Peter Nekrasov

Carlos Borges, Rafael Ceja Ayala, Peter Nekrasov

We study the inverse problem of qualitatively recovering a supported cavity in a thin elastic plate governed by the flexural (biharmonic) wave equation, using far-field pattern measurements. We derive a reciprocity principle and a factorization of the far-field operator for the supported plate boundary conditions, and we analyze its range properties to justify both the linear sampling method (LSM) and the direct sampling method (DSM). Numerical experiments assess the performance of LSM and DSM under noise, a limited amount of data, multiple scattering, and variations in the Poisson's ratio. The results show that both methods robustly recover the obstacle's location and convex hull, with DSM offering improved stability and reduced computational cost.

Peter Nekrasov, Jessica Freeze, Victor Batista

Precise physical descriptions of molecules can be obtained by solving the Schrodinger equation; however, these calculations are intractable and even approximations can be cumbersome. Force fields, which estimate interatomic potentials based on empirical data, are also time-consuming. This paper proposes a new methodology for modeling a set of physical parameters by taking advantage of the restricted Boltzmann machine's fast learning capacity and representational power. By training the machine on ab initio data, we can predict new data in the distribution of molecular configurations matching the ab initio distribution. In this paper we introduce a new RBM based on the Tanh activation function, and conduct a comparison of RBMs with different activation functions, including sigmoid, Gaussian, and (Leaky) ReLU. Finally we demonstrate the ability of Gaussian RBMs to model small molecules such as water and ethane.