Schneider, Naomi: Learning dictionaries for inverse problems on the sphereLernen von Dictionarys für inverse Probleme auf der SphäreLernen von Dictionarys für inverse Probleme auf der Sphäre. 2020
Inhalt
- Zusammenfassung
- Abstract
- Contents
- List of Figures
- List of Tables
- About dictionary learning in geomathematics
- Preparatory Work
- From geodesy to inverse problems
- Preliminaries
- Some aspects about polynomials on the sphere
- A geodetic reference model: the gravitational potential
- The exterior Dirichlet problem for satellite orbits
- An overview of inverse problems
- Particular real-valued trial functions on the sphere
- A few aspects of scalar Slepian functions
- An overview of spherical Sobolev spaces
- From radial basis functions to low pass filters
- Radial basis wavelets as band pass filters
- Inner products and upward continued values
- An algorithmic approach: matching pursuits
- A learning approach for spherical inverse problems
- An introduction to learning
- Towards learning dictionaries
- From the status-quo to the particular situation
- Optimal, near-optimal and well-working dictionaries
- An outlook
- A learning algorithm
- Idea and main structure
- Optimization problems in detail
- Formulation of parametrized optimization problems
- Regarding gradient-based optimization
- Inner products dependent on the operator
- Inner products of linear combinations of dictionary ele- ments
- Inner products of the penalty term
- Additional features
- Pseudo-codes for the LIPMP algorithms
- Theoretical aspects
- Numerical experiments
- Experiment setting in general
- The Earth Gravitational Model 2008 (EGM2008)
- The Gravity Recovery And Climate Experiment (GRACE)
- Further experiment setting
- Comparisons with the IPMP algorithms
- Previously published results
- Downward continuation of regularly distributed global data
- Downward continuation of monthly Data
- Learning a GRACE dictionary
- Further experiments with the LIPMP algorithms
- Summary
- Technical Appendix
- Computational aspects
- Point grids
- Legendre polynomials and the Clenshaw algorithm
- Associated Legendre functions and fully normalized spherical harmonics
- Aspects of optimization
- Documentation
- Bibliography
- Index