Convergence analysis of leapfrog for geodesics

Geodesics are of fundamental interest in mathematics, physics, computer science, and many other subjects. The so-called leapfrog algorithm was proposed in [L. Noakes, J. Aust. Math. Soc., 65 (1998), pp. 37-50] (but not named there as such) to find geodesics joining two given points x0 and x1 on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between x0 and x1 that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics { k}k ≥ 1 generated by this algorithm converges to a geodesic joining x0 and x1. The present paper investigates leapfrog\u27s convergence rate i,n of ith junction depending on the manifold M. A relationship is found with the maximal root n of a polynomial of degree n-3, where n (n \u3e 3) is the number of geodesic segments. That is, the minimal i,n is upper bounded by n(1 + c+), where c+ is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that n increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all 2 × 2 symmetric positive definite matrices

Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction

Active Manifold-Geodesics: A Riemannian View On Active Subspaces With Shape Sensitivity Applications

Aerospace designers routinely manipulate shapes in engineering systems toward design goals and study changes in the modeled system to facilitate new intuitions about the physical processes---e.g., shape optimization and parameter sensitivity analysis of an airfoil. The computational tools for such manipulation can include parameterized geometries, where the parameters provide a set of independent variables that control the geometry. Active subspaces provide an intuitive change of basis for studying differentiable functions with Euclidean domain of dimension greater than or equal to two. Recent work has developed and exploited active subspaces in the composition from geometry parameters to design quantities of interest (e.g., lift or drag of an airfoil); the active subspace is spanned by a set of directions in a parameter space which change the associated quantity of interest more, on average over the parameter design space, than orthogonal directions. Consequently, the active directions produce insight-rich geometry perturbations for a specific quantity of interest; however, these perturbations also depend on the chosen geometry parameterization. Several engineering applications explore this shape-parameterization dependency for optimization and sensitivity analysis. However, selection of a parameterization restricts any subsequent analysis to the class of chosen parameterization; including the approximation of an active subspace. Defining a precise calculus of shapes independent of engineering parameterizations requires a new interpretation of the domain of scalar-valued functions dependent on these shapes. The space of shapes admits a topological structure of a smooth manifold, a more general non-Euclidean domain for quantities of interest. This work extends the computation of active subspaces to differentiable functions defined on smooth manifolds M. We seek ordered geodesics defining submanifolds of a Riemannian manifold (M, g), endowed with a metric g, which change the differentiable function iv more, by an analogous globalizing notion of the average. These submanifolds representing analogous subspaces on a more general non-Euclidean domain are referred to as active manifold-geodesics. However, there are competing intrinsic and extrinsic perspectives regarding computations and approximations on Riemannian manifolds. Extrinsic perspectives rely on the existence of isometric embeddings of the manifold into an ambient Euclidean space while intrinsic perspectives work entirely with objects defined only on the manifold, i.e., not requiring an isometric embedding. The continuous form of an analogous average outer product of the gradient is presented from an intrinsic perspective. A discretization and approximation of the eigenspaces of the proposed intrinsic extension is applied to the sphere S2 &sub; R3 as an example which can be visualizedThese routines are then generalized to a matrix manifold of landmark-affine shapes to inform a global shape sensitivity analysis of transonic airfoils---independent of a shape-parameterization.</p

Active Manifold-Geodesics: A Riemannian View on Active Subspaces with Shape Sensitivity Applications

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System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques