PhasePack: A Phase Retrieval Library

Phase retrieval deals with the estimation of complex-valued signals solely from the magnitudes of linear measurements. While there has been a recent explosion in the development of phase retrieval algorithms, the lack of a common interface has made it difficult to compare new methods against the state-of-the-art. The purpose of PhasePack is to create a common software interface for a wide range of phase retrieval algorithms and to provide a common testbed using both synthetic data and empirical imaging datasets. PhasePack is able to benchmark a large number of recent phase retrieval methods against one another to generate comparisons using a range of different performance metrics. The software package handles single method testing as well as multiple method comparisons. The algorithm implementations in PhasePack differ slightly from their original descriptions in the literature in order to achieve faster speed and improved robustness. In particular, PhasePack uses adaptive stepsizes, line-search methods, and fast eigensolvers to speed up and automate convergence

Neural Differentiable Integral Control Barrier Functions for Unknown Nonlinear Systems with Input Constraints

In this paper, we propose a deep learning based control synthesis framework for fast and online computation of controllers that guarantees the safety of general nonlinear control systems with unknown dynamics in the presence of input constraints. Towards this goal, we propose a framework for simultaneously learning the unknown system dynamics, which can change with time due to external disturbances, and an integral control law for trajectory tracking based on imitation learning. Simultaneously, we learn corresponding safety certificates, which we refer to as Neural Integral Control Barrier Functions (Neural ICBF's), that automatically encode both the state and input constraints into a single scalar-valued function and enable the design of controllers that can guarantee that the state of the unknown system will never leave a safe subset of the state space. Finally, we provide numerical simulations that validate our proposed approach and compare it with classical as well as recent learning based methods from the relevant literature.Comment: 15 pages, 4 figure

Discontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials. (I) General Considerations. (II) An Example

This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related is sue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of "elastostatic shocks" -- elastostatic fields with continuous displacements and discontinuous deformation gradients -- is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipticity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material. In order to illustrate the occurrence of elastostatic shocks in a physical problem, a specific displacement boundary value problem is studied. Here, a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity is considered. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary value problem. In order to resolve this non-uniqueness, dissipativity and stability are investigated.</p