Session: Manifold optimization with special metrics
Chair: Max Pfeffer
Cluster: Optimization on Manifolds
Talk 1: Riemannian optimization methods for ground state computations of multicomponent Bose-Einstein condensates
Speaker: Tatjana Stykel
Abstract: In this talk, we address the computation of ground states of multicomponent Bose-Einstein condensates by solving the underlying energy minimization problem on the infinite-dimensional generalized oblique manifold. First, we discuss the existence and uniqueness of a ground state with non-negative components and its connection to the coupled Gross-Pitaevskii eigenvector problem. Then, we study the Riemannian structure of the generalized oblique manifold by introducing several Riemannian metrics and computing important geometric tools such as orthogonal projections and Riemannian gradients. This allows us to develop the Riemannian gradient descent methods based on different metrics. Exploiting first- and second-order information of the energy functional for the construction of appropriate metrics makes it possible to incorporate preconditioning into Riemannian optimization, which significantly improves the performance of the optimization schemes. A collection of numerical experiments demonstrates the computational efficiency of the proposed methods. (Joint work with R. Altmann, M. Hermann, and D. Peterseim)
Talk 2: Approximating maps into manifolds with multiple tangent spaces
Speaker: Hang Wang
Abstract: A manifold-valued function takes values from a Euclidean domain into a manifold. Approximating a manifold-valued function from input-output samples consists of modeling the relationship between an output on a Riemannian manifold and the Euclidean input vector. In this talk, I will present algorithms for building a surrogate model to approximate either a known or an unknown manifold-valued function. The proposed methods are based on pullbacks to multiple tangent spaces and the Riemannian center of mass, hereby relying on Riemannian optimization algorithms. The effectiveness of this scheme will be illustrated with numerical experiments for a few model problems.
Talk 3: The injectivity radii of the Stiefel manifold under a one-parameter family of deformation metrics
Speaker: Ralf Zimmermann
Abstract: The injectivity radius of a manifold is an important quantity, both from a theoretical point of view and in terms of numerical applications. It is the largest possible radius within which all geodesics are unique and length-minimizing. In consequence, it is the largest possible radius within which calculations in Riemannian normal coordinates are well-defined. A matrix manifold that arises frequently in a wide range of practical applications is the compact Stiefel manifold of orthogonal p-frames in the Euclidean n-space. Its geometry may be considered under a one-parameter family of deformation metrics. We observe that the associated geodesics are space curves of constant Frenet curvatures. In combination with tight sectional curvature bounds, this allows us to determine the injectivity radius of the Stiefel manifold for a large subset of the one-parameter family of metrics that includes the Euclidean metric.
