Natasha Flyer

Stable Computations with Gaussian Radial Basis Functions

Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Pade method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.

Magnetic field confinement in the corona: The role of magnetic helicity accumulation

A loss of magnetic field confinement is believed to be the cause of coronal mass ejections (CMEs), a major form of solar activity in the corona. The mechanisms for magnetic energy storage are crucial in understanding how a field may possess enough free energy to overcome the Aly limit and open up. Previously, we have pointed out that the accumulation of magnetic helicity in the corona plays a significant role in storing magnetic energy. In this paper, we investigate another hydromagnetic consequence of magnetic-helicity accumulation. We propose a conjecture that there is an upper bound on the total magnetic helicity that a force-free field can contain. This is directly related to the hydromagnetic property that force-free fields in unbounded space have to be self-confining. Although a mathematical proof of this conjecture for any field configuration is formidable, its plausibility can be demonstrated with the properties of several families of power-law, axisymmetric force-free fields. We put forth mathematical evidence, as well as numerical, indicating that an upper bound on the magnetic helicity may exist for such fields. Thus, the accumulation of magnetic helicity in excess of this upper bound would initiate a nonequilibrium situation, resulting in a CME expulsion as a natural product of coronal evolution.

A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere

The current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(10^5) nodes on the sphere. The RBF-FD method scaled as O(N) per time step, where N is the total number of nodes on the sphere. In Appendix A, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs.

Transport Schemes on a Sphere Using Radial Basis Functions

The aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularity free. Then, two classical test cases are conducted: (1) linear advection, where the initial condition is simply transported around the sphere and (2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large time-steps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.

A radial basis function method for the shallow water equations on a sphere

The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

A Primer on Radial Basis Functions with Applications to the Geosciences

Adapted from a series of lectures given by the authors, this monograph focuses on radial basis functions (RBFs), a powerful numerical methodology for solving PDEs to high accuracy in any number of dimensions. This method applies to problems across a wide range of PDEs arising in fluid mechanics, wave motions, astro- and geosciences, mathematical biology, and other areas and has lately been shown to compete successfully against the very best previous approaches on some large benchmark problems. Using examples and heuristic explanations to create a practical and intuitive perspective, the authors address how, when, and why RBF-based methods work. The authors trace the algorithmic evolution of RBFs, starting with brief introductions to finite difference (FD) and pseudospectral (PS) methods and following a logical progression to global RBFs and then to RBF-generated FD (RBF-FD) methods. The RBF-FD method, conceived in 2000, has proven to be a leading candidate for numerical simulations in an increasingly wide range of applications, including seismic exploration for oil and gas, weather and climate modeling, and electromagnetics, among others. This is the first survey in book format of the RBF-FD methodology and is suitable as the text for a one-semester first-year graduate class. Audience: The book is primarily written for graduate students and researchers in application areas such as atmospheric modeling and geosciences. It is also suited for numerical analysts and computational scientists interested in large-scale PDE-based simulations on modern computer architectures.

Solving PDEs with radial basis functions

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids

Abstract Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the accuracy of finite difference schemes. We find that the accuracy is directly linked to the decay rate, at large arguments, of the (generalized) Fourier transform of the radial function. Three different types of convergence rates can be distinguished as the node density increases – polynomial, spectral, and superspectral, as exemplified, for example, by thin plate splines, multiquadrics, and Gaussians respectively.

MAGNETIC FIELD CONFINEMENT IN THE SOLAR CORONA. I. FORCE-FREE MAGNETIC FIELDS

Axisymmetric force-free magnetic fields external to a unit sphere are studied as solutions to boundary value problems in an unbounded domain posed by the equilibrium equations. It is well known from virial considerations that stringent global constraints apply for a force-free field to be confined in equilibrium against expansion into the unbounded space. This property as a basic mechanism for solar coronal mass ejections is explored by examining several sequences of axisymmetric force-free fields of an increasing total azimuthal flux with a power-law distribution over the poloidal field. Particular attention is paid to the formation of an azimuthal rope of twisted magnetic field embedded within the global field, and to the energy storage properties associated with such a structure. These sequences of solutions demonstrate (1) the formation of self-similar regions in the far global field where details of the inner boundary conditions are mathematically irrelevant, and (2) the possibility that there is a maximum to the amount of azimuthal magnetic flux confined by a poloidal field of a fixed flux anchored rigidly to the inner boundary. The nonlinear elliptic boundary value problems we treat are mathematically interesting and challenging, requiring a specially designed solver, which is described in the Appendix.

Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

This paper presents parallelization strategies for the radial basis function-finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and scales as O ( N ) per time step, with N being with the total number of nodes. To our knowledge, this is the first implementation of the RBF-FD method to leverage GPU accelerators for the solution of PDEs. Additionally, this implementation is the first to span both multiple CPUs and multiple GPUs. OpenCL kernels target the GPUs and inter-processor communication and synchronization is managed by the Message Passing Interface (MPI). We verify our implementation of the RBF-FD method with two hyperbolic PDEs on the sphere, and demonstrate up to 9x speedup on a commodity GPU with unoptimized kernel implementations. On a high performance cluster, the method achieves up to 7x speedup for the maximum problem size of 27,556 nodes.