uv run python verify.py

B1 exit assessment: formulation, stability, and reduction

Time: 70 minutes, then a 5 minute live challenge.

Use the numerical lab and registered notation. Every mathematical claim must be paired with a calculation, limiting case, or counterexample.

Submission

1. Formulation and Hadamard assessment, B1.1

For the declared modal coefficient problem in the lab guide, state the spaces, admissible parameter set, source, model residual, observation map, solution map, and parameter-to-data map. Assess existence, uniqueness, and continuous dependence on those stated spaces. Do not use “ill posed” as an unqualified label.

2. Continuous versus discrete diagnosis, B1.2

At the declared reference parameter, compute normalized singular spectra and condition estimates for the reduced first-order operator on at least three discretization levels. Distinguish:

• an invertible but ill-conditioned finite system;
• evidence consistent with an unbounded inverse in the limiting problem;
• floating-point or implementation instability.

State what the finite experiment cannot prove.

3. Reduced derivative, B1.3

Derive the coupled tangent equations and the reduced parameter-to-data derivative for the same modal coefficient problem. Recover the diagonal operator used in the spectral experiment, then separate:

• terms neglected by first-order linearization;
• invertibility assumed during state elimination;
• explicit coupled structure hidden by composition.

Verify the derivative with a convergence table and state the smoothness needed for any claimed rate.

4. Parameter choice, B1.4

Freeze one regularization-parameter rule before opening the noisy linearized data. Compare its choice with one alternative on the same data. Report reconstruction, residual, nonlinear-remainder, and sensitivity evidence, then bound the claim.

Live challenge

The instructor changes the admissible set, source sequence, observation weights, discretization, state tangent, or known noise bound. Identify the smallest affected conclusion and run the decisive check.