Nonlinear Collapse in the Semilinear Wave Equation in AdS

Previous studies of the semilinear wave equation in Minkowski space have shown a type of critical behavior in which large initial data collapse to singularity formation due to nonlinearities while small initial data does not. Numerical solutions in spherically symmetric Anti-de Sitter (AdS) are presented here which suggest that, in contrast, even small initial data collapse eventually. Such behavior appears analogous to the recent result of Ref. [1] that found that even weak, scalar initial data collapse gravitationally to black hole formation via a weakly turbulent instability. Furthermore, the imposition of a reflecting boundary condition in the bulk introduces a cut-off, below which initial data fails to collapse. This threshold appears to arise because of the dispersion introduced by the boundary condition.Comment: 4 pages, 5 figures, Updated with higher resolution runs; Updated with suggestions from referee and an added panel in Fig. 4; Accepted at PR

Multiply Unstable Black Hole Critical Solutions

The gravitational collapse of a complex scalar field in the harmonic map is modeled in spherical symmetry. Previous work has shown that a change of stability of the attracting critical solution occurs in parameter space from the discretely self-similarity critical (DSS) solution originally found by Choptuik to the continuously self-similar (CSS) solution found by Hirschmann and Eardley. In the region of parameter space in which the DSS is the attractor, a family of initial data is found which finds the CSS as its critical solution despite the fact that it has more than one unstable mode. An explanation of this is proposed in analogy to families that find the DSS in the region where the CSS is the attractor.Comment: 8 pages, 7 figure

Semi-Blind Spatially-Variant Deconvolution in Optical Microscopy with Local Point Spread Function Estimation By Use Of Convolutional Neural Networks

We present a semi-blind, spatially-variant deconvolution technique aimed at optical microscopy that combines a local estimation step of the point spread function (PSF) and deconvolution using a spatially variant, regularized Richardson-Lucy algorithm. To find the local PSF map in a computationally tractable way, we train a convolutional neural network to perform regression of an optical parametric model on synthetically blurred image patches. We deconvolved both synthetic and experimentally-acquired data, and achieved an improvement of image SNR of 1.00 dB on average, compared to other deconvolution algorithms.Comment: 2018/02/11: submitted to IEEE ICIP 2018 - 2018/05/04: accepted to IEEE ICIP 201

Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion

We propose a robust autofocus method for reconstructing digital Fresnel holograms. The numerical reconstruction involves simulating the propagation of a complex wave front to the appropriate distance. Since the latter value is difficult to determine manually, it is desirable to rely on an automatic procedure for finding the optimal distance to achieve high-quality reconstructions. Our algorithm maximizes a sharpness metric related to the sparsity of the signal’s expansion in distance-dependent waveletlike Fresnelet bases. We show results from simulations and experimental situations that confirm its applicability

Threshold of Singularity Formation in the Semilinear Wave Equation

Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which the exponent of the nonlinear term is either $p=5$ or $p=7$. Near the threshold of singularity formation, numerical solutions suggest an approach to self-similarity for the $p=7$ case and an approach to a scale evolving static solution for $p=5$.Comment: 6 pages, 7 figure

The Nonlinear Sigma Model With Distributed Adaptive Mesh Refinement

An adaptive mesh refinement (AMR) scheme is implemented in a distributed environment using Message Passing Interface (MPI) to find solutions to the nonlinear sigma model. Previous work studied behavior similar to black hole critical phenomena at the threshold for singularity formation in this flat space model. This work is a follow-up describing extensions to distribute the grid hierarchy and presenting tests showing the correctness of the model.Comment: 6 pages, 5 figure