Discretization of variational regularization in Banach spaces

Abstract

Consider a nonlinear ill-posed operator equation F (u) = y where F is defined on a Banach space X. In general, for solving this equation numerically, a finite dimensional approximation of X and an approximation of F are required. Moreover, in general the given data y δ of y are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the L ∞ -space.

Finite-dimensional approximation of convex regularization via hexagonal pixel grids

“…Thus, one obtains a sequence of elements u α,δ m,n ∈ D(F m ) ∩ D(R), which minimizes the functionals u → F m (u) − v δ p + αR(u), p > 1 ( 2 ) over U n , respectively. The results in [1], which generalize the results of [2] from a Hilbert space setting to a Banach space setting (see also Sections 4.1.1-4.1.2 in [3] regarding the corresponding infinite-dimensional framework), show that (1) u α,δ m,n → u † in a weak sense and (2) R(u α,δ m,n ) → R(u † ) for m, n → ∞ and δ, α(δ, m, n) → 0 in an appropriate manner.…”

“…This work is concerned with a follow up to the work, [1] where we derived a convergence analysis for discrete approximations of minimizers of convex regularization functionals.…”

“…The regularization method with finite-dimensional approximations considered in this paper cannot be analyzed with previous settings [1,2] and requires a weaker form of convergence for n, m → ∞, which will be specified in Section 2. Important applications are finite-dimensional realizations of total variation minimization, which was introduced in [4].…”

“…Note that such realizations correspond to the most commonly used finite difference methods, for which our analysis now provides a general convergence framework. In contrast, implementation of regularization techniques with continuous finite element approximations can be analyzed with the general results of [1]. The particular case of total variation minimization discretized by continuous finite elements has been studied before in [5].…”

On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

“…Indeed, if α is sufficiently small, (19) or (18) could be satisfied. It follows that, the similar assertion is true for α ∈ B.…”