Multilevel convergence analysis : parallel-in-time integration for fluid-structure interaction problems with applications in cardiac flow modeling

Abstract

In this Ph.D. Thesis, multigrid-reduction-in-time (MGRIT) is considered as means to reduce the time-to-solution for numerical algorithms concerned with the solution of time-dependent partial differential equations (PDEs) arising in the field of fluid-structure interaction (FSI) modeling. As a parallel-in-time integration method, the MGRIT algorithm significantly increases the potential for parallel speedup by employing modern computer architectures, ranging from small-scale clusters to massively parallel high-performance computing platforms. In this work, the MGRIT algorithm is considered as a true multilevel method that can exhibit optimal scaling. Convergence of MGRIT is studied for the solution of linear and nonlinear (systems of) PDEs: from single- to multiphysics applications relevant to FSI problems in two and three dimensions. A multilevel convergence framework for MGRIT is derived that establishes a priori upper bounds and approximate convergence factors for a variety of cycling strategies (e.g., V- and F-cycles), relaxation schemes and parameter settings. The convergence framework is applied to a number of test problems relevant to FSI modeling, both linear and nonlinear as well as parabolic and hyperbolic in nature. An MGRIT variant is further proposed that exploits the time-periodicity that is present in many biomedical engineering applications, e.g., cyclic blood flow in the human heart. The time-periodic MGRIT algorithm proves capable of consistently reducing the time-to-solution of an existing simulation model with significant observed speedups.