the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

strict feedforward control systems linearisability and convergent normal forms this article discusses the feedback equivalence of multi inputs feedforward control systems via smooth resp analytic feedback transformations we first address the state resp feedback linearisation problem and provide easily computable algorithms that yield explicit state resp feedback linearising coordinates for systems in strict feedforward form the application of the algorithms does not require checking the commutativity resp involutivity of the distributions associated with the system and the algorithms fail after few steps if the system is not linearisable in the latter case the algorithms are extended to provide coordinate systems bringing the system into a normal form which is a smooth resp analytic counterpart of kang s formal normal form illustrative examples for both the linearisation and convergent normal form include the vertical take off and landing aircraft the multi vehicle wireless testbed among others

asymptotic tracking of a reference state for systems with a feedforward structure we consider the problem of asymptotic tracking for a system which can be written in a feedforward form the data of a bounded reference state trajectory is assumed our solution relies on a lyapunov construction the time varying state feedbacks obtained are bounded and ensure the global uniform asymptotic stability when the signals are periodic and the local exponential stability of the reference state trajectory of the closed loop system

subdivision algorithms converge quadratically this paper is concerned with determining the exact convergence rates of subdivision algorithms for various settings of polynomial and spline curve or surface representations

semi global stabilization of discrete time systems with bounded inputs using a periodic controller we show that semi global stabilization is achieved for linear discrete time systems with bounded controls the proposed scheme is a linear periodic controller and is applicable to systems having open loop eigenvalues in the closed unit disc

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

reduction model approach for linear time varying systems with delays we study stabilization problems for time varying linear systems with constant input delays our reduction method ensures input to state stability with respect to additive uncertainties under arbitrarily long delays it applies to rapidly time varying systems and gives a lower bound on the admissible rapidness parameters we also cover slowly time varying systems including upper bounds on the allowable slowness parameters we illustrate our work using a pendulum model

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

graph homology koszul and verdier duality we show that verdier duality for certain sheaves on the moduli spaces of graphs associated to differential graded operads corresponds to the cobar duality of operads which specializes to koszul duality for koszul operads this in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and lie types in computations of the cohomology of the outer automorphism group of a free group another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs thus characterizing to which extent these spaces are different from oriented orbifolds we also provide a relation between the cohomology of the space of metric ribbon graphs known to be homotopy equivalent to the moduli space of riemann surfaces and the cohomology of a certain sheaf on the space of usual metric graphs

fast inversion of the simplicial bernstein mass matrix we consider the mass matrix arising from bernstein polynomials as finite element shape functions in particular we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the bernstein representation of orthogonal polynomials we then derive a fast algorithm for solving linear systems involving the element mass matrix after establishing these results we describe the application of bernstein techniques to the discontinuous galerkin finite element method for hyperbolic conservation laws obtaining optimal complexity algorithms finally we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run time for the function evaluation needed in dg time stepping

conversion between triangular bezier patches and rectangular bezier patches abstract in this paper two explicit conversion formulae between triangular and rectangular bezier patches are derived using the formulae one triangular bezier patch of degree n can be converted into one rectangular bezier patch of degree n n and one rectangular bezier patch of degree m n can be converted into two triangular bezier patches of degree m n besides two stable recursive algorithms corresponding to the two conversion formulae are given using the algorithms when converting triangular bezier patches to rectangular bezier patches we can computer the relations between the control points of the two types of patches for any n 2 based on the relationships for n 1 when converting rectangular bezier patches to triangular bezier patches we can computer the relations between the control points of the two types of patches for any m 2 n n and n 2 m n based on the relationships for m n 1

bounds on spectral condition numbers of matrices arising in the p version of the finite element method we estimate condition numbers of p version matrices for tensor product elements with two choices of reference element degrees of freedom in one case lagrange elements the condition numbers grow exponentially in p whereas in the other hierarchical basis functions based on tchebycheff polynomials the condition numbers grow rapidly but only algebraically in p we conjecture that regardless of the choice of basis the condition numbers grow like p 4d or faster where d is the dimension of the spatial domain

time varying state feedback stabilisation of stochastic feedforward nonlinear systems with unknown growth rate abstractwe consider the time varying state feedback stabilisation problem for a class of stochastic feedforward nonlinear systems with unknown growth rate in this paper a new lasalle type theorem for stochastic time varying systems is firstly established by using the generalized weakly positive definite function as an application to deal with serious uncertainties in the unknown growth rate a time varying approach rather than an adaptive one is adopted to design the scheme of a state feedback controller for stochastic feedforward systems based on the established lasalle type theorem it is shown that all signals of the resulting closed loop system converge to zero almost surely illustrative examples are given to verify the theoretical findings

brief finite gain stabilization of discrete time linear systems subject to actuator saturation it is shown that for neutrally stable discrete time linear systems subject to actuator saturation finite gain l p stabilization can be achieved by linear output feedback for all p 1 an explicit construction of the corresponding feedback laws is given the feedback laws constructed also result in a closed loop system that is globally asymptotically stable and in an input to state estimate

on finite gain stabilizability of linear systems subject to input saturation this paper deals with global finite gain input output stabilization of linear systems with saturated controls for neutrally stable systems it is shown that the linear feedback law suggested by the passivity approach indeed provides stability with respect to every l p norm explicit bounds on closed loop gains are obtained and they are related to the norms for the respective systems without saturation these results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple size 1 jordan blocks contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state space sense this is shown in particular for the double integrator

tracking control of a class of non linear systems with applications to cruise control of air breathing hypersonic vehicles a new tracking control method for general non linear systems is proposed a virtual controller and some command references are introduced to asymptotically stabilise the system of the tracking error dynamics then the actual controller and command references are derived by solving a system of linear algebraic equations compared with other tracking control methods in the literature the tracking controller design in this paper is simple because it needs only to solve a system of linear algebraic equations the boundedness of the tracking controller and command references is guaranteed by the solvability of the terminal value problem tvp of an ordinary differential equation for non linear systems with minimum phase properties the tvp is automatically solvable a numerical example shows that the tracking control method is still available for some systems with non minimum phase properties to enhance the robustness of the tracking controller a non linear disturbance observer ndo is introduced to esti

