Massively parallel computations for the solution of the 3D-Helmholtz equation in the frequency domain.

The topic of our work is the solution of the three-dimensional Helmholtz equation in the frequency<br>domain on massively parallel computers modeled by the following partial differential<br>equation: (view PDF), with some absorbing boundary conditions, where u is the pressure of the wave, f its frequency,<br>c the propagation velocity of the subsurface and g is a Dirac function that represents the wave<br>source in the frequency domain. This equation is involved in an inverse problem modelling a<br>wave propagation under Earth. The solution of this inverse problem enables geophysicists to<br>deduce from experimental data the structure of the subsoil. An explicit solution method (time<br>domain) is often considered because it keeps the memory need acceptable. But working in time<br>domain supposes that stability conditions on the discretization scheme hold both in time and<br>space, that often leads to very small time steps (i.e. large simulation time) for real problems.<br>One of the great advantage of the frequency domain formulation is that stability conditions of<br>the discretization scheme only rely on the frequency. The frequency formulation is yet much<br>greedier in memory than the time one’s, because standard discretization methods such as finite<br>difference and finite element methods leads to linear systems of size depending linearly on the<br>frequency. Nevertheless, according to recent trends concerning massively parallel architectures,<br>solving the implicit Helmholtz equation seems now feasible, because large distributed memories<br>and efficient interconnect become available. We shall show that linear systems of size more than<br>one billion can be solved by present supercomputers in a few minutes.