Well-Conditioned Fractional Collocation Methods Using Fractional Birkhoff Interpolation Basis

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order $\mu\in (0,1)$ to compute that of any order $k+\mu$ with integer $k\ge 0,$ while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order $\mu\in (0,1).$ Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.Comment: 30 pages, 10 figures and 1 tabl

Robust Decoding from 1-Bit Compressive Sampling with Least Squares

In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: $y = \eta \odot\textrm{sign} (\Psi x^* + \epsilon)$, where $x^{*}\in \mathcal{R}^{n}, y\in \mathcal{R}^{m}$, $\Psi \in \mathcal{R}^{m\times n}$, and $\epsilon$ is the random error before quantization and $\eta\in \mathcal{R}^{n}$ is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For $m>n$, we show that, up to a constant $c$, with high probability, the least squares solution $x_{\textrm{ls}}$ approximates $x^*$ with precision $\delta$ as long as $m \geq\widetilde{\mathcal{O}}(\frac{n}{\delta^2})$. For $m< n$, we prove that, up to a constant $c$, with high probability, the $\ell_1$-regularized least-squares solution $x_{\ell_1}$ lies in the ball with center $x^*$ and radius $\delta$ provided that $m \geq \mathcal{O}( \frac{s\log n}{\delta^2})$ and $\|x^*\|_0 := s < m$. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm

Suppression of the emittance growth induced by coherent synchrotron radiation in triple-bend achromats

The coherent synchrotron radiation (CSR) effect in a bending path plays an important role in transverse emittance dilution in high-brightness light sources and linear colliders, where the electron beams are of short bunch length and high peak current. Suppression of the emittance growth induced by CSR is critical to preserve the beam quality and help improve the machine performance. It has been shown that the CSR effect in a double-bend achromat (DBA) can be analyzed with the two-dimensional point-kick analysis method. In this paper, this method is applied to analyze the CSR effect in a triple-bend achromat (TBA) with symmetric layout, which is commonly used in the optics designs of energy recovery linacs (ERLs). A condition of cancelling the CSR linear effect in such a TBA is obtained, and is verified through numerical simulations. It is demonstrated that emittance preservation can be achieved with this condition, and to a large extent, has a high tolerance to the fluctuation of the initial transverse phase space distribution of the beam.Comment: 9 pages, 4 figure

Approximation Algorithm for Fault-Tolerant Virtual Backbone in Wireless Sensor Networks

To save energy and alleviate interferences in a wireless sensor network, the usage of virtual backbone was proposed. Because of accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone, which can be modeled as a $k$-connected $m$-fold dominating set (abbreviated as $(k,m)$-CDS) in a graph. A node set $C\subseteq V(G)$ is a $(k,m)$-CDS of graph $G$ if every node in $V(G)\backslash C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$-connected. In this paper, we present an approximation algorithm for the minimum $(3,m)$-CDS problem with $m\geq3$. The performance ratio is at most $\gamma$, where $\gamma=\alpha+8+2\ln(2\alpha-6)$ for $\alpha\geq4$ and $\gamma=3\alpha+2\ln2$ for $\alpha<4$, and $\alpha$ is the performance ratio for the minimum $(2,m)$-CDS problem. Using currently best known value of $\alpha$, the performance ratio is $\ln\delta+o(\ln\delta)$, where $\delta$ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. This is the first performance-guaranteed algorithm for the minimum $(3,m)$-CDS problem on a general graph. Furthermore, applying our algorithm on a unit disk graph which models a homogeneous wireless sensor network, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the $(3,m)$-CDS problem on a unit disk graph.Comment: IEEE/ACM Transactions on Networking, 201

SNAP: A semismooth Newton algorithm for pathwise optimization with optimal local convergence rate and oracle properties

We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients along the solution path with warm start. At each knot in the path, SNAP converges locally superlinearly for the Enet criterion and achieves an optimal local convergence rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of two matrix-vector multiplication per iteration. Under certain regularity conditions on the design matrix and the minimum magnitude of the nonzero elements of the target regression coefficients, we show that SNAP hits a solution with the same signs as the regression coefficients and achieves a sharp estimation error bound in finite steps with high probability. The computational complexity of SNAP is shown to be the same as that of LARS and coordinate descent algorithms per iteration. Simulation studies and real data analysis support our theoretical results and demonstrate that SNAP is faster and accurate than LARS and coordinate descent algorithms

