Browsing by Author "de Hoop, Maarten V"

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    Discontinuous Galerkin method with a modified penalty flux for the modeling of acousto-elastic waves, coupled to rupture dynamics, in a self gravitating Earth
    (2018-03-29) Ye, Ruichao; de Hoop, Maarten V
    We present a novel method to simulate the propagation of seismic waves in realistic fluid-solid materials, coupled with dynamically evolving faults, in the self-gravitating prestressed Earth. A discontinuous Galerkin method is introduced, with a modified penalty numerical flux dealing with various boundary conditions, in particular with discontinuities. This numerical scheme allows general heterogeneity and anisotropy in the materials, by avoiding the diagonalization into polarized wave constituents such as in the approach based on solving elementwise Riemann problems, while maintains the numerical accuracy with mesh and polynomial refinements. We also include the interior slip boundary conditions for dynamic ruptures coupling with nonlinear friction laws, as an approach to simulate spontaneously cracking faults. We show the well-posedness for the system of particle motion coupled with gravitation field and its perturbation, by proving the coercivity of the bilinear operator, both in the continuous and discretized polynomial space, and therefore the convergence results. A multi-rate iterative scheme is proposed to address the challenging of solving the large implicit nonlinear system, and to allow different time steps for distinct physical processes in the overall coupling problem. We give rigorous proof for the well-posedness of mathematical model and moreover the stability of the numerical methods. Numerical experiments show the convergence as well as robustness in both well-established benchmark examples and realistic simulations.
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    Structured matrix algorithms for solving Helmholtz equation
    (2019-04-12) Liu, Xiao; de Hoop, Maarten V
    In this dissertation, novel solution algorithms are developed for large structured linear systems such as the discretized Helmholtz equation. Firstly, we develop two parallel randomized algorithms for constructing hierarchically semiseparable (HSS) matrices. Randomized sampling reduces the computational complexity and the communication cost simultaneously, and the resulting method is suitable for solving dense systems arising from multiple scattering problems. Secondly, a direct factorization update algorithm is proposed for solving the Helmholtz equation with changing wavespeed. The data dependency among a set of interior and exterior sub-problems is exploited to maximize the data reuse and to minimize the propagation of changes. Thirdly, interconnected HSS structures are designed to improve the efficiency of rank-structured factorization of PDE problems. Intermediate Schur complements at different levels share the same set of HSS bases during the factorization. Finally, we propose a contour integration preconditioner for solving 3D high-frequency Helmholtz equation. By solving systems with complex shifts, the problem is projected in subspaces with faster GMRES convergence. The shifted problems are solved by a polynomial fixed-point iteration, which is robust when the shift changes.