Lavrentiev Regularization methods for Ill-posed equations

Abstract

In this chapter, we consider the problem of approximately solving the nonlinear ill-posed operator equation of the form F(x) = y, (9.1) where F : D(F) ⊂ X → X is a monotone operator and X is a real Hilbert space. We denote the inner product and the corresponding norm on a Hilbert space by ⟨., .⟩ and ||.||, respectively. Let U(x, r) stand for the open ball in X with center x ∈ X and radius r > 0. Recall that F is said to be a monotone operator if it satisfies the relation ⟨F(x1)− F(x2), x1 − x2⟩ ≥ 0 (9.2) for all x1, x2 ∈ D(F).