Multi-resolution and multi-frequency FWI¶

Multi-resolution FWI changes the model grid; multi-frequency continuation changes the observations included in the objective. SimpleFWI's MultiscaleInversion schedules either or both. A typical sequence begins with a coarse, low-frequency problem, refines the physical model and then introduces additional frequencies.

Run the three-stage Marmousi example¶

From an installed checkout:

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 \
  python examples/fwi2d/multiscale_marmousi.py \
  --device cpu --quick --output /tmp/multiscale-cpu.json

The quick schedule uses a small crop of the bundled Marmousi data:

For CUDA, install the matching CUDA PyTorch wheel and require an actual device:

python -c "import torch; assert torch.cuda.is_available(), 'CUDA device required'"
OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 \
  python examples/fwi2d/multiscale_marmousi.py \
  --device cuda --quick --output /tmp/multiscale-cuda.json

The JSON output records model/data hashes, stage losses, physical PML widths, timing and device memory. The example uses synthetic fine-grid observations and short, uncalibrated PML. It is an integration experiment, not demonstrated full-model recovery.

What happens between stages?¶

1. Validate the full schedule. Stages must preserve physical node extent, origin, acquisition coordinates and top-boundary type. Refinement is coarse-to-fine; the highest frequency cannot decrease. Dispersion is screened at the minimum declared velocity before optimization begins.
2. Transfer physical squared slowness. The accepted m = 1 / velocity**2 is interpolated with physical nodes aligned. Same-grid stages preserve accepted velocity values exactly. Observations are selected by frequency; measured data are not spatially interpolated.
3. Create fresh stage state. The model, solver, optimizer and scheduler belong to that stage. Adam moments, momentum and L-BFGS history do not cross grid or objective transitions.
4. Keep memory bounded by declared operations. Retain scalar stage history and the final runner, clear obsolete solver caches, and process source batches sequentially. Observation copies and optimizer history still contribute to memory use.

Prepare the first coarse initial model explicitly. The workflow rejects coarsening and does not silently decimate a fine model. Halving spacing requires doubling PML cells to preserve physical width. Refinement creates computational resolution, not new geological information.

Use native PyTorch optimizers¶

Each stage accepts an optimizer factory, for example:

import torch

optimizer_factory = lambda model, stage: torch.optim.Adam(
    model.parameters(), lr=0.03
)

Use this factory in the complete stage-declaration example. The inversion runner supplies the objective and gradients; native torch.optim controls updates. Fifteen CPU optimizer recipes are checked, including Adam, RMSprop, L-BFGS, Muon and SparseAdam with their documented shape and gradient constraints. That does not establish every optimizer configuration on CUDA. The optimizer guide states supported contracts and examples.

Optional factories declare masks, fixed physical priors and stage-local recorders. Use separate output paths for each stage and close writers explicitly. The final runner supports curvature analysis at the final objective; its Hessian is not a derivative through the entire optimization trajectory.

Interpret a continuation result¶

Adding frequencies changes the objective. Compare initial and final loss within each stage, not raw losses across different bands. In the recorded CPU/A40 comparison, every stage's final loss was lower than its initial loss, but final model RMSE increased from 33.41 to 34.84 m/s. Individual Adam steps also need not decrease monotonically. Lower data misfit alone does not establish better geological recovery.

See validation and hardware evidence for the exact revision, CPU/GPU parity and timing. The full multiscale guide documents frequency weights, transfer roundoff, priors, masks, schedulers and failure handling.