Alfa R. H. Heryudono

Adaptive residual subsampling methods for radial basis function interpolation and collocation problems

We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains.

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions

Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation

In this paper, Radial Basis Function (RBF) method for interpolating two dimensional functions with localized features defined on irregular domain is presented. RBF points located inside the domain and on its boundary are chosen such that they are the image of conformally mapped points on concentric circles on a unit disk. On the disk, a fast RBF solver to compute RBF coefficients developed by Karageorghis et al. (Appl. Numer. Math. 57(3):304–319, 2007) is used. Approximation values at desired points in the domain can be computed through the process of conformal transplantation. Some numerical experiments are given in a style of a tutorial and MATLAB code that solves RBF coefficients using up to 100,000 RBF points is provided.

Preconditioning for Radial Basis Function Partition of Unity Methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF–PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.

A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

We propose and test the first Reduced Radial Basis Function Method for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.

A least squares radial basis function partition of unity method for solving PDEs

Recently, collocation-based radial basis function (RBF) partition of unity methods (PUMs) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUMs. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to ...

Implementation of Neumann boundary condition with influence matrix method for viscous annular flow using pseudospectral collocation

The flowfield of an annular Couette flow is predicted numerically from an unsteady initial condition using the Chebyshev-Fourier collocation method. The numerical solution is obtained from the vorticity-velocity formulation of the unsteady Navier-Stokes equations. In this formulation the velocity boundary conditions are overspecified while vorticity boundary conditions are unspecified. This difficulty is resolved by a matrix influence method to convert the overspecified velocity boundary conditions to sufficiently specified Dirichlet boundary conditions for both velocity and vorticity. These boundary conditions are implemented by considering three methods: the traditional row replacement method, Fornbergs fictitious points method, and Driscoll and Hales rectangular collocation method. The solution is advanced in time using the third-order Adams-Bashforth semi-implicit backward differentiation scheme. The accuracy of the numerical solutions are assessed in two ways. In one case, the accuracy of the converged steady state solution is compared to the analytical solution. In the second case, the residual of the continuity equation for the vorticity-velocity method is assessed in comparison to the vorticity streamfunction formulation that inherently satisfies the continuity equation. All methods demonstrate high accuracy for the continuity equation residual at the domain interior; however, the row replacement and fictitious points methods exhibit poor accuracy at the boundaries during the transient. The rectangular projection method exhibits excellent accuracy throughout the domain and boundaries at all times using the resampling points. On the other hand, for evaluating secondary quantities such as the wall shear stress, the rectangular projection method demonstrates several orders of magnitude less accuracy when evaluated using the resampling points, but high accuracy when evaluated directly at the original collocation points.

Numerical Solution of the Viscous Flow Past a Cylinder with a Non-global Yet Spectrally Convergent Meshless Collocation Method

The flow of a viscous fluid past a cylinder is a classical problem in fluid-structure interaction and a benchmark for numerical methods in computational fluid dynamics. We solve it with the recently introduced radial basis function-based partition of unity method (RBF-PUM), which is a spectrally convergent collocation meshless scheme well suited to this kind of problem. The resulting discrete system of nonlinear equations is tackled with a trust-region algorithm, whose performance is much enhanced by the analytic Jacobian which is provided alongside. Preliminary results up to Re = 60 with just 1292 nodes are shown.

An adaptive interpolation scheme for molecular potential energy surfaces

The calculation of potential energy surfaces for quantum dynamics can be a time consuming task-especially when a high level of theory for the electronic structure calculation is required. We propose an adaptive interpolation algorithm based on polyharmonic splines combined with a partition of unity approach. The adaptive node refinement allows to greatly reduce the number of sample points by employing a local error estimate. The algorithm and its scaling behavior are evaluated for a model function in 2, 3, and 4 dimensions. The developed algorithm allows for a more rapid and reliable interpolation of a potential energy surface within a given accuracy compared to the non-adaptive version.

Adaptive Methods for Analysis of Composite Plates with Radial Basis Functions

Driscoll and Heryudono [1] developed an adaptive method for radial basis functions method. This article addresses the adaptive analysis of composite plates in bending with radial basis multiquadric functions using Driscoll and Heryudonos technique. In this article, various laminates, thickness to side length ratios, and boundary conditions are considered. The method allows for a more natural and automatic selection of the problem grid, where the user must only define the error tolerance. The results obtained show an interesting and promising approach to the static analysis of composite laminates.