Approximation of Lyapunov functions from noisy data

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/ statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm

A Stable Algorithm for Divergence-Free and Curl-Free Radial Basis Functions in the Flat Limit

Radial basis functions (RBFs) were originally developed in the 1970s for interpolating scattered topographic data. Since then they have become increasingly popular for other applications involving the approximation of scattered, scalar-valued data in two and higher dimensions, especially data collected on the surface of a sphere. In the late 2000s, matrix-valued RBFs were introduced for approximating divergence-free and curl-free vector fields on the surface of a sphere from scattered samples, which arise naturally in atmospheric and oceanic sciences. The intriguing property of these RBFs is that the resulting vector-valued approximations analytically preserve the divergence-free or curl-free properties of the field. The most commonly used RBFs feature a shape parameter that controls how peaked or flat the basis functions are, with the choice of this parameter greatly affecting the accuracy of the RBF approximation to the underlying data. Flatter basis functions, which correspond to small shape parameters, generally result in more accurate approximations when the sampled data comes from a smooth function or vector-field. However, the direct method for computing the resulting RBF approximation becomes horribly ill-conditioned as the basis functions are made flatter and flatter. For scalar-valued RBF approximation, this was a fundamental issue until the mid-2000s when researchers started to develop stable algorithms for flat RBFs. One of the most successful of these is the RBF-QR algorithm, which completely bypasses the ill-conditioning associated with flat scalar-valued RBFs on the sphere using a clever change of basis. In this thesis, we extend the RBF-QR algorithm to flat matrix-valued RBFs for approximating both divergence-free and curl-free vector fields on the sphere. We give numerical results illustrating the effectiveness of this new algorithm and also show that in the limit where the matrix-valued RBFs become entirely flat, the resulting approximations converge to vector spherical harmonic approximants. This is the first algorithm that allows for stable computations of divergence-free and curl-free matrix-valued RBFs in the flat limit

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$ showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure