Communication modes in vector diffraction

The communication modes, which mathematically correspond to singular value decomposition, have proven a useful concept in optical scalar-field diffraction, with applications in resolution studies, image synthesis, and wave propagation. For optical near-field geometries the communication modes have to be extended to electromagnetic field accounting for the polarization properties. In this paper we present the vector-valued communication modes method based on the rigorous electric-field diffraction integral. As a special case the transverse-electric scalar field modes are obtained. The intensity and polarization properties of the leading electromagnetic communication modes in near-field arrangements with rectangular apertures are discussed in terms of the Stokes parameters. For small separations between the transmitting and receiving apertures the fundamental mode possesses a ring-shaped hollow structure. The polarization properties of the near-field modes show features on spatial scales smaller than the wavelength of light. The system symmetries lead to degenerate communication modes.     

A direct solver with <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) complexity for integral equations on one-dimensional domains

An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes’ equations. The scaling coefficient suppressed by the “big-O” notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the ?- and ? ²-matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.     

Segregating Markov Chains

Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There are, however, discrete finite (reducible) Markov chains, for which two copies started in different states can be coupled to meet almost surely in finite time, yet their distributions keep a total variation distance bounded away from 0, even in the limit as time tends to infinity. We show that the supremum of total variation distance kept in this context is \(\tfrac{1}{2}\).     

An accelerated Poisson solver based on multidomain spectral discretization

This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with \(O(N^{1.5})\) complexity for the factorization stage and \(O(N \log N)\) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.     

Parametric estimation of ultrasonic phase velocity and attenuation in dispersive media

In ultrasonic characterization of liquids, gases, and solids, accurate estimation of frequency dependent attenuation and phase velocity is of great importance. Non-parametric methods, such as Fourier analysis, suffers from noise sensitivity, and the variance of the estimated quantities is limited by the signal-to-noise ratio. In this paper we present a parametric method for estimation of these properties. Pulse echo experiments in ethane, oxygen and mixtures of the two show that the proposed method can estimate phase velocity and attenuation with up to 50 times lower variance than standard non-parametric methods.