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We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions $${N \geq 2}$$ via the approach of transformation optics. There are four major ingredients in our proposed theory: (1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation that blows up a subset $${K_\varepsilon}$$ in the virtual space, where $${\varepsilon \ll 1}$$ is an asymptotic regularization parameter. $${K_\varepsilon}$$ will degenerate to K as $${\varepsilon \rightarrow +0}$$ , and in our theory K could be any convex compact set in $${\mathbb{R}^N}$$ , or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. (2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. (3) The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. (4) In order to achieve a cloaking device of compact size, particularly for the case when $${K_\varepsilon}$$ is not 'uniformly small', an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field $${u_\varepsilon}$$ corresponding to a cloaking problem will converge to u as $${\varepsilon \rightarrow +0}$$ , with u being the scattered wave field corresponding to a sound-hard K. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers numerical design of finite sound-hard lining (FSHL)-based approximate cloaks with arbitrary shapes. Regarding the complexity of the shape, two new approaches are proposed to design the transformation map from the virtual space to the physical space via transformation optics. For star-shaped geometry, we propose an explicit global transformation map which can be easily differentiated and inverted. For more general shapes, an Initialize-Untangle-Extend (IUE) approach is initiated to build locally piecewise differentiable deformations, which can be locally inverted with the help of an approximate triangulation. With the locally piecewise-constructed transformation, the parameters of acoustic scattering models in physical space can be determined in both approaches based on the transformation invariance of the Helmholtz system. Then the cloaking effects for an arbitrary shape FSHL-based cloak can be realized following Li et al. (2012) [5] . Extensive numerical experiments are presented to illustrate both the effectiveness of cloak design and the efficiency of the proposed FSHL-based cloaks with arbitrary shapes. [ABSTRACT FROM AUTHOR]

This work is concerned with an inverse problem of identifying the current source distribution of the time-harmonic Maxwell's equations from multi-frequency measurements. Motivated by the Fourier method for the scalar Helmholtz equation and the polarization vector decomposition, we propose a novel method for determining the source function in the full vector Maxwell's system. Rigorous mathematical justifications of the method are given and numerical examples are provided to demonstrate the feasibility and effectiveness of the method. [ABSTRACT FROM AUTHOR]