Regularization matrices for discrete ill-posed problems in several space-dimensions

Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right-hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions

Eigenvalue condition numbers: Zero-structured versus traditional

AbstractWe discuss questions of eigenvalue conditioning. We study in some depth relationships between the classical theory of conditioning and the theory of the zero-structured conditioning, and we derive from the existing theory formulae for the mathematical objects involved. Then an algorithm to compare the zero-structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented. Numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero-structure. Patterned matrices (Toeplitz and Hankel) will be investigated in a forthcoming paper (Eigenvalue patterned condition numbers: Toeplitz and Hankel cases, Tech. Rep. 3, Mathematics Department, University of Rome ‘ La Sapienza’ , 2005.)

Network analysis with the aid of the path length matrix

Let a network be represented by a simple graph G with n vertices. A common approach to investigate properties of a network is to use the adjacency matrix A=[aij]i,j=1n∈Rn×n associated with the graph G , where aij> 0 if there is an edge pointing from vertex vi to vertex vj , and aij= 0 otherwise. Both A and its positive integer powers reveal important properties of the graph. This paper proposes to study properties of a graph G by also using the path length matrix for the graph. The (ij) th entry of the path length matrix is the length of the shortest path from vertex vi to vertex vj ; if there is no path between these vertices, then the value of the entry is ∞ . Powers of the path length matrix are formed by using min-plus matrix multiplication and are important for exhibiting properties of G . We show how several known measures of communication such as closeness centrality, harmonic centrality, and eccentricity are related to the path length matrix, and we introduce new measures of communication, such as the harmonic K-centrality and global K-efficiency, where only (short) paths made up of at most K edges are taken into account. The sensitivity of the global K-efficiency to changes of the entries of the adjacency matrix also is considered

A spectral method for bipartizing a network and detecting a large anti-community

Relations between discrete quantities such as people, genes, or streets can be described by networks, which consist of nodes that are connected by edges. Network analysis aims to identify important nodes in a network and to uncover structural properties of a network. A network is said to be bipartite if its nodes can be subdivided into two nonempty sets such that there are no edges between nodes in the same set. It is a difficult task to determine the closest bipartite network to a given network. This paper describes how a given network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. The algorithm also produces a node permutation which makes the possible bipartite nature of the initial adjacency matrix evident, and identifies the two sets of nodes. We finally show how the same procedure can be used to detect the presence of a large anti-community in a network and to identify it.Comment: 30 page

Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz-type matrices

The sensitivity of eigenvalues of structured matrices under general or structured perturbations of the matrix entries has been thoroughly studied in the literature. Error bounds are available and the pseudospectrum can be computed to gain insight. Few investigations have focused on analyzing the sensitivity of eigenvectors under general or structured perturbations. The present paper discusses this sensitivity for tridiagonal Toeplitz and Toeplitz-type matrices.Comment: 21 pages, 4 figure

Two-year observations of the Jupiter polar regions by JIRAM on board Juno

We observed the evolution of Jupiter's polar cyclonic structures over two years between February 2017 and February 2019, using polar observations by the Jovian InfraRed Auroral Mapper, JIRAM, on the Juno mission. Images and spectra were collected by the instrument in the 5‐μm wavelength range. The images were used to monitor the development of the cyclonic and anticyclonic structures at latitudes higher than 80° both in the northern and the southern hemispheres. Spectroscopic measurements were then used to monitor the abundances of the minor atmospheric constituents water vapor, ammonia, phosphine and germane in the polar regions, where the atmospheric optical depth is less than 1. Finally, we performed a comparative analysis with oceanic cyclones on Earth in an attempt to explain the spectral characteristics of the cyclonic structures we observe in Jupiter's polar atmosphere

Turbulence Power Spectra in Regions Surrounding Jupiter's South Polar Cyclones from Juno/JIRAM

We present a power spectral analysis of two narrow annular regions near Jupiter's South Pole derived from data acquired by the Jovian Infrared Auroral Mapper instrument onboard NASA's Juno mission. In particular, our analysis focuses on the data set acquired by the Jovian Infrared Auroral Mapper M‐band imager (hereafter IMG‐M) that probes Jupiter's thermal emission in a spectral window centered at 4.8 μm. We analyze the power spectral densities of circular paths outside and inside of cyclones on images acquired during six Juno perijoves. The typical spatial resolution is around 55 km pixel ⁻¹. We limited our analysis to six acquisitions of the South Pole from February 2017 to May 2018. The power spectral densities both outside and inside the circumpolar ring seem to follow two different power laws. The wave numbers follow average power laws of −0.9 ± 0.2 (inside) and −1.2 ± 0.2 (outside) and of −3.2 ± 0.3 (inside) and −3.4 ± 0.2 (outside), respectively, beneath and above the transition in slope located at ~2 × 10 ⁻³ km ⁻¹ wave number. This kind of spectral behavior is typical of two‐dimensional turbulence. We interpret the 500 km length scale, corresponding to the transition in slope, as the Rossby deformation radius. It is compatible with the dimensions of a subset of eddy features visible in the regions analyzed, suggesting that a baroclinic instability may exist. If so, it means that the quasi‐geostrophic approximation is valid in this context

Turbulence Power Spectra in Regions Surrounding Jupiter's South Polar Cyclones from Juno/JIRAM

We present a power spectral analysis of two narrow annular regions near Jupiter's South Pole derived from data acquired by the Jovian Infrared Auroral Mapper instrument onboard NASA's Juno mission. In particular, our analysis focuses on the data set acquired by the Jovian Infrared Auroral Mapper M‐band imager (hereafter IMG‐M) that probes Jupiter's thermal emission in a spectral window centered at 4.8 μm. We analyze the power spectral densities of circular paths outside and inside of cyclones on images acquired during six Juno perijoves. The typical spatial resolution is around 55 km pixel ⁻¹. We limited our analysis to six acquisitions of the South Pole from February 2017 to May 2018. The power spectral densities both outside and inside the circumpolar ring seem to follow two different power laws. The wave numbers follow average power laws of −0.9 ± 0.2 (inside) and −1.2 ± 0.2 (outside) and of −3.2 ± 0.3 (inside) and −3.4 ± 0.2 (outside), respectively, beneath and above the transition in slope located at ~2 × 10 ⁻³ km ⁻¹ wave number. This kind of spectral behavior is typical of two‐dimensional turbulence. We interpret the 500 km length scale, corresponding to the transition in slope, as the Rossby deformation radius. It is compatible with the dimensions of a subset of eddy features visible in the regions analyzed, suggesting that a baroclinic instability may exist. If so, it means that the quasi‐geostrophic approximation is valid in this context