L-curve

Remarks

The residual norm $\norm{A x_\lambda - b}$ is non-decreasing in $\lambda$, and the seminorm $\norm{L x_\lambda}$ is non-increasing. So the curve runs from upper left (small $\lambda$, noise-dominated) to lower right (large $\lambda$, over-smoothed). Example: for $A = L = I$ and $b \ne 0$, one gets $x_\lambda = b/(1 + \lambda^2)$. Then $\norm{A x_\lambda - b} = \lambda^2 \norm{b}/(1 + \lambda^2)$ and $\norm{x_\lambda} = \norm{b}/(1 + \lambda^2)$, so the first increases and the second decreases with $\lambda$, as stated. Non-example: the curve $\lambda \mapsto (\lambda, \norm{A x_\lambda - b})$ is not the L-curve, since it plots the parameter itself instead of the solution seminorm. Some authors use the squared norms or linear axes, but the corner is clearest on log-log axes. The criterion is a heuristic: it is not a convergent regularization parameter choice in the worst-case sense, yet it works well on many practical problems.