A counterexample to convergence for multiscale decompositions
Simone Rebegoldi, Luca Rondi
TL;DR
The paper addresses the convergence of the classical multiscale decomposition for inverse problems and builds a concrete linear counterexample to show that, even under strong assumptions, the standard multiscale scheme can diverge. It constructs a framework with $X=l_1$, $H=l_2$, and a weighted $\,F$-norm, along with a carefully designed bounded injective operator $\Lambda:l_1\to l_2$ and a growth law $\lambda_n=\alpha_0 M^n$. The main finding is that the sequence $\sigma_n$ can satisfy the optimality conditions yet satisfy $\|\sigma_n\|_X\to\infty$, while $\Lambda(\sigma_\infty)=\Lambda(e_1)$ in $H$, demonstrating failure of convergence in the classical multiscale method and motivating the tighter multiscale variant. This highlights fundamental limitations of classical multiscale decompositions in inverse problems and informs the design of robust, convergent schemes.
Abstract
We discuss the convergence of the multiscale procedure by Modin, Nachman and Rondi, Adv. Math. (2019), which extended to inverse problems the multiscale decomposition of images by Tadmor, Nezzar and Vese, Multiscale Model. Simul. (2004). We show that, for the classical multiscale procedure, the multiscale decomposition might fail even for the linear case with a Banach norm as regularization.