Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severely ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in a curved waveguide geometry.
The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the…
The Radial Point Interpolation Mixed Collocation (RPIMC) method is proposed in this paper for transient analysis of diffusion problems. RPIMC is an efficient purely meshless method where the solution of the field variable is obtained through collocation. The field function and its gradient are both interpolated (mixed collocation approach) leading to reduced $C$-continuity requirement compared to strong-form collocation schemes. The method's accuracy is evaluated in heat conduction benchmark problems. The RPIMC convergence is compared against the Meshless Local Petrov-Galerkin Mixed Collocatio…
Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretization matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore the extent to which oversampled collocation can improve both robustness and convergence rates. We show that in some cases convergence rates can actually be higher than the corresponding Galerkin…
Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite difference method in a discrete least-squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least-squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.
This paper presents a comparison of two methods for the forward uncertainty quantification (UQ) of complex industrial problems. Specifically, the performance of Multi-Index Stochastic Collocation (MISC) and adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF) surrogates is assessed for the UQ of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties, namely the ship speed and draught. The estimation of expected value, standard deviation, and probability density function of the (model-scale) resistance is presented and discussed; the…
Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spatial frequency domain, are the only existing methods for solving the equation. For the earth media with curved boundaries, the pseudospectral methods could be less attractive to handle the irregular computational domains. In the paper, we propose a radial basis function collocation method that can easily tackle the irregular domain problems. Unlike the pseudospectral methods, the proposed method solves the equation i…
This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Col…
Universidad Adolfo Ibáñez（阿道夫·伊瓦涅斯大学）；Data Observatory Foundation（数据观测站基金会）；Center of Applied Ecology and Sustainability (CAPES)（应用生态与可持续性中心（CAPES））
We consider physics-informed neural networks (PINNs) [Raissi et al., J.~Comput. Phys. 278 (2019) 686-707] for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter optimization (HPO) procedure via Gaussian processes-based Bayesian optimization. We apply the HPO to Helmholtz equation for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density $r$ and (iii) the frequency $κ$, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimens…
In this article, a formulation of a point-collocation method in which the unknown function is approximated using global expansion in tensor product Bernstein polynomial basis is presented. Bernstein polynomials used in this study are defined over general interval [a,b]. Method incorporates several ideas that enable higher numerical efficiency compared to Bernstein polynomial methods that have been previously presented. The approach is illustrated by a solution of Poisson, Helmholtz and Biharmonic equations with Dirichlet and Neumann type boundary conditions. Comparisons with analytical solutio…
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
University of Basel（巴塞尔大学）；Institute for Numerical Simulation, Bonn University（波恩大学数值模拟研究所）；Fraunhofer Institute for Algorithms and Scientific Computing SCAI（弗劳恩霍夫算法与科学计算研究所）；RWTH Aachen University（亚琛工业大学）
This paper is concerned with the cavity scattering problem in an infinite thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. Based on an operator splitting, the scattering problem is recast into a coupled boundary value problem for the Helmholtz and modified Helmholtz equations. A novel boundary integral formulation is proposed for the coupled problem. By introducing an appropriate regularizer, the well-posedness is established for the system of boundary integral equations. Moreover, the convergence analysis is carried out for the semi-…
In this work, we develop a high-order collocation method using radial basis function (RBF) for the incompressible Navier-Stokes equation (NSE) on the rotating sphere. The method is based on solving the projection of the NSE on the space of divergence-free functions. For that, we use matrix valued kernel functions which allow an analytically divergence-free approximation of the velocity field. Using kernel functions which lead to rotation-free approximations, the pressure can be recovered by a simple kernel exchange in one of the occurring approximations, without solving an additional Poisson p…
Application of Meshfree Method Based on Compactly Supported Radial Basis Function for Solving Unsteady Isothermal Gas Through a Micro-Nano Porous Medium
The technique of complex scaling for time harmonic wave type equations relies on a complex coordinate stretching to generate exponentially decaying solutions. In this work, we use a Galerkin method with ansatz functions with infinite support to discretize complex scaled Helmholtz resonance problems. We show that the approximation error of the method decays super algebraically with respect to the number of unknowns in radial direction. Numerical examples underline the theoretical findings and show the superior efficiency of our method compared to a standard perfectly matched layer method.
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 enfo…
We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify…
A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanic…
On fast computation of finite-time coherent sets using radial basis functions
School of Mathematics and Statistics, The University of New South Wales（新南威尔士大学数学与统计学院）；Zentrum Mathematik - M3 Technische Universität München（慕尼黑工业大学数学中心-M3）
The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape par…
Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related methods based on boundary integral equations. In this paper we introduce a local integral boundary domain method with a stable calculation based on Radial Basis Functions (RBF) approximations, in the context of wave chaos in acoustics and dielectric microresonator problems. RBFs have been gaining popularity recently for solving partial differential equations nu…
An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels
Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校）；Jozef Stefan Institute（约瑟夫斯蒂芬研究所）；Institute for Geophysics, University of Texas at Austin（德克萨斯大学奥斯汀分校地球物理研究所）
