Stability of numerical methods on Riemannian manifolds

Abstract: Stability of numerical integrators play a crucial role in approximating the flow of differential equations. Issues related to convergence and step size limitations have been successfully resolved by studying the stability properties of numerical schemes. Stability also plays a role in the existence and uniqueness to the solution of the nonlinear algebraic equations that need to be solved in each time step for an implicit method. However, very little has up to now been known about stability properties of numerical methods on manifolds, such as Lie group integrators. An interest in these questions has recently been sparked by the efforts in constructing ODE based neural networks that are robust against adversarial attacks. In this talk we shall discuss a new framework for B-stability on Riemannian manifolds. A method is B-stable if the numerical method exhibits a non-expansive behaviour in the Riemannian distance measure when applied to problems which have non-expansive solutions. We shall in particular see how the sectional curvature of the manifold plays a role, and show some surprising results regarding the non-uniqueness of geodesic implicit integrators for positively curved spaces.

numerical analysisoptimization and control

Audience: researchers in the topic

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CAM seminar

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