| name | inverse-problems-regularization |
| description | Formulate and solve inverse problems with stable regularization and calibrated uncertainty. Use for parameter or field recovery, deconvolution, tomography, system identification, data assimilation, Bayesian inversion, and learned inverse models. |
| license | MIT |
Inverse Problems and Regularization
Specify the inverse problem
State the forward operator, unknown, observation process, noise model, nuisance parameters, constraints, units,
and quantity actually identifiable from the data. Examine null spaces, conditioning, nonuniqueness, and model
discrepancy before selecting a solver. Distinguish recovering a latent field from predicting an outcome.
Write the complete objective and resolve whether a reported Tikhonov parameter enters as lambda or
lambda^2; include noise whitening and parameter scaling rather than relying on a method name alone.
Regularize
- Use scaling, truncated decompositions, Tikhonov penalties, sparsity, total variation, constraints, or priors only with a stated rationale.
- Choose regularization strength by a declared rule such as discrepancy, cross-validation, evidence, or an L-curve diagnostic.
For a discrepancy rule, use a whitened residual and calibrate its target to the noise law and effective degrees of freedom.
- Preserve a validation set or independent experiment; do not tune the prior or regularizer on the final evaluation data.
- Distinguish the observation being reconstructed from an external benchmark. Do not cross-validate across measurement
coordinates unless they are genuinely exchangeable; use independent or grouped acquisitions when available.
- For learned inverses, split by acquisition source, subject, time, or structure to prevent leakage through near duplicates.
Verify
- Perform synthetic recovery with known truth across noise and model-mismatch levels.
- Check forward residuals, but do not treat small residual as proof of correct recovery.
- Inspect singular values, posterior or profile geometry, resolution kernels, and parameter correlations.
- Test sensitivity to initialization, mesh, prior or penalty, regularization strength, and nuisance assumptions.
- Calibrate intervals or posterior predictive coverage and evaluate out-of-distribution failure.
Deliver
Report forward model, identifiability, data split, noise, regularizer or prior, selection rule, convergence,
recovery and forward-fit metrics, uncertainty, applicability domain, and unresolved nonuniqueness.
Source basis
Original synthesis informed by an openly licensed modern text on regularization of nonlinear inverse problems
and numerical-analysis sources recorded in ../../docs/TEXTBOOK_SOURCES.md.
