Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields

This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to prevent violations of local bounds for the minimal and maximal eigenvalues. In contrast to the flux-corrected transport (FCT) algorithm developed previously by the author and existing slope limiting techniques for stress tensors, the admissible eigenvalue range is defined implicitly and the limited antidiffusive terms are incorporated into the residual of the nonlinear system. In addition to scalar limiters that use a common correction factor for all components of a tensor-valued antidiffusive flux, tensor limiters are designed using spectral decompositions. The new limiter functions are analyzed using tensorial extensions of the existing AFC theory for scalar convection-diffusion equations. The proposed methodology is backed by rigorous proofs of eigenvalue range preservation and Lipschitz continuity. Convergence of pseudo time-stepping methods to stationary solutions is demonstrated in numerical studies.

Extreme Spectra Realization by Nonsymmetric Tridiagonal and Nonsymmetric Arrow Matrices

We consider the following inverse extreme eigenvalue problem: given the real numbers {λ1j,λjj}j=1n and the real vector x(n)=x1,x2,…,xn, to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that {λ1j,λjj}j=1n are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and x(n),λn(n) is an eigenpair of the matrix. We give sufficient conditions for the existence of such matrices. Moreover our results generate an algorithmic procedure to compute a unique solution matrix.

An Efficient Algorithm for Finding the Maximal Eigenvalue of Zero Symmetric Nonnegative Matrices

In this paper, we propose an improved power algorithm for finding maximal eigenvalues. Without any partition, we can get the maximal eigenvalue and show that the modified power algorithm is convergent for zero symmetric reducible nonnegative matrices. Numerical results are reported to demonstrate the effectiveness of the modified power algorithm. Finally, a modified algorithm is proposed to test the positive definiteness (positive semidefiniteness) of Z-matrices.

On inverse eigenvalue problems for two kinds of special banded matrices

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

Extremal Inverse Eigenvalue Problem for a Special Kind of Matrices

We consider the following inverse eigenvalue problem: to construct a special kind of matrix (real symmetric doubly arrow matrix) from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. Our results are constructive and they generate algorithmic procedures to construct such matrices.

The Applicability of Bipartite Graph Model for Thunderstorms Forecast over Kolkata

Single Spectrum Bipartite Graph (SSBG) model is developed to forecast thunderstorms over Kolkata(22∘32′N,88∘20′E)during the premonsoon season (April-May). The statistical distribution of normal probability is observed for temperature, relative humidity, convective available potential energy (CAPE), and convective inhibition energy (CIN) to quantify the threshold values of the parameters for the prevalence of thunderstorms. Method of conditional probability is implemented to ascertain the possibilities of the occurrence of thunderstorms within the ranges of the threshold values. The single spectrum bipartite graph connectivity model developed in this study consists of two sets of vertices; one set includes two time vertices (00UTC, 12UTC) and the other includes four meteorological parameters: temperature, relative humidity, CAPE, and CIN. Three distinct ranges of maximal eigen values are obtained for the three categories of thunderstorms. Maximal eigenvalues for severe, ordinary, and no thunderstorm events are observed to be(2.6±0.12),(1.88±0.09), and(1.26±.03), respectively. The ranges of the threshold values obtained using ten year data (1997–2006) are considered as the reference range and the result is validated with the IMD (India Meteorological Department) observation, Doppler Weather Radar (DWR) Products, and satellite images of 2007. The result reveals that the model provides 12- to 6-hour forecast (nowcasting) of thunderstorms with 96% to 98% accuracy.