Y.C. Hon

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.

On unsymmetric collocation by radial basis functions

Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system was still missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort.

A fundamental solution method for inverse heat conduction problem

Abstract In this paper, we develop a new meshless and integration-free numerical scheme for solving an inverse heat conduction problem. The numerical scheme is developed based on the use of the fundamental solution as a radial basis function. To regularize the resultant ill-conditioned linear system of equations, we apply successfully both the Tikhonov regularization technique and the L-curve method to obtain a stable numerical approximation to the solution. The approach is readily extendable to solve high-dimensional problems under irregular domain.

Multiquadric method for the numerical solution of a biphasic mixture model

A computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model. The model includes a set of constitutive equations for the fluid flows through the solid phase; a set of momentum equations for stress-strain equilibrium and a continuity equation for the solid phase and the fluid phase. The numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry. Numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading. The numerical results are compared with Spilkers penalty finite element method.

Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation

In this paper, the highly ill posed Cauchy problem for the Laplace equation is transformed to a classical moment problem whose numerical approximation can be achieved. Proofs on its convergence and stability estimates are given based on the Backus-Gilbert algorithm. For numerical verification, several examples which include random noise in the initial Cauchy data are presented.

Identification of source locations in two-dimensional heat equations

In this paper, we show the uniqueness of the identification of unknown source locations in two-dimensional heat equations from scattered measurements. Based on the assumption that the unknown source function is a sum of some known functions, we prove that one measurement point is sufficient to identify the number of sources and three measurement points are sufficient to determine all unknown source locations. For verification, we propose a numerical reconstruction scheme for recovering the number of unknown sources and all source locations.

Numerical Computation of a Cauchy Problem for Laplace's Equation

This paper investigates the numerical computation of a Cauchy problem for Laplaces equation which is a typical ill-posed problem. By using Greens formula, the problem is transformed to a moment problem. For numerical computation of the moment problem, an error estimation and several numerical examples for verification are presented. Necessary and sufficient conditions for the existence of the solution of the Cauchy problems for Laplaces equation in two-dimension are also given.

A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves

Abstract A new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations. The solutions obtained in this paper include soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.

Numerical comparisons of two meshless methods using radial basis functions

The recent advance in the development of various kinds of meshless methods for solving partial differential equations has drawn attention of many researchers in science and engineering. One of the domain-type meshless methods is obtained by simply applying the radial basis functions (RBFs) as a direct collocation, which has shown to be effective in solving complicated physical problems with irregular domains. More recently, a boundary-type meshless method that combines the method of fundamental solutions and the dual reciprocity method with the RBFs has been developed. In this paper, the performances of these two meshless methods are compared and evaluated. Numerical results indicate that these two methods provide a similar optimal accuracy in solving both 2D Poissons and parabolic equations.

The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations

The Laplace transform is applied to remove the time-dependent variable in the diffusion equation. For non-harmonic initial conditions this gives rise to a non-homogeneous modified Helmholtz equation which we solve by the method of fundamental solutions. To do this a particular solution must be obtained which we find through a method suggested by Atkinson. To avoid costly Gaussian quadratures, we approximate the particular solution using quasi-Monte-Carlo integration which has the advantage of ignoring the singularity in the integrand. The approximate transformed solution is then inverted numerically using Stehfests algorithm. Two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2-D and 3-D.