No abstract

A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. Computing, http://dx.doi.org/10.1007/s10915-014-9935-9 Journal of ScientificAccess to the published version may require subscription. Abstract Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF-PUM) proposed here. The objective of this paper is to establish that RBF-PUM is viable for parabolic PDEs of convection-diffusion type. The stability and accuracy of RBF-PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection-diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF-PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF-PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

Abstract. Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.Key words. radial basis function, flat limit, ill-conditioning, differentiation matrix, stencil weight, RBF, RBF-QR, RBF-FD AMS subject classifications. 65D15, 65D251. Introduction. Radial basis function (RBF) approximation [1,31,4] is emerging as an important method class for interpolation, approximation, and solution of partial differential equations (PDEs) for data given at scattered node locations, with non-trivial geometry, or with computational domains in higher dimensions. The main advantages are the spectral convergence rates that can be achieved using infinitely smooth basis functions, the geometrical flexibility, and the ease of implementation. However, in practical cases, convergence has often been hampered by ill-conditioning as the shape of the basis functions become flatter. The best accuracy for smooth and well resolved solutions is often found in this regime [18,13,19]. Therefore, moving to larger shape parameter values (less flat RBFs) is not a desirable solution to the conditioning problem.The first method that allowed stable computations in the flat RBF regime was the Contour-Padé method derived by Fornberg and Wright [13]. The method works in any number of dimensions, but for relatively low numbers of nodes. Except for the approach in [24], limited to Gaussian RBFs on an equispaced grid in one dimension, the next method that was developed was the RBF-QR method, which was first derived for nodes on the surface of the sphere by Fornberg and Piret [12] and then for general node distributions in up to three dimensions [9]. The RBF-QR methods can be employed for approximations over thousands of nodes.In [9], we gave examples of convergence and performance results in the case of i...

Abstract. Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) 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-PUM. 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 be 5-10 times faster than the collocation formulation for the same accuracy.

We consider model problems for the tear film over multiple blink cycles that utilize a single equation for the tear film; the single nonlinear partial differential equation (PDE) that governs the film thickness arises from lubrication theory. The two models that we consider arise from considering the absence of naturally occuring surfactant and the case when the surfactant is strongly affecting the surface tension. The film is considered on a time-varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. Realistic lid motion from observed blinks is included in the model with end fluxes specified to more closely match the blink cycle than those previously reported. Numerical computations show quantitative agreement with in vivo tear film thickness measurements under partial blink conditions. A transition between periodic and nonperiodic solutions has been estimated as a function of closure fraction and this may be a criterion for what is effectively a full blink according to fluid dynamics.

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.

No abstract

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.