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

efficient visualization of high order finite elements a general method for the post processing treatment of high order finite element fields is presented the method applies to general polynomial fields including discontinuous finite element fields the technique uses error estimation and h refinement to provide an optimal visualization grid some filtering is added to the algorithm in order to focus the refinement on a visualization plane or on the computation of one single iso zero surface 2d and 3d examples are provided that illustrate the power of the technique in addition schemes and algorithms that are discussed in the paper are readily available as part of an open source project that is developed by the authors namely gmsh

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

classes on the moduli space of riemann surfaces through a noncommutative batalin vilkovisky formalism abstract using the machinery of the batalin vilkovisky formalism we construct cohomology classes on compactifications of the moduli space of riemann surfaces from the data of a contractible differential graded frobenius algebra we describe how evaluating these cohomology classes upon a well known construction producing homology classes in the moduli space can be expressed in terms of the feynman diagram expansion of some functional integral by computing these integrals for specific examples we are able to demonstrate that this construction produces families of nontrivial classes

further remarks on strict input to state stable lyapunov functions for time varying systems we study the stability properties of a class of time varying non linear systems we assume that non strict input to state stable iss lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given lyapunov functions which guarantee the existence of strict iss lyapunov functions for our systems next we provide simple direct constructions of explicit strict iss lyapunov functions for our systems by applying an integral smoothing method we illustrate our constructions using a tracking problem for a rotating rigid body

on the p norm condition number of the multivariate triangular bernstein basis we show that the p norm condition number of the s variate triangular bernstein basis for polynomials of degree n grows at most as o n s 2 n for xed s and increasing n this is essentially the same growth as has already been established

an o p3 hp version fem in two dimensions preconditioning and post processing abstract the bernstein polynomials have been known for over a century and are widely used in the spline literature computer aided geometric design and computer graphics however the realization that the bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development for instance it is known ainsworth et al 2011 that the bernstein basis can be exploited to compute all of the entries in the load vector in o p 3 operations even in the case of non linear problems on curvilinear elements for a degree p approximation moreover the element matrices can be assembled in o 1 operations per entry we show that properties of the bernstein polynomials can also be exploited to obtain o p 3 complexity procedures for all of the main components needed to implement a high order finite element code including computation of the residuals needed for an iterative solution method evaluating the action of a preconditioner for the global mass matrices and visualization and post processing of the resulting finite element approximations the construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p at a cost of o p 3 operations is one of the main contributions of the present work the preconditioner is based on an abstract additive schwarz method recently developed by the authors the preconditioner can be implemented at a cost of o p 3 operations by exploiting properties of the bernstein polynomials in particular we present an algorithm which allows one to invert the interior block of the element mass matrix in o p 3 operations numerical examples are provided to illustrate the applicability of the bernstein basis to challenging non linear reaction diffusion problems non linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers

conditioning of hierarchic p version nedelec elements on meshes of curvilinear quadrilaterals and hexahedra the conditioning of a set of hierarchic basis functions for p version edge element approximation of the space h curl is studied theoretical bounds are obtained on the location of the eigenvalues and on the growth of the condition numbers for the mass curl curl and stiffness matrices that naturally arise from galerkin approximation of maxwell s equations the theory is applicable to meshes of curvilinear quadrilaterals or hexahedra in two and three dimensions respectively including the case in which the local order of approximation is nonuniform throughout the theory is illustrated with numerical examples that show that the theoretical asymptotic bounds are sharp and are attained within the range of practical computation

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

brief strict lyapunov functions for time varying systems uniformly asymptotically stable periodic time varying systems for which is known a lyapunov function with a derivative along the trajectories non positive and negative definite in the state variable on non empty open intervals of the time are considered for these systems strict lyapunov functions are constructed

asymptotic stabilization for feedforward systems with delayed feedbacks a problem of state feedback stabilization of time varying feedforward systems with a pointwise delay in the input is solved the approach relies on a time varying change of coordinates and lyapunov krasovskii functionals the result applies for any given constant delay and provides uniformly globally asymptotically stabilizing controllers of arbitrarily small amplitude although exact model knowledge and periodicity of the dynamics are required the closed loop systems enjoy input to state stability properties with respect to additive uncertainty on the controllers the work is illustrated through a tracking problem for a model for high level formation flight of unmanned air vehicles

the solution of the two dimensional sine gordon equation using the method of lines the method of lines is used to transform the initial boundary value problem associated with the two dimensional sine gordon equation in two space variables into a second order initial value problem the finite difference methods are developed by replacing the matrix exponential term in a recurrence relation with rational approximants the resulting finite difference methods are analyzed for local truncation error stability and convergence to avoid solving the nonlinear system a predictor corrector scheme using the explicit method as predictor and the implicit as corrector is applied numerical solutions for cases involving the most known from the bibliography line and ring solitons are given

tracking trajectories of feedforward systems the objective of this paper is to solve the problem of tracking trajectories of feedforward systems a family of time varying state feedbacks that globally uniformly asymptotically and locally exponentially stabilize trajectories which are not necessarily periodic functions of time is exhibited the control design is based on the construction of a strict lyapunov function