arXiv subjects

Hongpeng Sun

Publications and source records attributed to Hongpeng Sun.

24 records · Page 2Linked to original sources

arXiv2016

Accelerated Douglas-Rachford methods for the solution of convex-concave saddle-point problems

We study acceleration and preconditioning strategies for a class of Douglas-Rachford methods aiming at the solution of convex-concave saddle-point problems associated with Fenchel-Rockafellar duality. While the basic iteration converges weakly in Hilbert space with $\mathcal{O}(1/k)$ ergodic convergence of restricted primal-dual gaps, acceleration can be achieved under strong-convexity assumptions. Namely, if either the primal or dual functional in the saddle-point formulation is strongly convex, then the method can be modified to yield $\mathcal{O}(1/k^2)$ ergodic convergence. In case of both functionals being strongly convex, similar modifications lead to an asymptotic convergence of $\mathcal{O}({\vartheta}^k)$ for some $0 < {\vartheta} < 1$. All methods allow in particular for preconditioning, i.e., the inexact solution of the implicit linear step in terms of linear splitting methods with all convergence rates being maintained. The efficiency of the proposed methods is verified and compared numerically, especially showing competitiveness with respect to state-of-the-art accelerated algorithms.

math.OC↗

arXiv2013

Inverse Elastic Scattering for Multiscale Rigid Bodies with A Single Far-field Pattern

We develop three inverse elastic scattering schemes for locating multiple small, extended and multiscale rigid bodies, respectively. There are some salient and promising features of the proposed methods. The cores of those schemes are certain indicator functions, which are obtained by using only a single far-field pattern of the pressure (longitudinal) wave, or the shear (transversal) wave, or the total wave field. Though the inverse scattering problem is known to be nonlinear and ill-posed, the proposed reconstruction methods are totally "direct" and there are no inversions involved. Hence, the methods are very efficient and robust against noisy data. Both rigorous mathematical justifications and numerical simulations are presented in our study.

math.AP↗

arXiv2013

Two Single-shot Methods for Locating Multiple Electromagnetic Scatterers

We develop two inverse scattering schemes for locating multiple electromagnetic (EM) scatterers by the electric far-field measurement corresponding to a single incident/detecting plane wave. The first scheme is for locating scatterers of small size compared to the wavelength of the detecting plane wave. The multiple scatterers could be extremely general with an unknown number of components, and each scatterer component could be either an impenetrable perfectly conducting obstacle or a penetrable inhomogeneous medium with an unknown content. The second scheme is for locating multiple perfectly conducting obstacles of regular size compared to the detecting EM wavelength. The number of the obstacle components is not required to be known in advance, but the shape of each component must be from a certain known admissible class. The admissible class may consist of multiple different reference obstacles. The second scheme could also be extended to include the medium components if a certain generic condition is satisfied. Both schemes are based on some novel indicator functions whose indicating behaviors could be used to locate the scatterers. No inversion will be involved in calculating the indicator functions, and the proposed methods are every efficient and robust to noise. Rigorous mathematical justifications are provided and extensive numerical experiments are conducted to illustrate the effectiveness of the imaging schemes.

math.AP↗

arXiv2012

Enhanced Near-cloak by FSH Lining

We consider regularized approximate cloaking for the Helmholtz equation. Various cloaking schemes have been recently proposed and extensively investigated. The existing cloaking schemes in literature are (optimally) within $|\lnρ|^{-1}$ in 2D and $ρ$ in 3D of the perfect cloaking, where $ρ$ denotes the regularization parameter. In this work, we develop a cloaking scheme with a well-designed lossy layer right outside the cloaked region that can produce significantly enhanced near-cloaking performance. In fact, it is proved that the proposed cloaking scheme could (optimally) achieve $ρ^N$ in $\mathbb{R}^N$, $N\geq 2$, within the perfect cloaking. It is also shown that the limit of the proposed lossy layer corresponds to a sound-hard layer. We work with general geometry and arbitrary cloaked contents of the proposed cloaking device.

math.AP↗

arXiv2012

Singular perturbation of reduced wave equation and scattering from an embedded obstacle

We consider time-harmonic wave scattering from an inhomogeneous isotropic medium supported in a bounded domain $Ω\subset\mathbb{R}^N$ ($N\geq 2$). {In a subregion $D\SubsetΩ$, the medium is supposed to be lossy and have a large mass density. We study the asymptotic development of the wave field as the mass density $ρ\rightarrow +\infty$} and show that the wave field inside $D$ will decay exponentially while the wave filed outside the medium will converge to the one corresponding to a sound-hard obstacle $D\SubsetΩ$ buried in the medium supported in $Ω\backslash\bar{D}$. Moreover, the normal velocity of the wave field on $\partial D$ from outside $D$ is shown to be vanishing as $ρ\rightarrow +\infty$. {We derive very accurate estimates for the wave field inside and outside $D$ and on $\partial D$ in terms of $ρ$, and show that the asymptotic estimates are sharp. The implication of the obtained results is given for an inverse scattering problem of reconstructing a complex scatterer.}

math.AP↗

arXiv2011

Enhanced Approximate Cloaking by SH and FSH Lining

We consider approximate cloaking from a regularization viewpoint introduced in [13] for EIT and further investigated in [12] [17] for the Helmholtz equation. The cloaking schemes in [12] and [17] are shown to be (optimally) within $|\lnρ|^{-1}$ in 2D and $ρ$ in 3D of perfect cloaking, where $ρ$ denotes the regularization parameter. In this paper, we show that by employing a sound-hard layer right outside the cloaked region, one could (optimally) achieve $ρ^N$ in $\mathbb{R}^N,\ N\geq 2$, which significantly enhances the near-cloak. We then develop a cloaking scheme by making use of a lossy layer with well-chosen parameters. The lossy-layer cloaking scheme is shown to possess the same cloaking performance as the one with a sound-hard layer. Moreover, it is shown that the lossy layer could be taken as a finite realization of the sound-hard layer. Numerical experiments are also presented to assess the cloaking performances of all the cloaking schemes for comparisons.

math.AP↗