Nesterov’s accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional

Abstract

In this paper, we consider Nesterov's Accelerated Gradient method for solving Nonlinear Inverse and Ill-Posed Problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide a convergence analysis for ill-posed problems of this method based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

“…Landweber iteration is known to be notoriously slow. However, with the ingredients above, also state-of the art accelerated versions of Landweber iteration, such as the steepest descent or the minimal error method [41] or Nesterov iteration and versions thereof [21,39] can be implemented in a straightforward manner.…”

“…≤γ . Thus, the assumptions of Lemma 2.1 are satisfied and we can make use of estimates (20), (21) with f = r + 2κp 2 t cf. (18).…”

“…We now proceed to derive contractivity of T by applying the estimates (20), (21) to the pde (19). In the above, we have already shown that for any p ∈ M , under conditions (23), (25), (28), (29), (30), the coefficients α = 1 − 2κp t , β ≡ b, γ ≡ c 2 , δ ≡ 0, µ ≡ 0 satisfy the assumptions of Lemma 2.1 with 1 2 ≤ α ≤ 3 2 ,γ ≤γ < 1 16 , so that the estimates (20), (21) with u(0) = 0, u t (0) = 0 apply to (19). Using them together with Gronwall's inequality yields existence of constants C,C > 0 (depending only on the constants R 0 , R 1 , R 2 ,γ in the definition of M as well as b, c 2 and T ), such that…”

Optimal-order convergence of Nesterov acceleration for linear ill-posed problems*

“…For the Nesterov acceleration (1), a detailed analysis has been performed by Neubauer [15] with the result that, assuming a usual source condition (12) and an appropriate a priori stopping rule, the resulting iterative regularization scheme is of optimal order for μ 1 2 , and, for μ > 1 2 , the convergence rates improve with μ but in a suboptimal way. More precisely, the convergence rates proven in [15] are…”

“…In the realm of ill-posed problems, Hubmer and Ramlau [12] performed a convergence analysis for the nonlinear case, and showed the efficiency of the method.…”