SimpleFWI

FWI Hessian and Gauss–Newton uncertainty

SimpleFWI computes exact and Gauss–Newton Hessian-vector products without assembling a dense Hessian. These actions support curvature diagnostics and local uncertainty calculations around a fixed physical model. The resulting uncertainty depends on declared noise scaling, parameter coordinates, a proper prior and the local approximation.

Exact Hessian versus Gauss–Newton Hessian

For a weighted least-squares data objective, Gauss–Newton curvature uses the Jacobian action and its adjoint, Re(Jᴴ Wᴴ W J). The exact Hessian additionally includes residual-dependent second derivatives. It can be indefinite away from a suitable local minimum. They answer different questions and are not interchangeable positive precision operators.

Curvature freezes a physical reference m = 1 / v². Its exact_hvp and gn_hvp accept real vectors over an explicit active mask in [z, x] order. TransformedCurvature labels bounded parameter-coordinate actions separately; exact transformed curvature includes the nonlinear map's second derivative. Recreate curvature after changing the reference model or objective.

Use the curvature implementation and public API and uncertainty guide for coordinate conventions and complete examples.

What does local Gaussian uncertainty require?

LocalGaussian applies the inverse of data-only Gauss–Newton curvature plus an explicitly supplied proper prior precision, counted once. Objective weights must whiten the declared observational noise for a statistical interpretation. Numerical rescaling of the misfit changes the covariance.

  • A finite strictly positive diagonal prior is mandatory; optional smoothing may be added with a declared adjoint. No damping or pseudoinverse is silently inserted.
  • Covariance actions use preconditioned conjugate gradients with convergence checks and true residual reports. Nonconvergence and encountered nonpositive directions raise errors.
  • Streaming Rademacher probes estimate the diagonal of the inverse precision, which generally differs from the reciprocal precision diagonal. Sampling standard errors are reported separately from solver errors.
  • Bounded prior-metric Lanczos exploration returns information modes and Ritz residuals. A truncated basis does not certify the full global leading spectrum.

Physical squared-slowness covariance has units s⁴/m⁴. It is not velocity variance or unconstrained optimizer-parameter variance. Local Gaussian results do not capture all nonlinear effects, multimodality, unmodeled physics or uncertain noise and prior hyperparameters.

Verify the algebra on Camembert

From an installed checkout:

python examples/fwi2d/camembert_uncertainty.py --probes 8

The reference example uses a 48 × 48 Camembert domain at 5 Hz, eight surrounding sources, 32 receivers and a 2 × 2 active parameter window. Four active parameters permit a guarded explicit 4 × 4 covariance comparison while retaining the wave physics on the full grid. Dense assembly is a small-test diagnostic, not the production strategy for a large model.

This example uses same-grid synthetic observations and an initial homogeneous reference, not an optimized MAP estimate. It verifies local operator algebra; it does not demonstrate calibrated geological uncertainty or reconstruction. Read the Camembert Hessian and uncertainty notebook for figures and exact/Gauss–Newton comparisons, then the validation record for evidence boundaries.