Narrow Results

We present three ideas to accelerate the filtering process used in the multilayered Spectral Element Ocean Model (SEOM). We define and analyze a Schur complement preconditioner, a lumping of small entries and an algebraic multigrid (AMG) algorithm. and a algebraic multigrid with patch smoothing algorithm. Finally, we analyze the impact of variations of the Schur complement and AMG methods on memory and computer time.

This paper is dedicated to the optimal convergence properties of a domain decomposition method involving two-Lagrange multipliers at the interface between the subdomains and additional augmented interface operators. Most methods for optimizing these augmented interface operators are based on the discretization of continuous approximations of the optimal transparent operators.^1–5 Such approach is strongly linked to the continuous equation, and to the discretization scheme. At the discrete level, the optimal transparent operator can be proved to be equal to the Schur complement of the outer subdomain. Our idea consists of approximating directly the Schur complement matrix with purely algebraic techniques involving local condensation of the subdomain degree of freedom on small patch defined on the interface between the subdomains. The main advantage of such approach is that it is much more easy to implement in existing code without any information on the geometry of the interface and of the finite element formulation used. Such technique leads to a so-called "black box" for the users. Convergence results and parallel efficiency of this new and original algebraic optimization technique of the interface operators are presented for acoustics applications.

Let $G_1$ and $G_2$ be two graphs on disjoint sets of $n_1$ and $n_2$ vertices, respectively. The corona of graphs $G_1$ and $G_2$, denoted by $G_1\circ G_2$, is the graph formed from one copy of $G_1$ and $n_1$ copies of $G_2$ where the $i$th vertex of $G_1$ is adjacent to every vertex in the $i$th copy of $G_2$. The neighborhood corona of $G_1$ and $G_2$, denoted by $G_1◇G_2$, is the graph obtained by taking one copy of $G_1$ and $n_1$ copies of $G_2$ and joining every neighbor of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$ by a new edge. In this paper, the Laplacian generalized inverse for the graphs $G_1\circ G_2$ and $G_1◇G_2$ is investigated, based on which the resistance distances of any two vertices in $G_1\circ G_2$ and $G_1◇G_2$ can be obtained. Moreover, some examples as applications are presented, which illustrate the correction and efficiency of the proposed method.

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of $2\times 2$ block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics.

In teaching a course in linear statistical models to first year graduate students or to final year undergraduate students, say, there is no way to proceed smoothly without matrices and related concepts of linear algebra; their use is really essential. Our experience is that making some particular matrix tricks familiar to students can increase their insight into linear statistical models (and also multivariate statistical analysis). In matrix algebra, there are handy, sometimes even very simple "tricks" to simplify and clarify the problem treatment — both for the student and for the researcher. Of course, the concept of trick is not uniquely defined: by trick we simply mean here a central important handy result. In this paper we collect together our Top Fourteen favourite matrix tricks for linear statistical models. We merely state our tricks with some references; a more comprehensive report including proofs, examples and full references is in progress [see Isotalo, Puntanen &, Styan (2005)].

Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constants.