##
**Meshfree approximation methods with Matlab. With CD-ROM.**
*(English)*
Zbl 1123.65001

Interdisciplinary Mathematical Sciences 6. Hackensack, NJ: World Scientific (ISBN 978-981-270-633-1/hbk; 978-981-270-634-8/pbk). xviii, 500 p. (2007).

The book is a new contribution to the theory and the applications of approximation theory and interpolation in multiple dimensions. It is written by a well-known expert in the area. Its title refers explicitly to the applications by Matlab routines.

This book is indeed an excellent contribution to the numerical analysis of approximations which includes both theory and Matlab algorithms for the purpose of approximation without meshes. Much of the historical development of and the topical themes in meshless approximations is linked to the theory and practice of radial basis functions, a most successful tool in modern approximation theory and computational mathematics. The key point here is that most polynomial and piecewise polynomial approaches require the generation of meshes for their piecewise structure. Theoretically this is very useful, and many wonderful results are known about this approach, and it is so also in practice – unless the dimension of the underlying space becomes large. This is precisely the point where meshless methods come into action.

The author’s own research background is closely related to the radial basis function story and this fact is clearly reflected in his choice of topics in the book.

Generally, this huge subject-area is covered so comprehensively that we cannot mention all what is handled in the book, but we shall write about the most important areas which are contained in the volume.

It begins with the theory of radial basis functions and – more generally – positive definite (kernel) functions with many useful examples provided. Most radial basis functions came into the focus as completely monotone functions when used for scattered data interpolation and this is his next subject. Much of the theoretical interest in radial basis functions and many results reply on the idea of complete monotonicity.

The usefulness of the radial basis function approach is then greatly enlarged by admitting conditionally positive definite functions which include such ubiquitous functions like the multiquadrics or the linear radial basis function or the so-called thin-plate spline, one of the first radial basis functions used in practice and analysed by Jean Duchon. Before continuing with the theory of reproducing kernels and then with convergence proofs and error bounds, he also deals with the compactly supported radial basis functions. This is a property which turns out to be useful from time to time when radial basis functions are in action, which is not fulfilled by either multiquadrics, linears or by thin-plate splines.

The error estimates and convergence results in the classical way require the power function, native spaces and stability issues. These complicated issues are dealt with in the book accurately and well-presented.

Not only interpolation is employed in the context of radial basis functions, but also least-squares approximations, and the author is currently a much asked-for expert in the moving least-squares community. Therefore both least-squares and moving least-squares are also featured prominently in the book.

Much like moving-least squares are important for applications in solving partial differential equations numerically by radial basis function approaches, so are partition of unity methods and adaptive methods. Both are also treated in the book. It should be mentioned that there is a CD included with the book where very many matlab routines are provided for illustration of the algorithms as well as for practical use of the reader. This is an integral part of the book as the title indicates. The same is true for the already mentioned collocation methods as well as for the spectral methods and the standard Galerkin ansatz for solving partial differential equations.

The discussion of the Garlerkin ansatz for radial basis functions and numerical solutions of partial differential equations is the topic of the last chapter with which the book ends (plus an appendix with, e.g., a list of commonly used radial basis functions and derivarives). It combines to a most competently written piece of work which contains theoretical treatment of the radial basis functions comme il faut and interesting, useful practical work with matlab software provided. The author can only be congratulated for this work which will provide useful material for readers for years to come.

This book is indeed an excellent contribution to the numerical analysis of approximations which includes both theory and Matlab algorithms for the purpose of approximation without meshes. Much of the historical development of and the topical themes in meshless approximations is linked to the theory and practice of radial basis functions, a most successful tool in modern approximation theory and computational mathematics. The key point here is that most polynomial and piecewise polynomial approaches require the generation of meshes for their piecewise structure. Theoretically this is very useful, and many wonderful results are known about this approach, and it is so also in practice – unless the dimension of the underlying space becomes large. This is precisely the point where meshless methods come into action.

The author’s own research background is closely related to the radial basis function story and this fact is clearly reflected in his choice of topics in the book.

Generally, this huge subject-area is covered so comprehensively that we cannot mention all what is handled in the book, but we shall write about the most important areas which are contained in the volume.

It begins with the theory of radial basis functions and – more generally – positive definite (kernel) functions with many useful examples provided. Most radial basis functions came into the focus as completely monotone functions when used for scattered data interpolation and this is his next subject. Much of the theoretical interest in radial basis functions and many results reply on the idea of complete monotonicity.

The usefulness of the radial basis function approach is then greatly enlarged by admitting conditionally positive definite functions which include such ubiquitous functions like the multiquadrics or the linear radial basis function or the so-called thin-plate spline, one of the first radial basis functions used in practice and analysed by Jean Duchon. Before continuing with the theory of reproducing kernels and then with convergence proofs and error bounds, he also deals with the compactly supported radial basis functions. This is a property which turns out to be useful from time to time when radial basis functions are in action, which is not fulfilled by either multiquadrics, linears or by thin-plate splines.

The error estimates and convergence results in the classical way require the power function, native spaces and stability issues. These complicated issues are dealt with in the book accurately and well-presented.

Not only interpolation is employed in the context of radial basis functions, but also least-squares approximations, and the author is currently a much asked-for expert in the moving least-squares community. Therefore both least-squares and moving least-squares are also featured prominently in the book.

Much like moving-least squares are important for applications in solving partial differential equations numerically by radial basis function approaches, so are partition of unity methods and adaptive methods. Both are also treated in the book. It should be mentioned that there is a CD included with the book where very many matlab routines are provided for illustration of the algorithms as well as for practical use of the reader. This is an integral part of the book as the title indicates. The same is true for the already mentioned collocation methods as well as for the spectral methods and the standard Galerkin ansatz for solving partial differential equations.

The discussion of the Garlerkin ansatz for radial basis functions and numerical solutions of partial differential equations is the topic of the last chapter with which the book ends (plus an appendix with, e.g., a list of commonly used radial basis functions and derivarives). It combines to a most competently written piece of work which contains theoretical treatment of the radial basis functions comme il faut and interesting, useful practical work with matlab software provided. The author can only be congratulated for this work which will provide useful material for readers for years to come.

Reviewer: Martin D. Buhmann (Gießen)

### MSC:

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

41A15 | Spline approximation |

65D05 | Numerical interpolation |

65D07 | Numerical computation using splines |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

41A05 | Interpolation in approximation theory |

41A63 | Multidimensional problems |

41A30 | Approximation by other special function classes |