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- Contributions of the Geomathematics Group to the GAMM 76th Annual Meeting (2005)
- The following three papers present recent developments in nonlinear Galerkin schemes for solving the spherical Navier-Stokes equation, in wavelet theory based on the 3-dimensional ball, and in multiscale solutions of the Poisson equation inside the ball, that have been presented at the 76th GAMM Annual Meeting in Luxemburg. Part A: A Nonlinear Galerkin Scheme Involving Vectorial and Tensorial Spherical Wavelets for Solving the Incompressible Navier-Stokes Equation on the Sphere The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. This article introduces a wavelet based nonlinear Galerkin method applied to the Navier-Stokes equation on the rotating sphere. In detail, this scheme is implemented by using divergence free vectorial spherical wavelets, and its convergence is proven. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations for the nonlinear interaction of three vortices are presented. Part B: Methods of Resolution for the Poisson Equation on the 3D Ball Within the article at hand, we investigate the Poisson equation solved by an integral operator, originating from an ansatz by Greens functions. This connection between mass distributions and the gravitational force is essential to investigate, especially inside the Earth, where structures and phenomena are not sufficiently known and plumbable. Since the operator stated above does not solve the equation for all square-integrable functions, the solution space will be decomposed by a multiscale analysis in terms of scaling functions. Classical Euclidean wavelet theory appears not to be the appropriate choice. Ansatz functions are chosen to be reflecting the rotational invariance of the ball. In these terms, the operator itself is finally decomposed and replaced by versions more manageable, revealing structural information about itself. Part C: Wavelets on the 3–dimensional Ball In this article wavelets on a ball in R^3 are introduced. Corresponding properties like an approximate identity and decomposition/reconstruction (scale step property) are proved. The advantage of this approach compared to a classical Fourier analysis in orthogonal polynomials is a better localization of the used ansatz functions.
- Multiscale Signal-to-Noise Thresholding (2000)
- The basic idea behind selective multiscale reconstruction of functions from error-affected data is outlined on the sphere. The selective reconstruction mechanism is based on the premise that multiscale approximation can be well-represented in terms of only a relatively small number of expansion coefficients at various resolution levels. An attempt is made within a tree algorithm (pyramid scheme) to remove the noise component from each scale coefficient using a priori statistical information (provided by an error covariance kernel of a Gaussian, stationary stochastic model).