Generalized moving least squares vs. radial basis function finite difference methods for approximating surface derivatives

Approximating differential operators defined on two-dimensional surfaces is an important problem that arises in many areas of science and engineering. Over the past ten years, localized meshfree methods based on generalized moving least squares (GMLS) and radial basis function finite differences (RBF-FD) have been shown to be effective for this task as they can give high orders of accuracy at low computational cost, and they can be applied to surfaces defined only by point clouds. However, there have yet to be any studies that perform a direct comparison of these methods for approximating surface differential operators (SDOs). The first purpose of this work is to fill that gap. For this comparison, we focus on an RBF-FD method based on polyharmonic spline kernels and polynomials (PHS+Poly) since they are most closely related to the GMLS method. Additionally, we use a relatively new technique for approximating SDOs with RBF-FD called the tangent plane method since it is simpler than previous techniques and natural to use with PHS+Poly RBF-FD. The second purpose of this work is to relate the tangent plane formulation of SDOs to the local coordinate formulation used in GMLS and to show that they are equivalent when the tangent space to the surface is known exactly. The final purpose is to use ideas from the GMLS SDO formulation to derive a new RBF-FD method for approximating the tangent space for a point cloud surface when it is unknown. For the numerical comparisons of the methods, we examine their convergence rates for approximating the surface gradient, divergence, and Laplacian as the point clouds are refined for various parameter choices. We also compare their efficiency in terms of accuracy per computational cost, both when including and excluding setup costs

Meshfree Methods for PDEs on Surfaces

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software. In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost. The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative methods for solving linear systems that arise in numerical PDEs. The key components for multilevel methods: \grid coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution. We finish with several application problems. We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures

Interpolating point spread function anisotropy

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require very accurate modeling and correction. While numerous methods have been devised to carry out the PSF correction itself, modeling of the PSF shape and its spatial variations across the instrument field of view has, so far, attracted much less attention. This step is nevertheless crucial because the PSF is only known at star positions while the correction has to be performed at any position on the sky. A reliable interpolation scheme is therefore mandatory and a popular approach has been to use low-order bivariate polynomials. In the present paper, we evaluate four other classical spatial interpolation methods based on splines (B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and ordinary Kriging (OK). These methods are tested on the Star-challenge part of the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and are compared with the classical polynomial fitting (Polyfit). We also test all our interpolation methods independently of the way the PSF is modeled, by interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are known exactly at star positions). We find in that case RBF to be the clear winner, closely followed by the other local methods, IDW and OK. The global methods, Polyfit and B-splines, are largely behind, especially in fields with (ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all interpolators reach a variance on PSF systematics $\sigma_{sys}^2$ better than the $1\times10^{-7}$ upper bound expected by future space-based surveys, with the local interpolators performing better than the global ones

Solution of the two-dimensional telegraph equation via the local radial basis functions method

In the present study, we utilize the local meshless method for solving second order hyperbolic partial differential equation in two dimensions. First we apply the Crank-Nicolson difference scheme for the time derivative and the local radial basis functions (LRBFs) collocation method for the spatial derivative. The local approach breaks down the problem domain into subdomains and results small matrix system for each data. Some numerical examples are included to verify the computational efficiency of the proposed method.Publisher's Versio

Accurate Stabilization Techniques for RBF-FD Meshless Discretizations with Neumann Boundary Conditions

A major obstacle to the application of the standard Radial Basis Function-generated Finite Difference (RBF-FD) meshless method is constituted by its inability to accurately and consistently solve boundary value problems involving Neumann boundary conditions (BCs). This is also due to ill-conditioning issues affecting the interpolation matrix when boundary derivatives are imposed in strong form. In this paper these ill-conditioning issues and subsequent instabilities affecting the application of the RBF-FD method in presence of Neumann BCs are analyzed both theoretically and numerically. The theoretical motivations for the onset of such issues are derived by highlighting the dependence of the determinant of the local interpolation matrix upon the boundary normals. Qualitative investigations are also carried out numerically by studying a reference stencil and looking for correlations between its geometry and the properties of the associated interpolation matrix. Based on the previous analyses, two approaches are derived to overcome the initial problem. The corresponding stabilization properties are finally assessed by succesfully applying such approaches to the stabilization of the Helmholtz-Hodge decomposition

Lagrangian Flow Field Reconstruction Based on Constrained Stable Radial Basis Function

