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Daniel Potts
Name: Daniel Potts
E-mail: potts@math.mu-luebeck.de
Homepage: http://www.math.mu-luebeck.de/potts
Host institution: SINTEF Applied Mathematics (SINTEF), Norway
Period of visit: April 2001 - September 2001
During his visit, Daniel has been working on the following projects:
  • Fast Algorithms for Discrete Polynomial Transforms on arbitrary Grids

    Consider the Vandermonde-like matrix $\mbox{\boldmath {${P}$ }}:=(P_k(x_{M,l}))_{l,k=0}^{M,N}$, where the polynomials Pk satisfy a three-term recurrence relation and $x_{M,l}\in[-1,1]$ are arbitrary nodes. If Pk are the Chebyshev polynomials Tk, then $\mbox{\boldmath {${P}$ }}$ coincides with $\mbox{\boldmath {${A}$ }}_{}:=\left(T_k(x_{M,l})\right)_{l=0,k=0}^{M,N}$. We presents a fast algorithm for the computation of the matrix-vector product $\mbox{\boldmath {${P}$ }} a$ in ${\cal O}(N \log^2\!N)$ arithmetical operations. The algorithm divides into a fast transform which replaces $\mbox{\boldmath {${P}$ }} a$ with ${\mbox{\boldmath {${A}$ }}}_{} {\tilde a}$ and a fast cosine transform on arbitrary nodes (NDCT). Our considerations are completed by numerical tests.

  • Approximation of scattered data by trigonometric polynomials on the torus and the sphere

    Fast, efficient and reliable algorithms for the discrete least-square approximation of scattered points on the torus ${\cal T}^d$ and the 2-sphere ${\cal S}^2$ by trigonometric polynomials are presented. The algorithms are based on iterative CG-type methods in combination with fast Fourier transforms for nonequispaced data and fast cosine transform on arbitrary node. The emphasis is on the numerical aspects in order to solve large scale problems. Numerical examples are presented that show the efficiency of the new algorithms. The algorithms were e.g. applied to Data Sets given on the Mingle Homepage.

    Spock


    		Reconstructed topography data




    I would like to thank N. Papenberg (MU-Lübeck) for producing the animated gif.

  • Spherical Wavelets with an application in preferred crystallographic orientation

    with Helmut Schaeben and Jürgen Prestin

    Several useful representations of a function $f: \Omega_p \mapsto \mathbb R$exist which are usually related to specific purposes: (i) series expansion into spherical harmonics to do mathematics, (ii) series expansion into (unimodal) radial basis functions to do probability and statistics, (iii) series expansion into spline functions to do numerics. In many practical applications the common problem is to reconstruct an approximation of f from sampled data $({\bf r}_i, f({\bf r}_i)), i=1,\ldots,n$, with some convenient properties using one of the above representations. Their critical parameter, e.g. (i) the degree of the harmonic series expansion, (ii) the spherical dispersion of unimodal radial functions, (iii) the choice of the knots, may to some extent be adjusted to the total number and/or the geometric arrangement of the measurement locations. However, these representations are in no way involved in the sampling process itself. Spherical wavelets are well suited to render functions defined on a sphere. Moreover, it will be demonstrated that wavelets are well apt to allow for locally varying spatial resolution, thus providing a digital device to zoom into those spherical areas where the function f is of special interest. Such a device seems to be required to increase the spatial resolution by a factor of 1000 or greater locally. Thus, spherical wavelets provide the means to control the sampling process to gradually adapt automatically to a local refinement of the spatial resolution. In particular, it is shown that spherical wavelets apply to X-ray pole intensity data as well as to crystallographic orientation density functions, and that the multiscale resolution easily transfers from pole spheres to orientation space.

Publications
The following publications summarize some of the results of the research work that Daniel has participated in during his stay:
Conferences
Daniel was sponsered by MINGLE to participate in the following conferences and workshops, presenting recent research results:
  • Fast Algorithms in Mathematics, Computer Science and Engineering, South Hadley, USA, August 5-9, 2001
    giving a talk about "Fourier reconstruction of functions from their nonstandard sampled Radon transform "

  • GAMM Workshop, Numerical Linear Algebra with special emphasis on Numerical Methods for Structured and Random Matrices, Technical University of Berlin, Germany, September 7-8, 2001,
    giving a talk about "Fast Algorithms for Discrete Polynomial Transforms on arbitrary Grids "


Prepared by: SINTEF Applied Mathematics & TECHNION, Computer Science Dept.
Last update: April 24, 2003 | Vitaly Surazhsky