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Approximate continuation of harmonic functions in geodesy : a weighted least-squares approach based on splines with extension to the multiscale adaptive case / Gabriela Jager. 2010
Inhalt
Introduction
Motivation
Structure of the Work
Harmonic Functions
Preliminaries
Some Basic Notation
Harmonic Functions in Natural Sciences
Spherical Harmonics
Orthogonal Expansions: Laplace Series
The Earth's Gravitational Field
Preliminaries
Fundamental Relationships
Gravity Acceleration and Gravity Potential
Differential and Integral Formulas for the Gravity Potential
Geometry of the Gravity Field
Classical Gravity Field Model: The Spherical Harmonics Representation
Determination of the Model
Coefficients Significance
Convergence Issues
Deficiencies of the Spherical Harmonic Representation
Boundary Problems in Geodesy
Classical Upward Continuation of the Potential Field
Interpretation and Non-uniqueness of Gravity Data
New Approach: Least Squares with Regularization
General Setup, Justification
Construction of the Solution
Data Fitting
Enforcing Harmonicity
Resulting System of Equations
Concrete Choices
B-Splines
Tensor Product Spaces
Tensor Product Splines
Computing the Solution
Alternative Formulations for Comparison Purposes
Finite Elements
Finite Differences
Two Dimensional Illustration of the Method
From the Boundary Data to the System of Equations
Variation of the Weight Parameter vs. Boundary Data Configurations
Variation of the Basis Cardinality vs. Strongly Incomplete Boundary Data
Numerical Results
Algorithm
Numerical Setup
Input
Tensor Product of B-Splines Basis
Measures of Error and Validation
Experiments with Synthetic Harmonic Functions
Test Data
Variation of the Boundary Data Set
Comparison to Finite Differences
Variation of the Weight Parameter
Experiments with Earth's Potential Field Data
Some Approximate Continuations
Comparison to Finite Differences and Finite Elements
Constraints of the Least-Squares-Approach for Potential Field Datasets
Estimating the Weight Parameter
Classical Regularization Strategies
Estimators Based on the System Matrices
Final Strategy
Working with Iterative Solvers
Adaptivity: Experiments with the Hierarchical Cubic B-spline Basis
Prerequisites
Background on Multivariate and Multiscale Constructions
Multiscale Constructions
Monovariate Hierarchical and Wavelet Basis
Tensored Nested Spaces
Stability Issues
General Adaptive Set-up
Coarse--to--Fine Strategy
Test Data
3D Grid Visualization
Refinement Process
Alternative Refinement Strategies
The Isotropic Refinement
The Sparse Refinement
Comparative Draw
Conclusions
Separable Solution of the Laplace Equation
Notation
Bibliography
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