Recent advances in three-dimensional (3D) high seeding density time-resolved Lagrangian particle tracking (LPT) techniques have made diagnosing fluid flows at high resolution in space and time under a Lagrangian framework feasible and practical. But challenges persist in developing LPT data processing methods. A promising processing method should accurately and robustly reconstruct, for example, particle trajectories, velocities, and differential quantities from noisy experimental data. Despite numerous algorithms available in the LPT community, they may suffer from some issues, such as unfavorable accuracy and robustness, lack of physical constraints, and unnecessary projection from Lagrangian data onto Eulerian meshes. These challenges may limit the application of the 3D high seeding density time-resolved LPT techniques. In this thesis, a novel 3D Lagrangian flow field reconstruction method is proposed to address these challenges. The proposed method is based on a stable radial basis function (RBF) and constrained least squares (CLS). The stable RBF serves as a model function to approximate trajectories and velocity fields. The CLS provides a framework to facilitate regression and enforce physical constraints, further enhancing the reconstruction performance. The stable RBF and CLS work together to reconstruct 3D Lagrangian flow fields with high accuracy and robustness. The proposed method offers several advantages over the algorithms currently used in the LPT community. First, it accurately reconstructs particle trajectories, velocities, and differential quantities in 3D, even from noisy experimental data, while satisfying physical constraints such as divergence-free for incompressible flows. Second, it does not require projecting Lagrangian data onto Eulerian meshes, allowing for direct flow field reconstruction at scattered data locations. Third, it effectively mitigates experimental noise in particle locations. Last, the proposed method enables smooth spatial and temporal super-resolution with ease. These advantages exhibit that the proposed method is promising for LPT data processing and further applications in data assimilation and machine learning. Systematic tests were conducted to validate and verify the proposed method. Two-dimensional and 3D validations were performed using synthetic data based on exact solutions of the Taylor-Green vortex and Arnold-Beltrami-Childress flow with added artificial noise. The validations show that the proposed method outperforms baseline algorithms (e.g., finite difference methods and polynomial fittings) under different flow conditions. The method was then verified using experimental data from a 3D low-speed pulsing jet, showing its reliable performance. Based on these validations and verification, it is demonstrated that the proposed method can process experimental LPT data and reconstruct Lagrangian flow fields with accuracy and robustness

Mesh-Free Hydrodynamic Stability

A specialized mesh-free radial basis function-based finite difference (RBF-FD) discretization is used to solve the large eigenvalue problems arising in hydrodynamic stability analyses of flows in complex domains. Polyharmonic spline functions with polynomial augmentation (PHS+poly) are used to construct the discrete linearized incompressible and compressible Navier-Stokes operators on scattered nodes. Rigorous global and local eigenvalue stability studies of these global operators and their constituent RBF stencils provide a set of parameters that guarantee stability while balancing accuracy and computational efficiency. Specialized elliptical stencils to compute boundary-normal derivatives are introduced and the treatment of the pole singularity in cylindrical coordinates is discussed. The numerical framework is demonstrated and validated on a number of hydrodynamic stability methods ranging from classical linear theory of laminar flows to state-of-the-art non-modal approaches that are applicable to turbulent mean flows. The examples include linear stability, resolvent, and wavemaker analyses of cylinder flow at Reynolds numbers ranging from 47 to 180, and resolvent and wavemaker analyses of the self-similar flat-plate boundary layer at a Reynolds number as well as the turbulent mean of a high-Reynolds-number transonic jet at Mach number 0.9. All previously-known results are found in close agreement with the literature. Finally, the resolvent-based wavemaker analyses of the Blasius boundary layer and turbulent jet flows offer new physical insight into the modal and non-modal growth in these flows

A Robust RBF-FD Formulation based on Polyharmonic Splines and Polynomials

We introduce a local method based on radial basis function-generated finite differences (RBF-FD) for interpolation and the numerical solution of partial differential equations (PDEs). The method uses polyharmonic spline (PHS) RBFs together with polynomials to derive differentiation weights on different node configurations. The formulation is explored in three directions: (i) Interpolation and approximation of differential operators, (ii) Elliptic PDEs, and (iii) Hyperbolic PDEs. In particular, the novel RBF-FD methodology is applied to standard test cases in numerical weather prediction, modeled by the compressible Navier-Stokes equations in 2D. Furthermore, the evaluation of the method on different node layouts, Cartesian, hexagonal, and scattered, is studied. The RBF-FD implementation acts as an extension of conventional finite-differences, achieving high accuracy on scattered nodes with no need for a computational mesh