Multiquadrics--a scattered data approximation scheme with applications to computational fluid-dynamics-- ii solutions to parabolic, hyperbolic and elliptic partial differential equations
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In this paper, the authors used MQ as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation, and showed 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.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 Poisson's 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.read more
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References
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Journal ArticleDOI
Multiquadric equations of topography and other irregular surfaces
TL;DR: In this paper, a method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described, and procedures are given for solving multiquadric equations of topography that are based on coordinate data.
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Scattered data interpolation: tests of some methods
TL;DR: In this paper, the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods are presented. But the evaluation process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation.
Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates
TL;DR: In this article, the authors presented an enhanced multiquadrics (MQ) scheme for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.
Journal ArticleDOI
Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates
TL;DR: In this article, the authors presented a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.
Book ChapterDOI
Interpolation of scattered data: Distance matrices and conditionally positive definite functions
TL;DR: In this paper, it was shown that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke, which is a conjecture that was later proved in the present paper.