Title: On the optimality of convergence conditions for multiscale decompositions in imaging and inverse problems Show Chinese title

Title: 关于成像和反问题中多尺度分解收敛条件的最优性

Abstract: We consider the multiscale procedure developed by Modin, Nachman and Rondi, Adv. Math. (2019), for inverse problems, which was inspired by the multiscale decomposition of images by Tadmor, Nezzar and Vese, Multiscale Model. Simul. (2004). We investigate under which assumptions this classical procedure is enough to have convergence in the unknowns space without resorting to use the tighter multiscale procedure from the same paper. We show that this is the case for linear inverse problems when the regularization is given by the norm of a Hilbert space. Moreover, in this setting the multiscale procedure improves the stability of the reconstruction. On the other hand, we show that, for the classical multiscale procedure, convergence in the unknowns space might fail even for the linear case with a Banach norm as regularization. Show Chinese abstract

Abstract: 我们考虑由Modin、Nachman和Rondi在Adv. Math. (2019)中开发的多尺度过程，用于反问题，该过程受到Tadmor、Nezzar和Vese在Multiscale Model. Simul. (2004)中对图像的多尺度分解的启发。我们研究在哪些假设下，这种经典的多尺度过程足以在不使用同一篇论文中的更严格的多尺度过程的情况下，在未知空间中实现收敛。我们证明，当正则化由Hilbert空间的范数给出时，这种情况适用于线性反问题。此外，在这种情况下，多尺度过程提高了重建的稳定性。另一方面，我们表明，对于经典的多尺度过程，即使在使用巴拿赫范数作为正则化的线性情况下，未知空间中的收敛也可能失败。