Tangente (géométrie) | EPFL Graph Search

Tangente vient du latin tangere, toucher : en géométrie, la tangente à une courbe en un de ses points est une droite qui « touche » la courbe au plus près au voisinage de ce point. La courbe et sa tangente forment alors un angle nul en ce point. La notion de tangente permet d'effectuer des approximations : pour la résolution de certains problèmes qui demandent de connaître le comportement de la courbe au voisinage d'un point, on peut assimiler celle-ci à sa tangente. Ceci explique la parenté entre la notion de tangente et le calcul différentiel. Se contenter comme on le fait parfois de définir la tangente comme une droite qui « touche la courbe sans la traverser » serait incorrect, puisque

rien n'empêche la courbe de retraverser une de ses tangentes un peu plus loin (le concept de tangente au point M ne décrit bien la situation que dans un petit voisinage du point M).
il y a des situations exceptionnelles où la tangente en M traverse la courbe justement au point M. On dit alors

An Accelerated First-Order Method for Non-convex Optimization on Manifolds

Nicolas Boumal, Christopher Arnold Criscitiello

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Dynamical low rank approximation for uncertainty quantification of time-dependent problems

Eva Vidlicková

The quantification of uncertainties can be particularly challenging for problems requiring long-time integration as the structure of the random solution might considerably change over time. In this respect, dynamical low-rank approximation (DLRA) is very appealing. It can be seen as a reduced basis method, thus solvable at a relatively low computational cost, in which the solution is expanded as a linear combination of a few deterministic functions with random coefficients. The distinctive feature of the DLRA is that both the deterministic functions and random coefficients are computed on the fly and are free to evolve in time, thus adjusting at each time to the current structure of the random solution. This is achieved by suitably projecting the dynamics onto the tangent space of a manifold consisting of all random functions with a fixed rank. In this thesis, we aim at further analysing and applying the DLR methods to time-dependent problems.Our first work considers the DLRA of random parabolic equations and proposes a class of fully discrete numerical schemes.Similarly to the continuous DLRA, our schemes are shown to satisfy a discrete variational formulation.By exploiting this property, we establish the stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a ``parabolic'' type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions. The discrete variational formulation is further applied in our second work, which derives a-priori and a-posteriori error estimates for the discrete DLRA of a random parabolic equation obtained by the three newly-proposed schemes. Under the assumption that the right-hand side of the dynamical system lies in the tangent space up to a small remainder, we show that the solution converges with standard convergence rates w.r.t. the time, spatial, and stochastic discretization parameters, with constants independent of singular values.We follow by presenting a residual-based a-posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The a-posteriori error estimate consists of four parts: the finite element method error, the time discretization error, the stochastic collocation error, and the rank truncation error. These estimators are then used to drive an adaptive choice of FE mesh, collocation points, time steps, and time-varying rank.The last part of the thesis examines the idea of applying the DLR method in data assimilation problems, in particular the filtering problem. We propose two new filtering algorithms. They both rely on complementing the DLRA with a Gaussian component. More precisely, the DLR portion captures the non-Gaussian features in an evolving low-dimensional subspace through interacting particles, whereas each particle carries a Gaussian distribution on the whole ambient space. We study the effectiveness of these algorithms on a filtering problem for the Lorenz-96 system.

EPFL2022

Arm-hand motion-force coordination for physical interactions with non-flat surfaces using dynamical systems: Toward compliant robotic massage

Aude Billard, Karl Gustav Henriks, Mahdi Khoramshahi, Seyed Sina Mirrazavi Salehian, Aileen Charlotte Naef

Many manipulation tasks require coordinated motions for arm and fingers. Complexity increases when the task requires to control for the force at contact against a non-flat surface; This becomes even more challenging when this contact is done on a human. All these challenges are regrouped when one, for instance, massages a human limb. When massaging, the robotic arm is required to continuously adapt its orientation and distance to the limb while the robot fingers exert desired patterns of forces and motion on the skin surface. To address these challenges, we adopt a Dynamical System (DS) approach that offers a unified motion-force control approach and enables to easily coordinate multiple degrees of freedom. As each human limb may slightly differ, we learn a model of the surface using support vector regression (SVR) which enable us to obtain a distance-to-surface mapping. The gradient of this mapping, along with the DS, generates the desired motions for the interaction with the surface. A DS-based impedance control for the robotic fingers allows to control separately for force along the normal direction of the surface while moving in the tangential plane. We validate our approach using the KUKA IIWA robotic arm and Allegro robotic hand for massaging a mannequin arm covered with a skin-like material.We show that our approach allows for 1) reactive motion planning to reach for an unknown surface, 2) following desired motion patterns on the surface, and 3) exerting desired interaction forces profiles. Our results show the effectiveness of our approach; especially the robustness toward uncertainties for shape and the given location of the surface.

2020

Dynamical low rank approximation for uncertainty quantification of time-dependent problems

Eva Vidlicková

EPFL2022