schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

realizing rotation vectors for torus homeomorphisms we consider the rotation set p f for a lift f of a homeomorphism f t2 t2 which is homotopic to the identity our main result is that if a vector v lies in the interior of p f and has both coordinates rational then there is a periodic point x e t2 with the property that

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

asymptotic homotopy cycles for flows and π de rham theory we define the asymptotic homotopy of trajectories of flows on closed manifolds these homotopy cycles take values in the 2 step nilpotent lie group which is associated to the fundamental group by means of malcev completion the cycles are an asymptotic limit along the orbit of the product integral of a lie algebra valued 1 form propositions 5 1 5 7 show how the formal properties of our theory parallel the properties of the asymptotic homology cycles of sol schwartzman in particular asymptotic homotopy is an invariant of topological conjugacy and in certain cases of topological equivalence we compute the asymptotic homotopy of those measure preserving flows on heisenberg manifolds which lift from the torus t 2 theorem 8 1 and then show how this invariant distinguishes up to topological equivalence certain of these flows which are indistinguishable homologically theorem 9 1

standard and goal oriented adaptive mesh refinement applied to radiation transport on 2d unstructured triangular meshes standard and goal oriented adaptive mesh refinement amr techniques are presented for the linear boltzmann transport equation a posteriori error estimates are employed to drive the amr process and are based on angular moment information rather than on directional information leading to direction independent adapted meshes an error estimate based on a two mesh approach and a jump based error indicator are compared for various test problems in addition to the standard amr approach where the global error in the solution is diminished a goal oriented amr procedure is devised and aims at reducing the error in user specified quantities of interest the quantities of interest are functionals of the solution and may include for instance point wise flux values or average reaction rates in a subdomain a high order up to order 4 discontinuous galerkin technique with standard upwinding is employed for the spatial discretization the discrete ordinates method is used to treat the angular variable

minimising the error in eigenvalue calculations involving the boltzmann transport equation using goal based adaptivity on unstructured meshes this article presents a method for goal based anisotropic adaptive methods for the finite element method applied to the boltzmann transport equation the neutron multiplication factor k e f f is used as the goal of the adaptive procedure the anisotropic adaptive algorithm requires error measures for k e f f with directional dependence general error estimators are derived for any given functional of the flux and applied to k e f f to acquire the driving force for the adaptive procedure the error estimators require the solution of an appropriately formed dual equation forward and dual error indicators are calculated by weighting the hessian of each solution with the dual and forward residual respectively the hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence the two indicators are combined to form a single error metric that is used to adapt the finite element mesh the residual is approximated using a novel technique arising from the sub grid scale finite element discretisation two adaptive routes are demonstrated i a single mesh is used to solve all energy groups and ii a different mesh is used to solve each energy group the second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh the k e f f goal based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

an adaptive hanging node discontinuous isogeometric analysis method for the first order form of the neutron transport equation with discrete ordinate sn angular discretisation abstract in this paper a discontinuous hanging node isogeometric analysis iga method is developed and applied to the first order form of the neutron transport equation with a discrete ordinate s n angular discretisation in two dimensional space the complexities involved in upwinding across curved element boundaries that contain hanging nodes have been addressed to ensure that the scheme remains conservative a robust algorithm for cycle breaking has also been introduced in order to develop a unique sweep ordering of the elements for each discrete ordinates direction the convergence rate of the scheme has been verified using the method of manufactured solutions mms with a smooth solution heuristic error indicators have been used to drive an adaptive mesh refinement amr algorithm to take advantage of the hanging node discretisation the effectiveness of this method is demonstrated for three test cases the first is a homogeneous square in a vacuum with varying mean free path and a prescribed extraneous unit source the second test case is a radiation shielding problem and the third is a 3 3 supercell featuring a burnable absorber in the final test case comparisons are made to the discontinuous galerkin finite element method dgfem using both straight sided and curved quadratic finite elements

an adaptive mesh refinement algorithm for the radiative transport equation the discrete ordinates form of the radiative transport equation rte is spatially discretized and solved using an adaptive mesh refinement amr algorithm this technique permits local grid refinement to minimize spatial discretization error of the rte an error estimator is applied to define regions for local grid refinement overlapping refined grids are recursively placed in these regions and the rte is then solved over the entire domain the procedure continues until the spatial discretization error has been reduced to a sufficient level the following aspects of the algorithm are discussed error estimation grid generation communication between refined levels and solution sequencing this initial formulation employs the step scheme and is valid for absorbing and isotropically scattering media in two dimensional enclosures the utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm for two simple benchmark problems the amr algorithm maintains the convergence characteristics of the standard single grid algorithm but it does not provide any efficiency gains due to a lack of disparate spatial scales in a third more localized problem however the amr algorithm demonstrates significant memory and cpu time reductions

