Non linear moving least squares and applications
- Autores: José Manuel Ramón García
- Directores de la Tesis: Dionisio F. Yáñez Avendaño (dir. tes.)
- Lectura: En la Universitat de València ( España ) en 2025
- Idioma: español
- Tribunal Calificador de la Tesis: Pep Mulet Mestre (presid.), Juan Ruiz Álvarez (secret.), María Santágueda Villanueva (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universitat de València (Estudi General) y la Universitat Politècnica de València
- Materias:
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- Resumen
1.1 The Weighted Essentially Non-Oscillatory Shepard method This section is dedicated to constructing a non-linear Shepard's method.
We build upon the fundamental principles of the Weighted Essentially Non-Oscillatory (WENO) interpolation method to achieve this. The primary aim of our proposed method is to enhance the accuracy and mitigate the smearing effect inherent in the traditional Shepard's method. We accomplish this by integrating WENO's adaptive and non-linear weighting mechanism. Crucially, our approach involves a non-linear modification of the general weight function in Shepard's method, offering flexibility beyond simply relying on the inverse of the squared distance. This approach effectively reduces the smearing of discontinuities providing a sharper approximation. The numerical experiments presented demonstrate the superior performance of the new method close to the discontinuities and confirm the theoretical results exposed in it.
1.2 Data-Dependent Moving Least Squares method This section addresses a data-dependent modification of the moving least squares (MLS) problem. We propose a novel approach by replacing the traditional weight functions with new functions. These new functions assign smaller weights to nodes that are close to discontinuities, while still assigning smaller weights to nodes that are far from the point of approximation. This adjustment allows us to mitigate the undesirable Gibbs phenomenon near discontinuities, a common issue in the classical MLS approach, and reduce the smearing of these discontinuities in the final data approximation. The core of our method resides in the precise identification of affected nodes through the use of smoothness indicators, a technique adapted from the data-dependent WENO (DD-WENO) approach.
Our formulation results in a data-dependent weighted least squares problem where the weights depend on two factors: the distances between nodes and the point of approximation, and the smoothness of the data in a region of predetermined radius around the nodes. We explore the design of the new data-dependent approximant, analyze its properties including polynomial reproduction, accuracy, and smoothness, and study its impact on diffusion and the Gibbs phenomenon.
Numerical experiments are conducted to validate the theoretical findings.
1.3 Non-linear Partition of Unity method This section introduces the Non-linear Partition of Unity Method (NL-PUM), a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. As far as we know, this is the first time that a non-linear PUM method is introduced in the literature. The main advance of this proposal is providing an algorithm that keeps the properties of the PUM at smooth zones, while introducing a non-linear modification close to the discontinuities to avoid oscillations. This is done automatically by computing estimations of the smoothness of the data and replacing the PUM by Sheppard method when all the data is affected by a discontinuity. Thus, the computation of smoothness indicators and the use of compactly supported base functions ensure precision in regions affected by the presence of discontinuities. Error bounds are calculated and validate the effectiveness of the new method, showing improved interpolation capabilities at discontinuity regions as well as at smooth zones. A series of experiments are presented to check the theoretical results provided.
