High-order Gauss–Lobatto formulae

Currently, the method of choice for computing the (n+2)-point Gauss–Lobatto quadrature rule for any measure of integration is to first generate the Jacobi matrix of order n+2 for the measure at hand, then modify the three elements at the right lower corner of the matrix in a manner proposed in 1973 by Golub, and finally compute the eigenvalues and first components of the respective eigenvectors to produce the nodes and weights of the quadrature rule. In general, this works quite well, but when n becomes large, underflow problems cause the method to fail, at least in the software implementation provided by us in 1994. The reason is the singularity (caused by underflow) of the 2×2 system of linear equations that is used to compute the modified matrix elements. It is shown here that in the case of arbitrary Jacobi measures, these elements can be computed directly, without solving a linear system, thus allowing the method to function for essentially unrestricted values of n. In addition, it is shown that all weights of the quadrature rule can also be computed explicitly, which not only obviates the need to compute eigenvectors, but also provides better accuracy. Numerical comparisons are made to illustrate the effectiveness of this new implementation of the method.     

A stochastic Gauss–Bonnet–Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle \(E\) over compact manifold \(M\) canonically defines a metric on \(E\) together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle \(E\) and the manifold \(M\) are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

High-order generalized Gauss–Radau and Gauss–Lobatto formulae for Jacobi and Laguerre weight functions

The generation of generalized Gauss–Radau and Gauss–Lobatto quadrature formulae by methods developed by us earlier breaks down in the case of Jacobi and Laguerre measures when the order of the quadrature rules becomes very large. The reason for this is underflow resp. overflow of the respective monic orthogonal polynomials. By rescaling of the polynomials, and other corrective measures, the problem can be circumvented, and formulae can be generated of orders as high as 1,000. In memoriam Gene H. Golub.     

Integration of the Gauss–Codazzi Equations

The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same rotation coefficients. Each equivalence class is defined by a function of two variables (master function of a surface). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.     

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.     

The Gauss–Bonnet–Chern mass for graphic manifolds

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss–Bonnet–Chern mass $m_2$ for the graphic manifold with flat normal bundle.     

Hypersurfaces with Prescribed Affine Gauss–Kronecker Curvature

We extend a procedure for solving particular fourth order PDEs by splitting them into two linked second order Monge–Ampère equations. We use this for the global study of Blaschke hypersurfaces with prescribed Gauss–Kronecker curvature.     

Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

Legendre–Gauss–Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive the main properties of the associated dyadic grids needed for preconditioning the systems arising from \(hp\)- or even spectral (conforming or Discontinuous Galerkin type) discretizations for second order elliptic problems in a way that is fully robust with respect to varying polynomial degrees. To establish these properties requires revisiting some refined properties of LGL grids and their relatives.     

Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations

Using the covering theory approach (zero-curvature representations with the gauge group SL), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Chebyshev and geodesic coordinates. For each choice, four integrable systems are obtained.     

A correction term for Gauss–Gegenbauer quadrature

By means of Chebyshev polynomials of the first kind a correction term for Gauss–Gegenbauer quadrature is derived.     

Performance Optimization for the Parallel Gauss–Seidel Smoother

Simulation, e.g., in the field of computational fluid dynamics, accounts for a major part of the computing time on highperformance systems. Many simulation packages still rely on Gauss–Seidel iteration, either as the main linear solver or as a smoother for multigrid schemes. Straight-forward implementations of this solver have efficiency problems on today's most common high-performance computers, i.e., multiprocessor clusters with pronounced memory hierarchies. In this work we present two simple techniques for improving the performance of the parallel Gauss–Seidel method for the 3D Poisson equation by optimizing cache usage as well as reducing the number of communication steps. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Legendre–Gauss collocation methods for ordinary differential equations

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.      

A Stereological Version of the Gauss–Bonnet Formula

We give a stereological version of the Gauss–Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in ³, by looking at contacts with a 'sweeping' plane.     

An iterative method for Bayesian Gauss–Markov image restoration

The purpose of this paper is to introduce a new method for the restoration of images that have been degraded by a blur and an additive white Gaussian noise. The model adopted here is assumed to be Bayesian Gauss–Markov linear model. By exploiting the structure of the blurring matrix and by using Kronecker product approximations, the image restoration problem is formulated as matrix equations which will be solved iteratively by projection methods onto Krylov subspaces. We give some theoretical and experimental results with applications to image restoration.     

Fast and reliable high-accuracy computation of Gauss–Jacobi quadrature

Numerical Algorithms - There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of...     

Statistical approximation by double complex Gauss–Weierstrass integral operators

In this paper, we introduce the complex Gauss–Weierstrass integral operators defined on a space of analytic functions in two variables on the Cartesian product of two unit disks. Then, we study the geometric properties and statistical approximation process of our operators.     

A Remark on Kleiman–Piene's Question for Gauss Maps

We discuss some algebraic properties of the monoid generated by (left) translations in left distributive structures.This furnishes methods for enriching the original structure with a compatible associative product.     

Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

Abstract

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ε > 0. As ε vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ε ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L ²(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.