Regularization parameter choice

A regularization parameter choice selects the amount of stabilization from the noise level, the measured data, or both. For Tikhonov regularization with deterministic error at most $\delta$, a sufficient a priori condition for convergence is $\alpha (\delta )\to 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$.

An error bound $A\delta /\alpha +Bc\alpha$, with positive $A,B,c$, is minimized at ${\alpha }_{\ast }=\sqrt{A\delta /(Bc)}$, with value $2\sqrt{ABc\delta }$. Differentiate the scalar bound, or apply the arithmetic-geometric mean inequality. This is an a priori regularization parameter choice for a linear regularization with inverse-parameter noise amplification and first-order approximation bias; the parameter must also lie in the method's allowed interval.

An a priori regularization parameter choice depends on a known noise bound and fixed problem information, but not on the observed values of the noisy data.