Fernando Pérez

Gregory Beylkin, Martin J. Mohlenkamp, Fernando Pérez

The wavefunction for the multiparticle Schrödinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions, and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrix-integral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.

Gregory Beylkin, Vani Cheruvu, Fernando Pérez

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class $(-Δ+μ^{2}I)^{-α}$, where $μ\geq0$ and $0<α<3/2$, and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has an advantage of being easily extendable to higher dimensions.

Ellianna Abrahams, Tasha Snow, Matthew R. Siegfried et al.

We propose a new tiling strategy, Flip-n-Slide, which has been developed for specific use with large Earth observation satellite images when the location of objects-of-interest (OoI) is unknown and spatial context can be necessary for class disambiguation. Flip-n-Slide is a concise and minimalistic approach that allows OoI to be represented at multiple tile positions and orientations. This strategy introduces multiple views of spatio-contextual information, without introducing redundancies into the training set. By maintaining distinct transformation permutations for each tile overlap, we enhance the generalizability of the training set without misrepresenting the true data distribution. Our experiments validate the effectiveness of Flip-n-Slide in the task of semantic segmentation, a necessary data product in geophysical studies. We find that Flip-n-Slide outperforms the previous state-of-the-art augmentation routines for tiled data in all evaluation metrics. For underrepresented classes, Flip-n-Slide increases precision by as much as 15.8%.