Regularization Parameter Selection via L-curve for Discrete Ill-Posed Problems¶
- Task ID:
math.ill_posed_inverse_problem - Domain:
math - Subdomain:
inverse_problems - Status:
test - Tags:
ill-posed,regularization,l-curve,tikhonov,tsvd,inverse-problems
Public Summary¶
This page is generated from task metadata and selected public-safe excerpts.
Example B1 Prompt Excerpt¶
**Role:** You are a numerical analysis engineer tasked with solving an ill-conditioned linear discrete ill-posed problem.
**Task:** Given an ill-conditioned matrix `K` (saved as `data/K.npy`) and a noisy observation vector `y` (saved as `data/y_obs.csv`), solve the linear system $K x = y$. Because the matrix is ill-conditioned and the data contain noise, the direct least squares solution will dramatically amplify the noise. Your task is to use **Tikhonov regularization** combined with the **L-curve criterion** to automatically select the optimal regularization parameter $\lambda$ and produce a stable approximate solution.
---
### I. Core Method and Formulas
1. **Tikhonov regularized solution**
For a given regularization parameter $\lambda > 0$, the solution is:
$$
x_\lambda = (K^T K + \lambda^2 I)^{-1} K^T y
Notes¶
- This page is a generated site artifact.
- Higher-level prompt details and internal benchmark specifics may remain intentionally undisclosed.