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

Analysis of moving least squares approximation revisited

In this article the error estimation of the moving least squares approximation is provided for functions in fractional order Sobolev spaces. The analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details. An application to Galerkin method for partial differential equations is also supplied.Comment: Journal of Computational and Applied Mathematics, 2015 Journal of Computational and Applied Mathematic

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

Meshless methods for Maxwell’s equations with applications to magnetotelluric modelling and inversion

The first part of thesis presents new meshless methods for solving time harmonic electromagnetic fields in closed two- or three-dimensional volumes containing heterogeneous materials. This new methods will be used to simulate magnetotelluric experiments, when an Earth conductivity model is given in advanced. Normally, classical approximation methods like finite elements or finite differences are used to solve this task. The algorithms here in this thesis, only need an unstructured point sampling in the modelling domain for the discretization and is able to gain a solution for the partial differential equation without a fixed mesh or grid. This is advantageous when complex model geometries have to be described, because no adapted mesh or grid need to be generated. The meshless methods, described here in this thesis, use a direct discretization technique in combination with a generalized approximation method. This allows to formulate the partial differential equations in terms of linear functionals, which can be approximated and directly form the discretization. For the two-dimensional magnetotelluric problem, a second-order accurate algorithm to solve the partial differential equations was developed and tested with several example calculations. The accuracy of the new meshless methods was compared to analytical solutions, and it was found, that a better accuracy can be achieved with less degrees of freedoms compared to previously published results. For the three-dimensional case, a meshless formulation was given and numerical calculations show the ability of the scheme to handle models with heterogeneous conductivity structures. In the second part of this thesis, the newly developed two-dimensional simulation method will be used in an inversion scheme. Here, the task is to recover the unknown Earth conductivity model with the help of data gained from a magnetotelluric experiment. Due to the previously developed meshless approximation algorithm, some numerical tasks during the inversion can be simplified by reusing the discretization defined on the point sampling from the forward simulation. The newly developed meshless inversion algorithm will be tested with synthetic data to reconstruct known conductivity anomalies. It can be shown, that the inverse algorithm produces correct results, even in the presence of topography

Generalized moving least squares interpolation for solution of partial differential equations

Methods for computing the solution of partial differential equation typically require three key ingredients, namely: (1) how to represent the simulation domain, (2) how to represent the approximate solution and (3) how to enforce the governing equation. For example, the Finite Element Method requires a mesh to satisfy conditions (1) and (2). Doing so, however, places strict requirements on the mesh that are difficult to meet in applications. This thesis mainly concentrates on utilizing the Generalized Moving Least Squares approximation in order to fulfill requirement (2) of the three key ingredients. Thereby, we reduce the requirements on the mesh to represent the unknown functions. Generalized Moving Least Squares builds a polynomial approximation for a function by minimizing the squared residual errors at specific locations throughout the domains. In the first part, we will fulfill condition (1) by a point-cloud (particle) representation of the simulation domain and condition (3) with a finite-difference-like collocation scheme on the strong form of the Partial Differential Equations at the particle locations. We apply this scheme to solve steady-state Stokes flow. Our results indicate that the error for both velocity and pressure field exhibits a high-order convergence rate. Additionally, the performance benchmarks suggest that our parallel implementation of the method is scalable for larger systems, and thus has potential to be executed on sizeable supercomputing clusters. In the second part, we will borrow the framework from the Finite Element Method and satisfy condition (1) with a mesh. The resulting method has compactly supported discontinuous shape functions which are generated from generalized moving least squares. These discontinuous polynomials are then applied within a Discontinuous Galerkin variational formulation with interior penalty to accomplish condition (3). Since the basic functions are separated from the shape of the underlying elements, the dependence on the mesh quality is, therefore, removed. We derive \textit{a priori} error bounds of this formulation, specifically for solving Poisson's boundary value problem and the linear elasticity problem. The numerical result demonstrates the expected convergence behavior even on poor-quality meshes. Moreover, we have found that this scheme is able to maintain much higher stability, when compared against the conventional Finite Element Methods

Spectral methods for solving elliptic PDEs on unknown manifolds

In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. Especially, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergent of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.Comment: 8 figure

Computational Multiscale Methods

Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments

Numerical analysis of reproducing kernel collocation method for linear nonlocal models

Hydraulic fracturing has played a major role in north America's “shale revolution” over the past decades. Modeling of the hydraulic fracture propagation is challenging. Peridynamics, a nonlocal theory of continuum mechanics, has been used to model complex hydraulic fracturing processes in recent years. While the peridynamics-based hydraulic fracturing model has shown promising simulations results, its current numerical discretization lacks any mathematical analysis. This dissertation is motivated by the numerical solution of the peridynamics-based hydraulic fracturing model. The major objective is to develop a robust numerical method, under the change of the modeling parameters, for linear nonlocal diffusion models and peridynamic Navier equation, which are decoupled models of the peridynamics-based hydraulic fracturing model. Reproducing kernel (RK) collocation method is of our interest due to its mesh-free nature. By choosing special RK support sizes, we have developed a RK collocation method for nonlocal models and numerical solutions converge to the nonlocal solution and also to the corresponding local limit independent of the modeling parameters as the nonlocal interactions vanish. Accurate evaluation of the stiffness matrix for nonlocal models is computationally prohibitive even for collocation method. To save computational costs, the concept of RK approximation is generalized to approximate integrals and the quasi-discrete nonlocal operator, which uses a finite number of symmetric quadrature points to evaluate the integral, is proposed. We have shown RK collocation on the quasi-discrete nonlocal diffusion and peridynamic Navier equation converge to their classical counterparts. Finally, for the pure displacement form of the classical linear elasticity model, finite element solutions deteriorate when the material is nearly incompressible. A common remedy is to introduce an additional variable, pressure, and rewrite the equation in a mixed formulation, but the discrete functional spaces need to satisfy the famous inf-sup condition. For the the mixed form of the quasi-discrete peridynamic Navier equation, the discretization obtained using RK collocation with equal order interpolation for displacements and pressure passes the inf-sup test; the solution does not suffer from instability. Hence, with the use of penalty techniques or artificial compressibility, the proposed RK collocation method is promising in solving the peridynamics-based hydraulic fracturing model, which has an embedded saddle-point problemPetroleum and Geosystems Engineerin