate diagonal matrix s the key feature of the hierarchy is that the inverse scattering data are given by a class of noncompact groups of matrices including o p q we give an explicit formula for the solution to the initial value problem of this hierarchy by generalizing the orthogonalization procedure of szego the formula also provides an explicit solution of the hr flows introduced by watkins based upon the explicit formula we also study the behaviors of the solutions generically there are two types of solutions having either sorting property or blowing up to infinity in finite time the function structure for the tridiagonal hierarchy is also studied

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

an sdr algorithm for the solution of the generalized algebraic riccati equation an algorithm is presented for solving certain classes of the algebraic riccati equation by using the mdr transformation and extending the symplectic qr like or sr method of a bunse gerstner linear alg applic vol 83 p 49 78 1986 and v mehrmann ph d diss univ bielefeld w germany to the generalized eigenvalue problem the riccati equations arise from linear systems of the form edx dt ax bu where e is nonsingular but for numerical reasons e sup 1 a and e sup 1 b cannot be computed reliably

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

symplectic householder transformations for a qr like decomposition a geometric and algebraic approaches the aim of this paper is to show how geometric and algebraic approaches lead us to a new symplectic elementary transformations the 2 d symplectic householder transformations their features are studied in details their interesting properties allow us to construct a new algorithm for computing a sr factorization this algorithm is based only on these 2 d symplectic householder transformations its new features are highlighted the study shows that in the symplectic case the new algorithm is the corresponding one to the classical qr factorization algorithm via the householder transformations some numerical experiments are given

a symmetric structure preserving γqr algorithm for linear response eigenvalue problems abstract in this paper we present an efficient qr algorithm for solving the linear response eigenvalue problem h x x where h is symmetric with respect to 0 diag i n i n based on newly introduced orthogonal transformations the qr algorithm preserves the symmetric structure of h throughout the whole process and thus guarantees the computed eigenvalues to appear pairwise as they should with the help of a newly established implicit orthogonality theorem we incorporate the implicit multi shift technique to accelerate the convergence of the qr algorithm numerical experiments are given to show the effectiveness of the algorithm

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

the shift inverted j lanczos algorithm for the numerical solutions of large sparse algebraic riccati equations abstract the goal of solving an algebraic riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left half plane the purpose of this paper is to propose a structure preserving lanczos type algorithm incorporated with shift and invert techniques named shift inverted j lanczos algorithm for computing the stable invariant subspace for large sparse hamiltonian matrices the algorithm is based on the j tridiagonalization procedure of a hamiltonian matrix using symplectic similarity transformations we give a detailed analysis on the convergence behavior of the j lanczos algorithm and present error bound analysis and paige type theorem numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high speed vehicles are reported

adaptive time discretization for retarded potentials in this paper we will present advanced discretization methods for solving retarded potential integral equations we employ a c infty c partition of unity method in time and a conventional boundary element method for the spatial discretization one essential point for the algorithmic realization is the development of an efficient method for approximation the elements of the arising system matrix we present here an approach which is based on quadrature for non analytic c infty c functions in combination with certain chebyshev expansions furthermore we introduce an a posteriori error estimator for the time discretization which is employed also as an error indicator for adaptive refinement numerical experiments show the fast convergence of the proposed quadrature method and the efficiency of the adaptive solution process

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

a galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface the transient acoustic scattered wave from a hard surface is computed using a space time integral equation a variational formulation of this integral equation is obtained from an energy identity leading to a galerkin type numerical procedure by choosing an appropriate basis for the discretized functional space this scheme is of the marching in time type with matrix coefficients this technique is applied to a sphere a box and dihedron targets