## Section: New Results

### Efficient use of sparsity by direct solvers applied to 3D controlled-source EM problems

Controlled-source electromagnetic (CSEM) surveying
becomes a widespread method for oil and gas exploration,
which requires
fast and efficient software for inverting large-scale EM datasets.
In this context, one often needs to solve
sparse systems of linear equations
with a *large* number of *sparse* right-hand sides,
each corresponding
to a given transmitter position.
Sparse direct solvers are very attractive for these problems,
especially when combined with
low-rank approximations which significantly reduce the complexity and the cost of the
factorization.
In the case of thousands of right-hand sides,
the time spent in the sparse triangular solve
tends to dominate the total simulation time and here we propose several approaches to reduce it.
A significant reduction is demonstrated for marine CSEM application
by utilizing the sparsity of the right-hand sides (RHS) and of the solutions that
results from the geometry of the problem.
Large gains are achieved by restricting computations at the forward substitution
stage to exploit the fact that the RHS matrix might have empty rows (*vertical sparsity*)
and/or empty blocks of columns within a non-empty row (*horizontal sparsity*).
We also adapt the parallel algorithms that were designed for the factorization to solve-oriented
algorithms and describe performance optimizations particularly relevant for
the very large numbers of right-hand sides of the CSEM application.
We show that both the operation count and the elapsed time for the solution phase can be significantly reduced.
The total time of CSEM simulation can be divided by approximately
a factor of 3 on all the matrices from our set
(from 3 to 30 million unknowns, and from 4 to 12 thousands RHSs).

These findings are described in a technical report [37] and will be submitted for publication.