Computational Riemannian Geometry: High-Order Methods, Analysis, and Structure Preservation

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Riemannian geometry plays a fundamental role in mathematical physics and geometric analysis, and computational methods for Riemannian geometry have surprisingly many practical applications. Equations that govern time-varying Riemannian metrics, for example, are prototypes for curvature-driven flows that arise in science and engineering like surface tension-driven flow. Such equations also underly several algorithms that are used widely in computer graphics, machine vision, and medical imaging. Examples include algorithms for surface parameterization, texture mapping, and surface registration. Computational Riemannian geometry also plays an essential role in gravitational wave astronomy, where accurate numerical simulations of Einstein’s equations are needed to make inferences about gravitational wave signals. Despite its importance, Riemannian geometry is, in certain respects, underserved by traditional tools of numerical analysis, which are tailored toward problems posed in Euclidean space. This project centers on developing novel computational methods for Riemannian geometry. The computational methods will be made freely available in a public repository, and graduate students will participate in their development. The main goal of this project is to design and analyze high-order methods for three families of problems in computational Riemannian geometry: (1) the numerical solution of intrinsic geometric flows with finite element methods, (2) intrinsic curvature approximation with finite elements, and (3) the efficient computation of interpolants, geodesics, Riemannian means, and the Riemannian exponential map on matrix manifolds. These three problems are tightly intertwined. The vast majority of geometric flows in Riemannian geometry are curvature-driven flows, so their discretization with finite elements goes hand in hand with the construction of finite element approximations of the Riemann curvature tensor and its contractions. In turn, tensor field and frame field discretizations play an important role in curvature approximation, underscoring the need for efficient algorithms for computations on matrix manifolds. This project aims to design numerical methods for the aforementioned problems that are high-order, provably convergent, and structure-preserving. This project will support one graduate student each year.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.