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Publication# Retraction-based numerical methods for continuation, interpolation and time integration on manifolds

Abstract

The goal of this thesis is the development and the analysis of numerical methods for problems where the unknown is a curve on a smooth manifold. In particular, the thesis is structured around the three following problems: homotopy continuation, curve interpolation and integration of ordinary differential equations. To accommodate the manifold constraint, all the proposed methods feature as a central ingredient the concept of retractions, as extensively developed in the context of Riemannian optimization methods. A retraction can be thought as a generic device for crafting portions of manifold-constrained curves which are in general computationally cheaper to evaluate than the geodesics defining the Riemannian exponential map. Yet, the axiomatic definition of a retraction reveals to be rich enough for algorithms originally defined on Euclidean spaces to be adapted to the manifold setting using retractions while maintaining properties that are analogous to their Euclidean ancestor. We provide this type of analysis for the methods proposed in the thesis and we showcase the performance of the algorithms with experiments involving matrix manifolds, notably the fixed-rank matrix manifold.First, we consider a generalization of numerical continuation methods for their application to Riemannian optimization problems. In practice, we propose a retraction-based path-following numerical continuation algorithm for efficiently solving a sequence of Riemannian optimization problems of which the last is the actual problem of interest. Then, we address the problem of Hermite interpolation, whereby a sequence of manifold points are interpolated by a manifold curve whose velocity is prescribed at each interpolation point. For this, we introduce a generalization of the de Casteljau algorithm where suitably chosen retraction curves replace the straight lines of the original algorithm. Lastly, we tackle numerical integration of manifold-constrained ordinary differential equations, in particular for equations evolving on low-rank matrix manifolds encountered in the field of dynamical low-rank approximation. We derive two methods defined using retractions which exhibit second-order convergence of the approximation error with respect to the time integration step.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

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