The De Casteljau Algorithm on Lie Groups and Spheres

LMS Journal of Computation and Mathematics, 2005

This paper presents a new geometric algorithm to construct a Ck-smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the De Casteljau procedure, this new algorithm is implemented in only three steps, independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function, which is defined as soon as the degree of smoothness is fixed.

Journal of Scientific Computing, 2013

The cubed-sphere grid is a spherical grid made of six quasi-cartesian square-like patches. It was originally introduced in . We extend to this grid the design of high-order nite-dierence compact operators . The present work is limitated to the design of a fourth-order accurate spherical gradient. The treatment at the interface of the six patches relies on a specic interpolation system which is based on using great circles in an essential way. The main interest of the approach is a fully symmetric treatment of the sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, including the cosine-bell test-case of Williamson et al. and a deformational test-case reported in .

2001

Symmetric spaces are well known in differential geometry from the study of spaces of constant curvature. The tangent space of a symmetric space forms a Lie triple system. Recently these objects have received attention in the numerical analysis community. A remarkable number of different algorithms can be understood and analyzed using the concepts of symmetric spaces and this theory unifies a range of different topics in numerical analysis, such as polar type matrix decompositions, splitting methods for computation of the matrix exponential, composition of self adjoint numerical integrators and time symmetric dynamical systems. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we are presenting new results related to time reversal symmetries, self adjoint numerical schemes and Yoshida type composition techniques.

Differential Geometry and Lie Groups A Computational Perspective, 2020

To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard and Jane. discuss three results, one of which being the Hadamard and Cartan theorem about complete manifolds of non-positive curvature. The goal of Chapter 18 is to understand the behavior of isometries and local isometries, in particular their action on geodesics. We also intoduce Riemannian covering maps and Riemannian submersions. If π : M → B is a submersion between two Riemannian manifolds, then for every b ∈ B and every p ∈ π -1 (b), the tangent space T p M to M at p splits into two orthogonal components, its vertical component V p = Ker dπ p , and its horizontal component H p (the orthogonal complement of V p ). If the map dπ p is an isometry between H p and T b B, then most of the differential geometry of B can be studied by lifting B to M , and then projecting down to B again. We also introduce Killing vector fields, which play a technical role in the study of reductive homogeneous spaces. In Chapter 19, we return to Lie groups. Not every Lie group is a matrix group, so in order to study general Lie groups it is necessary to introduce left-invariant (and rightinvariant) vector fields on Lie groups. It turns out that the space of left-invariant vector fields is isomorphic to the tangent space g = T I G to G at the identity, which is a Lie algebra. By considering integral curves of left-invariant vector fields, we define the generalization of the exponential map exp : g → G to an arbitrary Lie group. The notion of immersed Lie subgroup is introduced, and the correspondence between Lie groups and Lie algebra is explored. We also consider the special classes of semidirect products of Lie algebras and Lie groups, the universal covering of a Lie group, and the Lie algebra of Killing vector fields on a Riemannian manifold. Chapter 20 deals with two topics: 1. A formula for the derivative of the exponential map for a general Lie group (not necessarily a matrix group). 2. A formula for the Taylor expansion of µ(X, Y ) = log(exp(X) exp(Y )) near the origin. The second problem is solved by a formula known as the Campbell-Baker-Hausdorff formula. An explicit formula was derived by Dynkin (1947), and we present this formula. Chapter 21 is devoted to the study of metrics, connections, geodesics, and curvature, on Lie groups. Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. As a consequence, it is possible to find explicit formulae for the Levi-Civita connection and the various curvatures, especially in the case of metrics which are both left and right-invariant. In Section 21.2 we give four characterizations of bi-invariant metrics. The first one refines the criterion of the existence of a left-invariant metric and states that every bi-invariant metric on a Lie group G arises from some Ad-invariant inner product on the Lie algebra g. In Section 21.3 we show that if G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms Recall that a real symmetric matrix is called positive (or positive semidefinite) if its eigenvalues are all positive or null, and positive definite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n × n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive definite matrices by SPD(n). The next proposition shows that every symmetric positive definite matrix A is of the form e B for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) definite matrices is not a multiplicative group, so this result is of a different flavor as Theorem 2.6. Proposition 2.8. For every symmetric matrix B, the matrix e B is symmetric positive definite. For every symmetric positive definite matrix A, there is a unique symmetric matrix B such that A = e B . Proof. We showed earlier that e B = e B . If B is a symmetric matrix, then since B = B, we get e B = e B = e B , and e B is also symmetric. Since the eigenvalues λ 1 , . . . , λ n of the symmetric matrix B are real and the eigenvalues of e B are e λ 1 , . . . , e λn , and since e λ > 0 if λ ∈ R, e B is positive definite. To show the surjectivity of the exponential map, note that if A is symmetric positive definite, then by Theorem 12.3 from Chapter 12 of Gallier [50], there is an orthogonal matrix P such that A = P D P , where D is a diagonal matrix CHAPTER 2. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS

Journal of Computational and Applied Mathematics, 2006

This paper presents a simple geometric algorithm to generate splines of arbitrary degree of smoothness in Euclidean spaces. Unlike other existing methods, this simple geometric algorithm does not require a recursive procedure and, consequently, introduces a significant reduction in calculation time. The algorithm is then extended to other complete Riemannian manifolds, namely to matrix Lie groups and spheres.

2010

This essay shows the necessity and benefits of exploiting the geometric structure of the underlying manifold in which the solution of a differential equation evolves. I give an overview and first insight into the concept of solving differential equations on Lie groups and state relevant results. In the first section, I justified the need for new methods. After an introduction to differential geometry, that is needed for understanding the principles of Lie group methods, We discuss some methods that are typical for the Lie group concept, such as Crouch-Grossman, Runge-Kutta-Munthe-Kaas, Magnus and Fer expansions and -in less detail -a few others. We discuss properties of Magnus and Fer expansions, their order and convergence. After these parts, I return to general properties of Lie group methods and state results from backward error analysis. The next sections are devoted to the implementation of the discussed methods, in particular we see that the occuring integrations can be relatively cheaply computed and I present ideas about computing the exponential in an appropriate way for Lie group methods. The last section provides a brief summary about further Lie group methods.

Numerische Mathematik, 2009

Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres.

Acta Numerica 2000, 2000

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

2015

The main objective of this paper is to propose a new method to generate smooth interpolating curves on Stiefel manifolds. This method is obtained from a modification of the geometric Casteljau algorithm on manifolds and is based on successive quasi-geodesic interpolation. The quasi-geodesics introduced here for Stiefel manifolds have constant speed, constant covariant acceleration and constant geodesic curvature, and in some particular circumstances they are true geodesics.

Acta Applicandae Mathematicae, 2000

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by S. T. Smith is made explicit on the Grassmann manifold. Two applications -computing an invariant subspace of a matrix and the mean of subspaces-are worked out.