- Weighted Particles in the Finite Pointset method (1991)
- Using particle methods to solve the Boltzmann equation for rarefied gases numerically, in realistic streaming problems, huge differences in the total number of particles per cell arise. In order to overcome the resulting numerical difficulties the application of a weighted particle concept is well-suited. The underlying idea is to use different particle masses in different cells depending on the macroscopic density of the gas. Discrepance estimates and numerical results are given.
- Tensor Spherical Harmonics and Tensor Spherical Splines (1993)
- In this paper, we deal with the problem of spherical interpolation of discretely given data of tensorial type. To this end, spherical tensor fields are investigated and a decomposition formula is described. Tensor spherical harmonics are introduced as eigenfunctions of a tensorial analogon to the Beltrami operator and discussed in detail. Based on these preliminaries, a spline interpolation process is described and error estimates are presented. Furthermore, some relations between the spline basis functions and the theory of radial basis functions are developed.
- Spherical panel clustering and its numerical aspects (1997)
- In modern approximation methods linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulas on the uni sphere omega corresponding to prescribed nodes, spherical spline interpolation, and spherical wavelet approximation. the evaluation of such a linear combination is a time consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. This paper presents a generalization of the panel clustering method in a spherical setup. The economy and efficiency of panel clustering is demonstrated for three fields of interest, namely upward continuation of the earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.
- Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere (1996)
- Based on a new definition of delation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in case of band-limited wavelets.
- On the Local Multiscale Determination of the Earth`s Disturbing Potential From Discrete Deflections of the Vertical (2007)
- As a first approximation the Earth is a sphere; as a second approximation it may be considered an ellipsoid of revolution. The deviations of the actual Earth's gravity field from the ellipsoidal 'normal' field are so small that they can be understood to be linear. The splitting of the Earth's gravity field into a 'normal' and a remaining small 'disturbing' field considerably simplifies the problem of its determination. Under the assumption of an ellipsoidal Earth model high observational accuracy is achievable only if the deviation (deflection of the vertical) of the physical plumb line, to which measurements refer, from the ellipsoidal normal is not ignored. Hence, the determination of the disturbing potential from known deflections of the vertical is a central problem of physical geodesy. In this paper we propose a new, well-promising method for modelling the disturbing potential locally from the deflections of the vertical. Essential tools are integral formulae on the sphere based on Green's function of the Beltrami operator. The determination of the disturbing potential from deflections of the vertical is formulated as a multiscale procedure involving scale-dependent regularized versions of the surface gradient of the Green function. The modelling process is based on a multiscale framework by use of locally supported surface curl-free vector wavelets.
- On a New Condition for Strictly Positive Definite Functions on Spheres (1995)
- Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper, that even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results ofXu and Cheney.
- Nonorthogonal Expansions on the Sphere (1993)
- Discrete families of functions with the property that every function in a certain space can be represented by its formal Fourier series expansion are developed on the sphere. A Fourier series type expansion is obviously true if the family is an orthonormal basis of a Hilbert space, but it also can hold in situations where the family is not orthogonal and is overcomplete. Furthermore, all functions in our approach are axisymmetric (depending only on the spherical distance) so that they can be used adequately in (rotation) invariant pseudodifferential equations on the frames (ii) Gauss- Weierstrass frames, and (iii) frames consisting of locally supported kernel functions. Abel-Poisson frames form families of harmonic functions and provide us with powerful approximation tools in potential theory. Gauss-Weierstrass frames are intimately related to the diffusion equation on the sphere and play an important role in multiscale descriptions of image processing on the sphere. The third class enables us to discuss spherical Fourier expansions by means of axisymmetric finite elements.
- Multiresolution Analysis by Spherical Up Functions (2003)
- A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.
- Locally Supported Kernels for Spherical Spline Interpolation (1995)
- By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.