The geometry of algorithms using hierarchical tensors

In this paper, the differential geometry of the novel hierarchical Tucker format for tensors is derived. The set HT,k of tensors with fixed tree T and hierarchical rank k is shown to be a smooth quotient manifold, namely the set of orbits of a Lie group action corresponding to the non-unique basis representation of these hierarchical tensors. Explicit characterizations of the quotient manifold, its tangent space and the tangent space of HT,k are derived, suitable for high-dimensional problems. The usefulness of a complete geometric description is demonstrated by two typical applications. First, new convergence results for the nonlinear Gauss-Seidel method on HT, k are given. Notably and in contrast to earlier works on this subject, the task of minimizing the Rayleigh quotient is also addressed. Second, evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of HT,k. In addition, a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions. © 2013 Elsevier Inc.

Some applications of symmetries in differential geometry and dynamical systems

Oana Mihaela Dragulete

This thesis deals with applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of symmetries of cosphere bundles, the conservation properties of contact systems and their reduction. We generalise the results of [15] to the singular case making a complete topological and geometrical analysis of the reduced space. Applying the general theory of contact reduction developed by Lerman and Willett in [33] and [57], one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Based on the cotangent bundle reduction theorems, both in the regular and singular case, as well as regular cosphere bundle reduction, one expects additional bundle-like structure for the contact strata. The cosphere bundle projection to the base manifold descends to a continuous surjective map from the reduced space at zero to the orbit quotient of the configuration space, but it fails to be a morphism of stratified spaces if we endow the reduced space with its contact stratification and the base space with the customary orbit type stratification defined by the Lie group action. In this chapter we introduce a new stratification of the contact quotient at zero, called the C-L stratification (standing for the coisotropic or Legendrian nature of its pieces) which solves the above mentioned two problems. Its main features are the following. First, it is compatible with the contact stratification of the quotient and the orbit type stratification of the configuration orbit space. It is also finer than the contact stratification. Second, the natural projection of the C-L stratified quotient space to its base space, stratified by orbit types, is a morphism of stratified spaces. Third, each C-L stratum is a bundle over an orbit type stratum of the base and it can be seen as a union of C-L pieces, one of them being open and dense in its corresponding contact stratum and contactomorphic to a cosphere bundle. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. We also describe the relation between contact vector fields and the time dependent Hamilton-Jacobi equation. The reduction of contact systems and time dependent Hamiltonians is mentioned. In the second chapter we study geometric properties of Sasakian and Kähler quotients. We construct a reduction procedure for symplectic and Kähler manifolds using the ray preimages of the momentum map. More precisely, instead of taking as in point reduction the preimage of a momentum value μ, we take the preimage of ℝ+μ, the positive ray of μ. We have two reasons to develop this construction. One is geometric: non zero Kähler point reduction is not always well defined. The problem is that the complex structure may not leave invariant the horizontal distribution of the Riemannian submersion πμ : J-1(μ) → Mμ. The solution proposed in the literature is correct only in the case of totally isotropic momentum (i.e. Gμ = G). The other reason is that it provides invariant submanifolds for conformal Hamiltonian systems. They are usually non-autonomous mechanical systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map. We extend the class of conformal Hamiltonian systems already studied and complete the existing Lie Poisson reduction with the general ray one. As examples of symplectic (Kähler) and contact (Sasakian) ray reductions we treat the case of cotangent and cosphere bundles and we show that they are universal for ray reductions. Using techniques of A. Futaki, we prove that, under appropriate hypothesis, ray quotients of Kähler-Einstein or Sasaki-Einstein manifolds remain Kähler or Sasaki-Einstein. Note that it suffices to prove the Kähler case and the compatibility of ray reduction with the Boothby-Wang fibration. In the last chapter, we prove a stratification theorem for proper groupoids. First we find an equivalent way of describing the same result for a proper Lie group action, way which uses the theory of foliations and can be adapted to the language of Lie groupoids. We treat separately the case of free and proper groupoids. The orbit foliation of a proper Lie groupoid is a singular Riemannian foliation and we show this explicitly.

EPFL2007

Differential geometry: a natural tool for describing symmetry operations

Gervais Chapuis, Philippe Kocian, Kurt Schenk

Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the manifold. Manifold theory proves particularly powerful as a uniﬁed formalism describing symmetry operations of conventional as well as modulated crystals without requiring a higher-dimensional space. We show, in particular, that a modulated structure recovers a three-dimensional periodicity in any tangent space and that its point group consists of linear applications.

2009

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects.

EPFL2007

Some applications of symmetries in differential geometry and dynamical systems

Oana Mihaela Dragulete

EPFL2007

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

EPFL2007