goal based angular adaptivity applied to a wavelet based discretisation of the neutral particle transport equation a method for applying goal based adaptive methods to the angular resolution of the neutral particle transport equation is presented the methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution the angular resolution is adapted across both the spatial and energy dimensions the spatial domain is discretised using an inner element sub grid scale finite element method the goal based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution the goal based error estimators require the solution of an adjoint system to determine the importance to the specified functional the error estimators and the novel methods to calculate them are described several examples are presented to demonstrate the effectiveness of the methods it is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error the novelty of the work is the use of goal based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation wavelet angular discretisation used to solve transport equation adaptive method developed for the wavelet discretisation anisotropic angular resolution demonstrated through the adaptive method adaptive method provides improvements in computational efficiency

goal oriented hp adaptivity for elliptic problems we propose and test a fully automatic goal oriented hp adaptive strategy for elliptic problems the method combines two techniques the standard goal oriented adaptivity based on a simultaneous solution of a dual problem and a recently proposed hp strategy based on minimizing the projection based interpolation error of a reference solution the proposed strategy is illustrated with two numerical examples laplace equation in l shape domain and an axisymmetric maxwell problem involving radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium

topological equivalence of flows on homogeneous spaces and divergence of one parameter subgroups of lie groups soient et des treillis et et des sous groupes a un parametre des groupes de lie connexes g et g si une des conditions a b ou c est vraie alors le theoreme a etablit que si les flots induits sur les espaces homogenes g et g sont topologiquement equivalents alors ils sont topologiquement equivalents par une application affine

adaptive discontinuous galerkin finite element methods for nonlinear hyperbolic conservation laws we consider the a posteriori error analysis and adaptive mesh design for discontinuous galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws in particular we discuss the question of error estimation for general target functionals of the solution typical examples include the outflow flux local average and pointwise value as well as the lift and drag coefficients of a body immersed in an inviscid fluid by employing a duality argument we derive so called weighted or type i a posteriori error bounds these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated based on the resulting approximate type i bound we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest the performance of the proposed adaptive strategy and the quality of the approximate type i a posteriori error bound is illustrated by a series of numerical experiments in particular we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ type ii error indicators

convergence of a fully discrete scheme for two dimensional neutron transport we prove an error estimate for a fully discrete method for the numerical solution of a two dimensional model problem in neutron transport theory based on using the discrete ordinates method for the

an adaptive higher order method for solving the radiation transport equation on unstructured grids a new class of solution methods based on a short characteristics approach embedded in the finite element framework for the solution of the radiation transport equation is developed and tested the order of convergence and the error to runtime ratio are determined in numerical tests furthermore the efficiency of the scheme is demonstrated in the case of a challenging problem from the field of solar physics a quantitative comparison with other methods based either on the characteristics or on the finite element approach is carried out this emphasizes the efficiency and accuracy of our method two different adaptation mechanisms are studied grid adaptation and local variation of the order of the scheme

fast inverse qr factorization for toeplitz matrices fast orthogonalization schemes for m times n toeplitz matrices t introduced by bojanczyk brent and de hoog bbh and chun kailath and lev ari ckl are extended to compute directly an inverse qr factorization of t using only o mn operations an inverse factorization allows for an efficient parallel implementation and the algorithm is computationally less expensive for computing a solution to the toeplitz least squares problem than previously studied inverse qr methods in addition it is shown that regularization can be incorporated into the algorithm with virtually no extra work thus it is possible to compute regularized solutions to ill conditioned toeplitz least squares problems using only o mn operations an application to ill conditioned problems occurring in signal restoration is provided illustrating the effectiveness of this method

goal based h adaptivity of the 1 d diamond difference discrete ordinate method abstract the quantity of interest qoi associated with a solution of a partial differential equation pde is not in general the solution itself but a functional of the solution dual weighted residual dwr error estimators are one way of providing an estimate of the error in the qoi resulting from the discretisation of the pde this paper aims to provide an estimate of the error in the qoi due to the spatial discretisation where the discretisation scheme being used is the diamond difference dd method in space and discrete ordinate s n method in angle the qoi are reaction rates in detectors and the value of the eigenvalue k eff for 1 d fixed source and eigenvalue k eff criticality neutron transport problems respectively local values of the dwr over individual cells are used as error indicators for goal based mesh refinement which aims to give an optimal mesh for a given qoi

zero entropy non integrable geodesic flows and a non commutative rotation vector let g be a 2 step nilpotent lie algebra we say g is non integrable if for a generic pair of points p p g the isotropy algebras do not commute g p g p 0 theorem if g is a simply connected 2 step nilpotent lie group g lie g is non integrable d g is a cocompact subgroup and g is a left invariant riemannian metric then the geodesic flow of g on t d g is neither liouville nor non commutatively integrable with c first integrals the proof uses a generalization of the rotation vector pioneered by benardete and mitchell