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Zbl 0955.65086
Kansa, E.J.; Hon, Y.C.
Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. (English)
[J] Comput. Math. Appl. 39, No.7-8, 123-137 (2000). ISSN 0898-1221

Summary: {\it W. R. Madych} and {\it S. A. Nelson} [Math. Comput. 54, No. 189, 211-230 (1990; Zbl 0859.41004)] 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.\par 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\par (1) replacement of global solvers by block partitioning, LU decomposition schemes,\par (2) matrix preconditioners,\par (3) variable MQ shape parameters based upon the local radius of curvature of the function being solved,\par (4) a truncated MQ basis function having a finite, rather than a full band-width,\par (5) multizone methods for large simulation problems, and\par (6) knot adaptivity that minimizes the total number of knots required in a simulation problem.\par The hybrid combination of these methods contribute to very accurate solutions.\par Even though the finite element method (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.

MSC 2000:: *65N30 Finite numerical methods (BVP of PDE)
65N55 Multigrid methods; domain decomposition (BVP of PDE)
65H10 Systems of nonlinear equations (numerical methods)
65F35 Matrix norms, etc. (numerical linear algebra)
35J65 (Nonlinear) BVP for (non)linear elliptic equations
65N12 Stability and convergence of numerical methods (BVP of PDE)

Keywords: ill-conditioned matrices; nonlinear Poisson equation; numerical examples; multiquadric radial basis functions; domain decomposition methods; multizone decomposition methods; exponential convergence; matrix preconditioners; finite element method; sparse coefficient matrices

Citations: Zbl 0859.41004

Cited in: Zbl 1219.41027

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