Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors

Bistatic Radar Cross Section (RCS) Characterization of Complex Objects

This research addresses some phenomenological aspects of bistatic scattering from a complex object with an emphasis on specular, shadowing, dihedral, and cavity effects. Five targets are investigated for their simplicity and ability to highlight certain scattering phenomena. Direct measurements of scattered electromagnetic energy and simulation data is gathered for a frequency bandwidth of 6-18 GHz. Both ray tracing and scattering center approaches are used to describe the bistatic mechanisms. An appraisal of the effectiveness and utility of the monostatic-to-bistatic equivalence theorems (Kell\u27s and Crispin\u27s) and several commercial scattering prediction codes is also accomplished. Simulation data is generated from two different electromagnetic scattering prediction codes, Xpatch and FISC. Xpatch is a physical optics (PO) based code while FISC is a more rigorous method of moments (MOM) based tool. This data is utilized to achieve three objectives: (1) study Kell\u27s and Crispin\u27s monostatic-to-bistatic equivalence theorems (MBET) for a complex object through theoretical derivations and comparison of measured and simulated data sets, (2) evaluate the performance of Xpatch and FISC through direct comparisons to measured data, and (3) gain insight into the bistatic scattering phenomenology while extracting appropriate rules-of-thumb for bistatic scattering predictions. These rules of thumb are proposed to help guide the reader in evaluating the bistatic RCS of complex shapes in general