IWR Colloquium Summer Semester 2026 Scalable Domain Decomposition Solvers for Frequency-Domain Full Waveform Inversion

Full Waveform Inversion (FWI) is a high-resolution imaging technique that reconstructs the internal properties of a heterogeneous medium — for example, the geological structure of the Earth's subsurface — by simulating wave propagation and matching the simulated wavefield to recorded measurements. Each emitting source (e.g., a seismic shot) gives rise to a separate wave-propagation problem, and realistic three-dimensional applications routinely involve hundreds to thousands of such sources, making FWI one of the most computationally demanding inverse problems in scientific computing.

In the frequency-domain formulation, each forward problem reduces to a large, sparse, complex symmetric indefinite linear system — a class notoriously difficult for iterative solvers. Sparse direct solvers such as MUMPS are well suited to this multi-source setting, since a single factorization can be reused across all right-hand sides; however, their memory and computational cost become prohibitive at the resolutions required for modern 3D imaging. Domain Decomposition Methods (DDM) offer a scalable alternative, splitting the global problem into smaller subproblems that can be solved concurrently, with a much smaller memory footprint and natural parallelism.

We present recent results comparing overlapping and non-overlapping DDM formulations, together with an evaluation of recent spectral coarse-grid methods on large-scale benchmarks. The latter substantially improve convergence at the price of a non-negligible setup cost, which can be amortized across right-hand sides and is therefore well suited to FWI. We further examine the interaction between the linear solver and the outer optimization scheme: truncated-Newton methods require solving the forward problem for considerably more right-hand sides than classical quasi-Newton schemes, so an effective coarse-grid preconditioner can broaden the range of optimization strategies that remain viable at large scale.