Finding apparent horizons and other two-surfaces of constant expansion

Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalisation of apparent horizons. I present an efficient method for locating CE surfaces. This method uses an explicit representation of the surface, allowing for arbitrary resolutions and, in principle, shapes. The CE surface equation is then solved as a nonlinear elliptic equation. It is reasonable to assume that CE surfaces foliate a spacelike hypersurface outside of some interior region, thus defining an invariant (but still slicing-dependent) radial coordinate. This can be used to determine gauge modes and to compare time evolutions with different gauge conditions. CE surfaces also provide an efficient way to find new apparent horizons as they appear e.g. in binary black hole simulations.Comment: 21 pages, 8 figures; two references adde

A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where a SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin's Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, tracking of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven Sub-Riemannian Geodesics in SE(2)

Fast Adaptive Robust Differentiator Based Robust-Adaptive Control of Grid-Tied Inverters with a New L Filter Design Method

In this research, a new nonlinear and adaptive state feedback controller with a fast-adaptive robust differentiator is presented for grid-tied inverters. All parameters and external disturbances are taken as uncertain in the design of the proposed controller without the disadvantages of singularity and over-parameterization. A robust differentiator based on the second order sliding mode is also developed with a fast-adaptive structure to be able to consider the time derivative of the virtual control input. Unlike the conventional backstepping, the proposed differentiator overcomes the problem of explosion of complexity. In the closed-loop control system, the three phase source currents and direct current (DC) bus voltage are assumed to be available for feedback. Using the Lyapunov stability theory, it is proven that the overall control system has the global asymptotic stability. In addition, a new simple L filter design method based on the total harmonic distortion approach is also proposed. Simulations and experimental results show that the proposed controller assurances drive the tracking errors to zero with better performance, and it is robust against all uncertainties. Moreover, the proposed L filter design method matches the total harmonic distortion (THD) aim in the design with the experimental result

Tracking icebergs with time-lapse photography and sparse optical flow, LeConte Bay, Alaska, 2016–2017

We present a workflow to track icebergs in proglacial fjords using oblique time-lapse photos and the Lucas-Kanade optical flow algorithm. We employ the workflow at LeConte Bay, Alaska, where we ran five time-lapse cameras between April 2016 and September 2017, capturing more than 400 000 photos at frame rates of 0.5–4.0 min−1. Hourly to daily average velocity fields in map coordinates illustrate dynamic currents in the bay, with dominant downfjord velocities (exceeding 0.5 m s−1 intermittently) and several eddies. Comparisons with simultaneous Acoustic Doppler Current Profiler (ADCP) measurements yield best agreement for the uppermost ADCP levels (∼ 12 m and above), in line with prevalent small icebergs that trace near-surface currents. Tracking results from multiple cameras compare favorably, although cameras with lower frame rates (0.5 min−1) tend to underestimate high flow speeds. Tests to determine requisite temporal and spatial image resolution confirm the importance of high image frame rates, while spatial resolution is of secondary importance. Application of our procedure to other fjords will be successful if iceberg concentrations are high enough and if the camera frame rates are sufficiently rapid (at least 1 min−1 for conditions similar to LeConte Bay).This work was funded by the U.S. National Science Foundation (OPP-1503910, OPP-1504288, OPP-1504521 and OPP-1504191).Ye