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The SVD of the linearized EIT problem on a disk
Full text linkAbstract: In this paper we calculate the right singular functions to the linearized EIT problem on homogeneous disk.
We note the similarity to Zernike disk functions and the dependence on the mesh
Quasi-triangularization of matrix polynomials over arbitrary fields
No full textIn \cite{TasTisZab}, Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial P(\la) over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree.
When P(\la) is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to P(\la),
in which the diagonal blocks are of size at most
. This paper generalizes these results to regular matrix polynomials P(\la) over arbitrary fields \bF, showing that any such P(\la) can be
quasi-triangularized to a spectrally equivalent matrix polynomial over \bF of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the \bF-irreducible factors in the Smith form for P(\la)
The dynamical functional particle method for the generalized Sylvester equation
Full text linkRecent years have seen a renewal of interest in generalized Sylvester equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all generalized Sylvester equations with Hermitian positive definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of generalized Sylvester equations. For the Sylvester equation AX-XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size
Model order reduction of layered waveguides via rational Krylov fitting
No full textNetwork-based data-driven reduced order models have recently emerged as an efficient numerical tool for forward and inverse problems of wave propagation. Currently, this technique is limited to two classes of problems: bounded inhomogeneous domains (with applications in multiscale simulation and imaging) and homogeneous halfspaces (for the solution of exterior forward problems). Here we relax the constant coefficient requirement for the latter by considering reduced order models (ROMs) of unbounded waveguides with layered inclusions, thereby giving rise to efficient discrete perfectly matched layers (PMLs) for nonhomogeneous media. Our approach is based on the solution of a nonlinear rational least squares problem using the RKFIT method [M. Berljafa and S. Güttel, SIAM J. Sci. Comput., 39(5):A2049--A2071, 2017]. We show how the solution of this least squares problem can be converted into an accurate sparse network approximation within a rational Krylov framework. Several numerical experiments are included. They indicate that RKFIT computes PMLs more accurately than previous analytic approaches and even works in regimes where the transfer functions to be approximated are highly irregular due to pronounced scattering resonances. Spectral adaptation effects allow for accurate ROMs with dimensions near or even below the Nyquist limit
Optimizing and Factorizing the Wilson Matrix
Full text linkThe Wilson matrix, , is a unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940s, owing to its mild ill-conditioning. We ask how close is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations with having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that has an integer factor and two rational factors, up to signed permutations. This little matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results
Anymatrix: An Extensible MATLAB Matrix Collection
Full text linkAnymatrix is a MATLAB toolbox that provides an extensible collection of matrices with the ability to search the collection by matrix properties. Each matrix is implemented as a MATLAB function and the matrices are arranged in groups. Compared with previous collections, Anymatrix offers three novel features. First, it allows a user to share a collection of matrices by putting them in a group, annotating them with properties, and placing the group on a public repository, for example on GitHub; the group can then be incorporated into another user's local Anymatrix installation. Second, it provides a tool to search for matrices by their properties, with Boolean expressions supported. Third, it provides organization into sets, which are subsets of matrices from the whole collection appended with notes, which facilitate reproducible experiments. Anymatrix v1.0 comes with 146 built-in matrices organized into 7 groups with 49 recognized properties. The authors continue to extend the collection and welcome contributions from the community
A comparison of LSTM and GRU networks for learning symbolic sequences
No full textWe explore relations between the hyper-parameters of a recurrent neural network (RNN) and the complexity of string sequences it is able to memorize. We compare long short-term memory (LSTM) networks and gated recurrent units (GRUs). We find that an increase of RNN depth does not necessarily result in better memorization capability when the training time is constrained. Our results also indicate that the learning rate and the number of units per layer are among the most important hyper-parameters to be tuned. Generally, GRUs outperform LSTM networks on low complexity sequences while on high complexity sequences LSTMs perform better
Some Information Geometric Aspects of Cyber Security by Face Recognition
No full textSecure user access to devices and datasets is widely enabled by fingerprint or face recognition.
Organization of the necessarily large secure digital object datasets, with objects
having content that may consist of images,
text, video or audio, involves efficient classification and feature
retrieval processing. This usually will require multidimensional methods applicable
to data that is represented through a family of probability distributions.
Then information geometry is an appropriate context
in which to provide for such analytic work, whether with maximum likelihood fitted
distributions or empirical frequency distributions. The important provision is of
a natural geometric measure structure on families of probability
distributions by representing them as Riemannian manifolds.
Then the distributions are points lying in this geometrical manifold, different features
can be identified and dissimilarities
computed, so that neighbourhoods of objects nearby a given example object can be
constructed. This can reveal clustering and projections onto smaller eigen-subspaces
which can make comparisons easier to interpret. Geodesic distances can be used as a natural
dissimilarity metric applied over data described by probability distributions. Exploring this
property, we propose a new face recognition method which scores dissimilarities between face
images by multiplying geodesic distance approximations between -variate RGB Gaussians
representative of colour face images, and also obtaining joint probabilities. The experimental results show that this new method
is more successful in recognition rates than published comparative state-of-the-art methods
Diffraction tomography inversion and the transverse ray transform
No full textWe show that a reciprocal space squared intensity map of a material can be recovered, for each characteristic length scale, from diffraction tomography data by a simple slice-by-slice reconstruction method. Moreover if the reciprocal space map can be represented by a finite sum of spherical harmonic components for each length scale then the coefficients of that expansion can be recovered from inverting the transverse ray transform (TRT), where the data are polynomial coefficients of the azimuthal diffraction pattern for each length scale
Two body problem on a sphere in the presence of a uniform magnetic field
Full text linkWe investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles, interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria