 Lavrentiev regularization

For a positive self-adjoint linear operator $A$, Lavrentiev regularization shifts $A$ itself rather than $A^{\ast }A$. In a finite-dimensional positive-definite problem, writing $y^{\delta }=Ax+e$ gives error coefficients $(e_i-\alpha x_i)/({\lambda }_i+\alpha )$ in its orthonormal eigenbasis. Consequently $\Vert x_{\alpha }^{\delta }-x\Vert \le \delta /({\lambda }_1+\alpha )+\alpha \Vert x\Vert /({\lambda }_1+\alpha )\le \delta /\alpha +\alpha \Vert x\Vert /{\lambda }_1$. A known bound $X\ge \Vert x\Vert$ permits $\alpha =\sqrt{\delta {\lambda }_1/X}$, giving a robust bound $2\sqrt{\delta X/{\lambda }_1}$. For fixed positive ${\lambda }_1$, the sharper finite-dimensional estimate can give a linear noise rate with a smaller parameter; regularization is then a conditioning choice, not a cure for nonexistence of the inverse.