E.J. Kansa

MULTIQUADRICS—A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS—II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract This paper is the second in a series of investigations into the benefits of multiquadrics (MQ). MQ is a true scattered data, multidimensional spatial approximation scheme. In the previous paper, we saw that MQ was an extremely accurate approximation scheme for interpolation and partial derivative estimates for a variety of two-dimensional functions over both gridded and scattered data. The theory of Madych and Nelson shows for the space of all conditionally positive definite functions to which MQ belongs, a semi-norm exists which is minimized by such functions. In this paper, MQ is used as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poissons equation. We show that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.

Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates

Abstract We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gived credence to our results.

Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations☆

Abstract Madych and Nelson [1] proved multiquadric (MQ) mesh-independent radial basis functions (RBFs) enjoy exponential convergence. The primary disadvantage of the MQ scheme is that it is global, hence, the coefficient matrices obtained from this discretization scheme are full. Full matrices tend to become progressively more ill-conditioned as the rank increases. In this paper, we explore several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. The methods that were investigated are 1. (1) replacement of global solvers by block partitioning, LU decomposition schemes, 2. (2) matrix preconditioners, 3. (3) variable MQ shape parameters based upon the local radius of curvature of the function being solved, 4. (4) a truncated MQ basis function having a finite, rather than a full band-width, 5. (5) multizone methods for large simulation problems, and 6. (6) knot adaptivity that minimizes the total number of knots required in a simulation problem. The hybrid combination of these methods contribute to very accurate solutions. Even though FEM gives rise to sparse coefficient matrices, these matrices in practice can become very ill-conditioned. We recommend using what has been learned from the FEM practitioners and combining their methods with what has been learned in RBF simulations to form a flexible, hybrid approach to solve complex multidimensional problems.

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems. In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature. In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.

Improved accuracy of multiquadric interpolation using variable shape parameters

Given N scattered data points, we examined the problem of finding N variable Multiquadric (MQ) shape-parameters, or R2 values. Because the problem of finding the optimal R2 values is a nonlinear one, we optimized these parameters numerically by minimizing the root-mean-square (RMS) errors. The resulting R2 values varied over many orders of magnitude. We have tested this approach on a number of univariate and bivariate (Frankes) problems, and found that the RMS error reduction was substantial.

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N^2) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/@/N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.

A least-squares preconditioner for radial basis functions collocation methods

Abstract Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring

Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme

\mathcal{O}(N^{3})

CONTINUATION FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS DISCRETIZED BY THE MULTIQUADRIC METHOD

flops. We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements. The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain. Our preconditioner requires

A strictly conservative spatial approximation scheme for the governing engineering and physics equations over irregular regions and inhomogeneously scattered nodes

\mathcal{O}(mN^{2})