Suppression of the emittance growth induced by CSR in a DBA cell

The Emittace growth induced by Coherent Synchrotron Radiation(CSR) is an important issue when electron bunches with short bunch length and high peak current are transported in a bending magnet. In this paper, a single kick method is introduced which could give the same result as the R-matrix method, and much easier to use. Then with this method, an optics design technique which could minimize the emittance dilution within a single achromatic cell.Comment: 7 pages, 6 figure

Measurements of Outflow Velocities in On-Disk Plumes from EIS Hinode Observations

The contribution of plumes to the solar wind has been subject to hot debate in the past decades. The EUV Imaging Spectrometer (EIS) on board Hinode provides a unique means to deduce outflow velocities at coronal heights via direct Doppler shift measurements of coronal emission lines. Such direct Doppler shift measurements were not possible with previous spectrometers. We measure the outflow velocity at coronal heights in several on-disk long-duration plumes, which are located in coronal holes and show significant blue shifts throughout the entire observational period. In one case, a plume is measured 4 hours apart. The deduced outflow velocities are consistent, suggesting that the flows are quasi-steady. Furthermore, we provide an outflow velocity profile along the plumes, finding that the velocity corrected for the line-of-sight effect can reach 10 km s$^{-1}$ at 1.02 $R_{\odot}$, 15 km s$^{-1}$ at 1.03 $R_{\odot}$, and 25 km s$^{-1}$ at 1.05 $R_{\odot}$. This clear signature of steady acceleration, combined with the fact that there is no significant blue shift at the base of plumes, provides an important constraint on plume models. At the height of 1.03 $R_{\odot}$, EIS also deduced a density of 1.3$\times10^{8}$ cm$^{-3}$, resulting in a proton flux of about 4.2$\times10^9$ cm$^{-2}$s$^{-1}$ scaled to 1AU, which is an order of magnitude higher than the proton input to a typical solar wind if a radial expansion is assumed. This suggests that, coronal hole plumes may be an important source of the solar wind.Comment: accepted for publication in ApJ, 13 pages, 9 figure

Real-Time Dense Stereo Embedded in A UAV for Road Inspection

The condition assessment of road surfaces is essential to ensure their serviceability while still providing maximum road traffic safety. This paper presents a robust stereo vision system embedded in an unmanned aerial vehicle (UAV). The perspective view of the target image is first transformed into the reference view, and this not only improves the disparity accuracy, but also reduces the algorithm's computational complexity. The cost volumes generated from stereo matching are then filtered using a bilateral filter. The latter has been proved to be a feasible solution for the functional minimisation problem in a fully connected Markov random field model. Finally, the disparity maps are transformed by minimising an energy function with respect to the roll angle and disparity projection model. This makes the damaged road areas more distinguishable from the road surface. The proposed system is implemented on an NVIDIA Jetson TX2 GPU with CUDA for real-time purposes. It is demonstrated through experiments that the damaged road areas can be easily distinguished from the transformed disparity maps.Comment: 9 pages, 8 figures, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 16-20, 2019, Long Beach, US

A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including $\ell^0$, bridge, smoothly clipped absolute deviation, capped $\ell^1$ and minimax concavity penalty. First we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in efficiency and accuracy compared with the existing sparse recovery methods

A Self-Training Method for Machine Reading Comprehension with Soft Evidence Extraction

Neural models have achieved great success on machine reading comprehension (MRC), many of which typically consist of two components: an evidence extractor and an answer predictor. The former seeks the most relevant information from a reference text, while the latter is to locate or generate answers from the extracted evidence. Despite the importance of evidence labels for training the evidence extractor, they are not cheaply accessible, particularly in many non-extractive MRC tasks such as YES/NO question answering and multi-choice MRC. To address this problem, we present a Self-Training method (STM), which supervises the evidence extractor with auto-generated evidence labels in an iterative process. At each iteration, a base MRC model is trained with golden answers and noisy evidence labels. The trained model will predict pseudo evidence labels as extra supervision in the next iteration. We evaluate STM on seven datasets over three MRC tasks. Experimental results demonstrate the improvement on existing MRC models, and we also analyze how and why such a self-training method works in MRC. The source code can be obtained from https://github.com/SparkJiao/Self-Training-MRCComment: 12 pages, accepted by ACL 202