PyTikhonov

PyTikhonov is a pure-Python toolkit for Tikhonov regularization of linear inverse problems, with a focus on regularization parameter selection and GSVD-based diagnostics.

It is primarily intended for small to moderate-scale problems (e.g., $N$ up to a few thousand) where matrix-like access to $A$ and $L$ is available. For larger problems, it provides a ProjectedTikhonovFamily interface designed to be combined with iterative methods (e.g., Krylov / model reduction bases).

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Installation

pip install pytikhonov

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Features

• Core object: TikhonovFamily(A, L, b, d, ...) representing the entire family $$x_\lambda = \arg\min_x |Ax - b|_2^2 + \lambda |Lx - d|_2^2$$ for $\lambda > 0$.

• GSVD-based implementation (via easygsvd):

  • Fast evaluation of $x_\lambda$ for many $\lambda$.
  • Efficient computation of $|Ax - b|_2^2$, $|Lx - d|_2^2$, and their derivatives.
  • Support for both $\lambda$ and reciprocal parameterization $\beta = 1/\lambda$.

• Diagnostic tools:

  • Picard plot (discrete Picard condition).
  • L-curve (and curvature-based “L-corner”).
  • Monitoring function and degrees of freedom.

• Regularization parameter selection:

  • L-corner heuristic.
  • Discrepancy principle (DP).
  • Generalized cross validation (GCV).
  • Convenience function to compare all methods on a given problem.
  • Randomization experiments to study robustness of parameter choices.

• Performing projections:

  • ProjectedTikhonovFamily for reduced problems on subspaces $\underline{x} + \mathrm{col}(V)$.
  • Designed to plug into Krylov / model-reduction pipelines (e.g., GKB, Arnoldi) for large-scale problems.

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Minimal example

import numpy as np
from pytikhonov import TikhonovFamily, lcorner

# Example problem (A, L, b, d as dense arrays)
A = ...
L = ...
b = ...
d = np.zeros(L.shape[0])

# Build the Tikhonov family
tf = TikhonovFamily(A, L, b, d)

# Select lambda via the L-corner heuristic
lcorner_data = lcorner(tf)
lambdah_star = lcorner_data["lambdah"]
x_star = lcorner_data["x_lambdah"]

For full details, mathematical background, and additional examples (Picard plots, monitoring function, IRLS, projected problems, etc.), see the documentation.