15/06/2023 - 11:30 Alexandre Hoffmann (CEA Grenoble) IMAG 106
Krylov methods play a major role in solving Partial Differential Equations (PDEs) that arise in classical engineering and academic problems. While much effort has been invested into developing direct methods for solving large sparse linear algebra problems, Krylov methods remain very attractive as they are scalable and relatively cheap, as regards to their memory requirements. However, Krylov methods exhibit slow convergence for difficult problems and may not always converge. Previous works have shown, in the SuiteSparse Matrix Collection that increasing the numerical precision improves the convergence rate of Krylov methods. In the current work, we evaluate the effect of variable precision on two frequency domain wave propagation problems. First in an homogeneous medium with a single source, then in a breast phantom with multiple sources positioned around the breast-like object. The PDE of interest was discretized with the Spectral Element Method (SEM) which has been widely used in both fluid dynamics and geosciences for many decades. SEM provides good accuracy with fewer degrees of freedom. However, high order SEM produces dense and poorly conditioned system of linear equations, that can be difficult to solve with Krylov methods. The single source problem is solved with both BiConjugate Gradient (BiCG) and the quasi minimal residual methods and the multiple sources problem is solved with the block-BiCG. We show that, on this relatively difficult PDE, increasing the numerical speeds up the convergence and improves residual accuracy. We should also mention that, on the second problem, the solver do not converge until we exceed quadruple precision.