Efficient Simulation of Multiple Sources Time-Harmonic Waves With Substructured Domain Decomposition Methods And Block Krylov Methods

Full Waveform Inversion in the frequency domain requires the resolution of sequences of Helmholtz-
like problems, each one of them with many right-hand sides (i.e. many sources). For large-scale 3D
problems, Domain Decomposition Methods are a popular choice when sparse direct solvers become
too expensive, but usual Krylov methods do not handle multiple right-hand sides efficiently.
Coupling Optimized Restrictive Additive Schwarz with Block Krylov Methods (e.g. Block GM-
RES) has proven to significantly reduce the iteration count [1], but with an overhead that mitigates
these benefits for large blocks. In this work, we investigate similar ideas for non-overlapping methods
that solve a substructured problem, i.e. with unknowns on the subdomain interfaces.
While this approach has less favorable convergence properties, it can handle larger blocks due to the
reduced size of the vectors managed by the Krylov method. This property makes substructured non-
overlapping methods particularly attractive when many sources are involved. Using PETSc profiling
tools and the PETSc interface to HPDDM, which implements Block Krylov methods, we aim to
compare these approaches by comparing the convergence rates of the methods with and without
blocks, as well as the arithmetic costs of using block versions, in realistic test cases. In particular,
we want to determine whether to use large blocks with substructured methods outweighs their slower
convergence.