We propose a preconditioned version of the Douglas-Rachford splitting method for solving convex-concave saddle-point problems associated with Fenchel-Rockafellar duality. Our approach makes it possible to use approximate solvers for the linear subproblem arising in this context. We prove weak convergence in Hilbert space under minimal assumptions. In particular, various efficient preconditioners are introduced in this framework for which only a few inner iterations are needed instead of computing an exact solution or controlling the error. The method is applied to a discrete total-variation denoising problem. Numerical experiments show that the proposed algorithms with appropriate preconditioners are very competitive with existing fast algorithms including the first-order primal-dual algorithm for saddle-point problems of Chambolle and Pock. Introduction.The Douglas-Rachford splitting method is a classical approach for finding a zero point of the sum of maximal set-valued monotone operators, a task which is quite common for the minimization of the sum of convex functions [12,20]. While originally designed for the discretized solution of the heat conduction problem [11], this method grew to be a powerful tool due to its general applicability and unconditional stability. Convergence analysis is given, for example, in [20,9,10]. The Douglas-Rachford splitting method can in particular be interpreted in the framework of proximal point algorithms [12], which provides a convenient method for analysis of this algorithm and is also our starting point.The paper focuses on the following widely used generic saddle-point problem [8]:
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. and sonar, non-destructive testing, remote sensing, geophysical exploration and medical imaging to name just a few; see [4,5,10,11,15,20,25,31,33] and the references therein. In the current article, we shall mainly consider the reconstruction scheme for this inverse scattering problem. There are extensive studies in the literature in this aspect and many imaging schemes have been developed by various authors, and we would like to refer to [1-6, 9, 12, 14, 17-19, 21, 30, 32] and the references therein. However, we would like to remark that in those schemes, one either needs to make use of many wave measurements or if only a few wave measurements are utilized, then one must require that the underlying scatterer is of small size compared to the detecting wavelength. In this work, we shall consider our study in a very practical setting by making use of a single electric far-field measurement. That is, we shall consider the reconstruction by measuring the far-field electric wave corresponding to a single time-harmonic plane wave. From a practical viewpoint, the inverse scattering method with a single far-field measurement would be of significant interests, but is highly challenging with very limited progress in the literature; we refer to [15,20,[24][25][26]33] for related discussion and surveys on some existing development. For more practical considerations, we shall work in a even more challenging setting by assuming very little a priori knowledge of the underlying scattering object, which might consist of multiple components, and both the number of the components and the physical property of each...
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 | ln ρ| −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 R N , N 2, within the perfect cloaking. It is also shown that the proposed lossy layer is a finite realization of a sound-hard layer. We work with general geometry and arbitrary cloaked contents of the proposed cloaking device. RésuméOn considére le problème d'invisibilité approchée pour l'équation d'Helmholtz. Diverses méthodes ont été récemment proposées et étudiées. Les techniques de quasi-invisibilité présentes dans la littérature approchent l'invisibilité parfaite avec une erreur proportionelle à | ln ρ| −1 dans R 2 et ρ dans R 3 , où ρ désigne le paramètre de régularisation. Dans cet article on développe un système d'invisibilité qui utilise une couche avec perte à l'extérieur de la région dissimulée et on améliore considérablement la quasi-invisibilité. On démontre que cette nouvelle technique de dissimulation approche l'invisibilité parfaite avec une erreur proportionelle à ρ N dans R N , N 2. On démontre également que cette couche avec perte est un cas particulier d'une couche rigide. Cette étude concerne des dispositifs de dissimulation avec une géométrie générale.
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