On the Optimality of Regularization Methods for Solving Linear Ill-Posed Problems

Abstract

In this paper we consider a general class of regularization methods for ill-posed problems Ax = y where A X-V is a linear operator between Hubert spaces X and V. The regularization methods have the general form x, = x+g0((AA)')(A*A)A(yo-Al) where y 6 are the available noisy data with flyy fl :5 5. Assuming x E M,5 = {x E X X-I = (AA)" 2 v, 11v11 < E, p > 0) we consider different functions g, and discuss the question how to choose the order s and the regularization parameter a = a(5, E,p) in order to obtain optimal estimates sup Ii xx li < E p+15P/(P+1) where the supremum is taken over x,E M,E, y E V and li Ar-v'il

The inverse source problem for time-fractional diffusion equation: stability analysis and regularization

“…Let us introduce the worst case error (δ, R) for identifying x from y δ as. [45][46][47][48] (δ, R) := sup{ Ry δ − x |x ∈ M, y δ ∈ Y, Ax − y δ ≤ δ}. The best possible error bound (or optimal error bound) is defined as the infimum over all mappings R : Y → X :…”

Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

“…Let us assume that : → is an arbitrary mapping to approximately recover from . Then the worst case error for under the a priori information ∈ , is [16,17]…”

“…For the method of generalized Tikhonov regularization, a regularized approximation is determined by solving the minimization problem [13,14,16,17]…”

An inverse problem for space‐fractional backward diffusion problem

“…Consider the following ill‐posed operator equation where F ∈ £ ( X , Y ) is a linear compact operator between Hilbert spaces x and Y . Assume that y δ ∈ Y are available noisy data with ∥ y δ − y ∥ ≤ δ .…”