Mathematics / advanced
Inverse problems and regularization
Recover hidden quantities from indirect observations while confronting instability, identifiability, noise, and prior structure.
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Module 01 advanced
Inverse problems and regularization
Recover hidden quantities from indirect observations while confronting instability, identifiability, noise, and prior structure.
- 01
Forward maps, identifiability, and ill-posedness
Separate existence, uniqueness, and stability before choosing a reconstruction algorithm or a prior.
29 min - 02
Singular values and the Picard condition
Characterize the range of a compact operator using coefficient summability, including its orthogonality and nullspace requirements.
31 min - 03
Tikhonov regularization and filter functions
Derive the quadratic estimator, its singular-value filter, and the assumptions behind a bias–noise convergence rate.
32 min - 04
The discrepancy principle and parameter choice
Choose a regularization level using an explicit noise bound and distinguish a residual target from an object-error guarantee.
31 min - 05
Iterative regularization and Landweber iteration
Treat stopping time as a regularization parameter and distinguish residual descent from reliable recovery of weak modes.
32 min - 06
Sparsity and total variation regularization
Choose what should be sparse, derive thresholding and a two-pixel TV solution, and expose the bias introduced by structural preferences.
32 min - 07
Bayesian inverse problems and posterior contraction
Compute a posterior distribution while distinguishing its model-dependent uncertainty from identifiability and repeated-data contraction.
33 min - 08
Tomography and Radon transforms
Derive the Fourier-slice relation and its inversion constant, then distinguish incomplete angular data from a simple finite nullspace story.
33 min - 09
Nonlinear inversion and Gauss–Newton
Regularize a local linearized step and distinguish local derivative information from global identifiability and convergence.
32 min - 10
Uncertainty quantification and resolution limits
Separate resolution bias, repeated-noise variation, and posterior spread, then derive a deterministic indistinguishability bound.
34 min