A Numerical framework to address the inverse problem of estimating Lamé parameters using HDG Runge-Kutta methods
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Date
2024
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Abstract
In engineering, the problem of characterizing the immediate subsoil of the earth’s crust is a subject of extensive research. A common approach for solving this problem is to induce elastic waves from the surface to the underground. So, when the propagating media is not homogeneous, we expect collisions of these waves with inclusions, represented by changes in the Lame parameters ´ (λ, µ) that uniquely determine the composition of the isotropic solid. An inverse problem arises from measuring the elastic waves that propagate back to the surface and compare with some direct model predictions. To address this problem, we rst deliver an accurate numerical characterization of the time-dependent elastic waves by solving the elastodynamics PDE using an HDG nite elements spatial discretization with weak symmetry. We also present several time-marching schemes with different orders of convergence and properties and prove optimal error estimates for every scheme, generating a wide variety of fully-discrete direct solvers. Furthermore, we provide suitable treatment for numerical artifacts, such as the locking phenomenon, and review relevant techniques to perform computational domain truncation, such as PMLs, including a damping term. Finally, we consider the inverse problem of minimizing a mis-t L2 loss between boundary measurements of the ground truth and the forward model numerical solution. We propose a rst-order algorithm that iteratively delivers descent directions by characterizing Frechet derivatives of the loss function in a discrete fashion. ´Moreover, we present a full adjoint analysis of the continuous PDE-restricted optimization problem; this provides a continuous characterization of the second-order information of the Loss function when L2 regularization is included.
Description
Tesis (Master of Science in Engineering)--Pontificia Universidad Católica de Chile, 2024.
Keywords
IInverse problems, Linear elasticity, HDG-FEM, Runge-Kutta, Multistep, First-order methods, Problemas inversos, Elasticidad lineal, HDG-FEM, Runge-Kutta, Métodos multi-paso, Métodos de primer orden