Research Spotlights

Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change of variables $s = \log(x)$ connects different areas of computational mathematics. Authors Daan Huybrechs and Lloyd “Nick” Trefethen show new relationships between smooth approximation, rational approximation theory, adaptive mesh refinement, and numerical quadrature. For example, the authors show that this change of variables can be naturally tied to a “linear tapering” effect near singularities, which is a common feature in both rational approximation and $hp$-mesh refinement. Through a number of effective examples, the authors illustrate the power of these relationships across areas that have seen relatively independent lines of development. In doing so, the authors suggest opportunities for developing and analyzing new methods by leveraging the new connections, including mesh refinement strategies, techniques for multivariate approximation, and hybrid approaches that combine the strengths of disparate methods. How well can information be recovered from water waves? This question is at the heart of this issue's second research spotlight, “Feynman's Inverse Problem.” Author Adrian Kirkeby is motivated by a thought experiment posed by the physicist and iconoclast Richard Feynman wherein an insect floating in a swimming pool wants to determine where and when others have jumped into the pool, causing the waves the insect observes. Kirkeby constructs and analyzes a linear 2D-3D system of partial differential equations (PDEs) for the forward model. Leveraging the nonlocality of this system of PDEs, Kirkeby shows conditions under which the insect can determine the source of the waves---in fact, uniquely---simply by observing the wave amplitude and water velocity in any small area of the surface. This model is then extended to capture settings where noisy observations and observations at a finite number of time and space points are collected, and establishes stability properties and error bounds for the reconstruction. The paper concludes with illustrative numerical experiments based on a nonharmonic Fourier inversion method. Kirkeby also highlights several avenues for future research, noting that inverse problems for water or other surface waves have received less attention than those involving acoustic or electromagnetic waves. As an added bonus, the referenced video of Feynman is not to be missed.