Computing the Sound of the Sea in a Seashell

Abstract

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $${{\mathcal {C}}}^2$$ C 2 . The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem

“…Our proof techniques for fundamental barriers in Theorem 2 stem from the mathematics behind the solvability complexity index (SCI) hierarchy ( 33 , 69 – 78 ). The SCI hierarchy generalizes the fundamental problems of Smale ( 79 , 80 ) on existence of algorithms and work by McMullen ( 81 ) and Doyle and McMullen ( 82 ).…”

On the complexity of the inverse Sturm–Liouville problem

“…Research related to this theory has gathered pace in recent years. We point out [17,13,16] where some of the theory of spectral computations has been further developed; [32] where this has been applied to certain classes of unbounded operators; [4,15] where solutions of PDEs were considered; [9] where we considered periodic spectral problems; [8,7] where we considered resonance problems; and [18,36,14,12] where the authors give further examples of how to perform certain spectral computations with error bounds. Let us summarize the main definitions of the SCI theory.…”

On the complexity of the inverse Sturm-Liouville problem

“…Research related to this theory has gathered pace in recent years. We point out [13,12] where some of the theory of spectral computations has been further developed; [27] where this has been applied to certain classes of unbounded operators; [4] where solutions of PDEs were considered; [9] where we considered periodic spectral problems; [8,7] where we considered resonance problems; and [14] where the authors give further examples of how to perform certain spectral computations with error bounds. Let us summarize the main definitions of the SCI theory.…”