= Lavrentiev regularization
{c}
{title2=$x_\alpha^\delta=(A+\alpha I)^{-1}y^\delta$}
For a positive self-adjoint <linear operator> $A$, Lavrentiev regularization shifts $A$ itself rather than $A^*A$. 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 $\|x_\alpha^\delta-x\|\leq\delta/(\lambda_1+\alpha)+\alpha\|x\|/(\lambda_1+\alpha)\leq\delta/\alpha+\alpha\|x\|/\lambda_1$. A known bound $X\geq\|x\|$ 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.
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