Abstract
Fast RBF foundation concepts are described in this chapter. Details are provided for an effective usage of the direct method using advanced linear algebra strategies and libraries. Methods for data compression which allow to represent the same information with a smaller and less expensive cloud are introduced. Space localization methods are successively considered firstly with compact supported RBF, whose interaction distance is limited in the RBF function itself, and then with partition of unity (POU) that consists of the superposition and blending of smaller clouds. Approximated evaluation of the far field of full supported RBF is described and an overview of the fast multipole method (FMM) is given. Information about iterative solvers and parallel computing are finally provided to complete the chapter.
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- 1.
“DSPTRF computes the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method (\(\text{A} = \text{U} \cdot \text{D} \cdot \text{U}^{\text{T}}\) or \(\text{A} = \text{L} \cdot \text{D} \cdot \text{L}^{\text{T}}\)) where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. DSPTRS solves a system of linear equations A*X = B with a real symmetric matrix A stored in packed format using the factorization \(\text{A} = \text{U} \cdot \text{D} \cdot \text{U}^{\text{T}}\) or \(\text{A} = \text{L} \cdot \text{D} \cdot \text{L}^{\text{T}}\) computed by DSPTRF.”
- 2.
“Suppose we are able to write the matrix A as a product of two matrices LU where L is lower triangular and U is upper triangular, we can solve the linear set \(\text{A} \cdot \text{x} = (\text{L} \cdot \text{U}) \cdot \text{x} = \text{L} \cdot (\text{U} \cdot \text{x}) = \text{b}\) by first solving \(\text{L} \cdot \text{y} = \text{b}\) and then solving \(\text{U} \cdot \text{x} = \text{y}\). The advantage is that the solution of a triangular set of equations is quite trivial. \(\text{L} \cdot \text{y} = \text{b}\) can be solved by forward-substitution and \(\text{U} \cdot \text{x} = \text{y}\) by back-substitution”.
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Biancolini, M.E. (2017). Fast RBF. In: Fast Radial Basis Functions for Engineering Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-75011-8_3
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DOI: https://doi.org/10.1007/978-3-319-75011-8_3
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