a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

An algebraic multigrid method is presented to solve large systems of linear equations. The coarsen- ing is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than four. The matching algorithm favors the strongest negative coupling(s), inducing a problem depen- dant coarsening. This aggregation is combined with piecewise constant (unsmoothed) prolongation, ensuring low setup cost and memory requirements. Compared with previous aggregation-based multigrid methods, the scalability is enhanced by using a so-called K-cycle multigrid scheme, providing Krylov subspace acceleration at each level. This paper is the logical continuation of (SIAM J. Sci. Comput., 30 (2008), pp. 1082-1103), where the analysis of a model anisotropic problem shows that aggregation-based two-grid methods may have optimal order convergence, and of (Numer. Lin. Alg. Appl., 15 (2008), pp. 473-487), where it is shown that K-cycle multigrid may provide optimal or near optimal convergence under mild assumptions on the two-grid scheme. Whereas in these papers only model problems with geometric aggregation were considered, here a truly algebraic method is presented and tested on a wide range of discrete second order scalar elliptic PDEs, including nonsymmetric and unstructured problems. Numerical results indicate that the proposed method may be significantly more robust as black box solver than the classical AMG method as implemented in the code AMG1R5 by K. Stuben. The parallel implementation is also discussed. Satisfactory speedups are obtained on a medium size multi-processor cluster that is typical of today com- puter market. A code implemanting the method is freely available for download both as a Fortran program and a Matlab function.

/pdf/an-aggregation-based-algebraic-multigrid-method-2o49oedlr7.pdf

An aggregation-based algebraic multigrid method

The purpose of this book is to offer an overview of the most popular domain decomposition methods for partial differential equations (PDEs). These methods are widely used for numerical simulations in solid mechanics, electromagnetism, flow in porous media, etc., on parallel machines from tens to hundreds of thousands of cores. The appealing feature of domain decomposition methods is that, contrary to direct methods, they are naturally parallel. The authors focus on parallel linear solvers. The authors present all popular algorithms, both at the PDE level and at the discrete level in terms of matrices, along with systematic scripts for sequential implementation in a free open-source finite element package as well as some parallel scripts. Also included is a new coarse space construction (two-level method) that adapts to highly heterogeneous problems. Audience: This book is intended for those working in domain decomposition methods, parallel computing, and iterative methods, in particular those who need to implement parallel solvers for PDEs. It will also be of interest to mechanical, civil, and aeronautical engineers, physical and environmental scientists, and physicists in need of parallel PDE solvers.

https://hal.archives-ouvertes.fr/cel-01100932/file/bookddm.pdf

An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation

The establishment of covalent junctions between carbon nanotubes (CNTs) and the modification of their straight tubular morphology are two strategies needed to successfully synthesize nanotube-based three-dimensional (3D) frameworks exhibiting superior material properties. Engineering such 3D structures in scalable synthetic processes still remains a challenge. This work pioneers the bulk synthesis of 3D macroscale nanotube elastic solids directly via a boron-doping strategy during chemical vapour deposition, which influences the formation of atomic-scale "elbowg" junctions and nanotube covalent interconnections. Detailed elemental analysis revealed that the "elbowg" junctions are preferred sites for excess boron atoms, indicating the role of boron and curvature in the junction formation mechanism, in agreement with our first principle theoretical calculations. Exploiting this materialĝ€™s ultra-light weight, super-hydrophobicity, high porosity, thermal stability, and mechanical flexibility, the strongly oleophilic sponge-like solids are demonstrated as unique reusable sorbent scaffolds able to efficiently remove oil from contaminated seawater even after repeated use.

/pdf/covalently-bonded-three-dimensional-carbon-nanotube-solids-1yaxo81fd6.pdf

Covalently bonded three-dimensional carbon nanotube solids via boron induced nanojunctions

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modeling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

Solvers for the cardiac bidomain equations.

We introduce spectral element-based algebraic multigrid ($\rho$AMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically "smooth" vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of "strong" connections (relatively large coefficients in the operator matrix). One aim of $\rho$AMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. $\rho$AMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for $\rho$AMGe along with numerical experiments demonstrating its robustness.

Spectral AMGe ($\rho$AMGe)

We conduct an analysis of the sensitivity of three linear algebra-based ranking methods: the Colley, Massey, and Markov methods. Our analysis employs reverse engineering, in that we start with a simple input ranking vector that we use to build a perfect season, and we then determine the output rating vectors produced by the three methods. This analysis shows that the PageRank rating vector is strongly nonuniformly spaced, while the Colley and Massey methods provide a uniformly spaced rating vector, which is more natural for a perfect season. We further extend our study of the sensitivity and rank stability of these three methods with a careful perturbation analysis of the same perfect season dataset. We find that the Markov method is highly sensitive to small changes in the data and show with an example from the NFL that the Markov method's ranking vector displays some odd unstable behavior.

Sensitivity and Stability of Ranking Vectors

The purpose of this note is to describe an algorithm resulting from the uniting of two ideas introduced and applied elsewhere. For many problems, AMG has always been difficult due to complexities whose natures are difficult to discern from the entries of matrix A alone. Element-based interpolation has been shown to be an effective method for some of these problems, but it requires access to the element matrices on all levels. One way to obtain these has been to perform element agglomeration to form coarse elements, but in complicated situations defining the coarse degrees of freedom (dofs) is not easy. The spectral approach to coarse dof selection is very attractive due to its elegance and simplicity. The algorithm presented here combines the robustness of element interpolation, the ease of coarsening by element agglomeration, and the simplicity of defining coarse dofs through the spectral approach. As demonstrated in the numerical results, the method does yield a reasonable solver for the problems described. It can, however, be an expensive method due to the number and cost of the local, small dense linear algebra problems; making it a generally competitive method remains an area for further research.

/pdf/spectral-element-agglomerate-amge-21wp9ompii.pdf

Spectral Element Agglomerate AMGe

This paper poses and solves a new problem, the rankability problem, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. Ranking is a fundamental data science ...

https://epubs.siam.org/doi/pdf/10.1137/18M1183595

Timothy P. Chartier

Papers

Spectral AMGe ($\rho$AMGe)

Sensitivity and Stability of Ranking Vectors

Spectral Element Agglomerate AMGe

The Rankability of Data

Adaptive algebraic smoothers