Accelerating Spectral Elements Method with Extended Precision: A Case Study

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.