482 research outputs found
A Spectral Theory for Tensors
In this paper we propose a general spectral theory for tensors. Our proposed
factorization decomposes a tensor into a product of orthogonal and scaling
tensors. At the same time, our factorization yields an expansion of a tensor as
a summation of outer products of lower order tensors . Our proposed
factorization shows the relationship between the eigen-objects and the
generalised characteristic polynomials. Our framework is based on a consistent
multilinear algebra which explains how to generalise the notion of matrix
hermicity, matrix transpose, and most importantly the notion of orthogonality.
Our proposed factorization for a tensor in terms of lower order tensors can be
recursively applied so as to naturally induces a spectral hierarchy for
tensors.Comment: The paper is an updated version of an earlier versio
Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography
The inverse problem of electrical impedance tomography is severely ill-posed,
meaning that, only limited information about the conductivity can in practice
be recovered from boundary measurements of electric current and voltage.
Recently it was shown that a simple monotonicity property of the related
Neumann-to-Dirichlet map can be used to characterize shapes of inhomogeneities
in a known background conductivity. In this paper we formulate a
monotonicity-based shape reconstruction scheme that applies to approximative
measurement models, and regularizes against noise and modelling error. We
demonstrate that for admissible choices of regularization parameters the
inhomogeneities are detected, and under reasonable assumptions, asymptotically
exactly characterized. Moreover, we rigorously associate this result with the
complete electrode model, and describe how a computationally cheap
monotonicity-based reconstruction algorithm can be implemented. Numerical
reconstructions from both simulated and real-life measurement data are
presented
An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions
This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of “thin” structures as they arise in certain applications (e.g., material coatings); in addition, it is also designed to be used in conjunction with existing low-order FFT-based codes to upgrade their order of accuracy through a suitable treatment of material interfaces. The high-order convergence of the new procedure is attained through a combination of changes of parametric variables (to resolve the singularities of the Green function) and “partitions of unity” (to allow for a simple implementation of spectrally accurate quadratures away from singular points). Accelerated evaluations of the interaction between degrees of freedom, on the other hand, are accomplished by incorporating (two-face) equivalent source approximations on Cartesian grids. A detailed account of the main algorithmic components of the scheme are presented, together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results
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