Discretization methods for the Lavrent’ev regularization

Abstract

In this paper we consider an ill-posed operator equation Lu = f in a Banach space X where L is a linear compact operator in X from itself and there exists a constant C > 0 such that $\left\| {{{{(\rho + L)}}^{{ - 1}}}} \right\| \leqslant \tfrac{C}{{{{\rho }^{k}}}}$ for some k ∈ Nand small p > 0. Let fδ be available data and polluted with noise level $\delta > 0:\left\| {f - {{f}_{\delta }}} \right\| \leqslant \delta$ ≤ δ. For reconstruction of u from f _δ , we adopt a discretization of the Lavrent’ev regularization: solve u _p , _δ , _h in

$$ \rho u + {{P}_{h}}Lu = {{P}_{h}}{{f}_{\delta }} $$

where p > 0 is a parameter and {P _h} _h>0 is a family of projections used for the discretization with discretization parameter h. With suitable a-priori choice of p and h for δ, we will establish optimal convergence rates of u _P , _δ , _h towards u as _δ → 0. Furthermore we discuss a similar regularization in the case where f _δ is replaced by discretely observated data f _{δ h} . Finally we show that the Lavrent’ev regularization can suppress the condition numbers of the resulting discretized problems.