Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Self-adaptive isogeometric spatial discretisations of the first and second-order forms of the neutron transport equation with dual-weighted residual error measures and diffusion acceleration

As implemented in a new modern-Fortran code, NURBS-based isogeometric analysis (IGA) spatial discretisations and self-adaptive mesh refinement (AMR) algorithms are developed in the application to the first-order and second-order forms of the neutron transport equation (NTE). These AMR algorithms are shown to be computationally efficient and numerically accurate when compared to standard approaches. IGA methods are very competitive and offer certain unique advantages over standard finite element methods (FEM), not least of all because the numerical analysis is performed over an exact representation of the underlying geometry, which is generally available in some computer-aided design (CAD) software description. Furthermore, mesh refinement can be performed within the analysis program at run-time, without the need to revisit any ancillary mesh generator. Two error measures are described for the IGA-based AMR algorithms, both of which can be employed in conjunction with energy-dependent meshes. The first heuristically minimises any local contributions to the global discretisation error, as per some appropriate user-prescribed norm. The second employs duality arguments to minimise important local contributions to the error as measured in some quantity of interest; this is commonly known as a dual-weighted residual (DWR) error measure and it demands the solution to both the forward (primal) and the adjoint (dual) NTE. Finally, convergent and stable diffusion acceleration and generalised minimal residual (GMRes) algorithms, compatible with the aforementioned AMR algorithms, are introduced to accelerate the convergence of the within-group self-scattering sources for scattering-dominated problems for the first and second-order forms of the NTE. A variety of verification benchmark problems are analysed to demonstrate the computational performance and efficiency of these acceleration techniques.Open Acces

Symmetry in Electromagnetism

Electromagnetism plays a crucial role in basic and applied physics research. The discovery of electromagnetism as the unifying theory for electricity and magnetism represents a cornerstone in modern physics. Symmetry was crucial to the concept of unification: electromagnetism was soon formulated as a gauge theory in which local phase symmetry explained its mathematical formulation. This early connection between symmetry and electromagnetism shows that a symmetry-based approach to many electromagnetic phenomena is recurrent, even today. Moreover, many recent technological advances are based on the control of electromagnetic radiation in nearly all its spectra and scales, the manipulation of matter–radiation interactions with unprecedented levels of sophistication, or new generations of electromagnetic materials. This is a fertile field for applications and for basic understanding in which symmetry, as in the past, bridges apparently unrelated phenomena―from condensed matter to high-energy physics. In this book, we present modern contributions in which symmetry proves its value as a key tool. From dual-symmetry electrodynamics to applications to sustainable smart buildings, or magnetocardiography, we can find a plentiful crop, full of exciting examples of modern approaches to electromagnetism. In all cases, symmetry sheds light on the theoretical and applied works presented in this book

GeomInt–Mechanical Integrity of Host Rocks

This open access book summarizes the results of the collaborative project “GeomInt: Geomechanical integrity of host and barrier rocks - experiment, modeling and analysis of discontinuities” within the Program: Geo Research for Sustainability (GEO: N) of the Federal Ministry of Education and Research (BMBF). The use of geosystems as a source of resources, a storage space, for installing underground municipal or traffic infrastructure has become much more intensive and diverse in recent years. Increasing utilization of the geological environment requires careful analyses of the rock–fluid systems as well as assessments of the feasibility, efficiency and environmental impacts of the technologies under consideration. The establishment of safe, economic and ecological operation of underground geosystems requires a comprehensive understanding of the physical, (geo)chemical and microbiological processes on all relevant time and length scales. This understanding can only be deepened on the basis of intensive laboratory and in-situ experiments in conjunction with reliable studies on the modeling and simulation (numerical experiments) of the corresponding multi-physical/chemical processes. The present work provides a unique handbook for experimentalists, modelers, analysts and even decision makers concerning the characterization of various types of host rocks (salt, clay, crystalline formations) for various geotechnical applications

Proceedings, MSVSCC 2016

Proceedings of the 10th Annual Modeling, Simulation & Visualization Student Capstone Conference held on April 14, 2016 at VMASC in Suffolk, Virginia

Kinetic Physics with Solar Wind Heavy Ions Measured at 1 AU

One possibility to study kinetic processes such as particle acceleration, transport and thermalization in the solar wind are in-situ measurements of ion velocity distribution functions (VDFs). In particular the extension of the VDF analysis to the wide range of heavy ions allows to investigate the dependence of the underlying processes on the particles’ mass and charge in a systematical way. In this thesis we analyze nonthermal signatures in the velocity distribution functions of solarwind heavy ions, in particular differential speeds and thermal speed ratios between the heavy ions and the solar wind protons. For our investigation we utilize measurements that were conducted with the Charge Element Isotope Analysis System (CELIAS) experiment onboard the Solar and Heliospheric Observatory (SOHO) that is located at the Lagrange Point L1 at a distance of about one astronomical unit (1 AU) from the Sun. The measurement data is recorded during a relatively short period around solar minimum between DOY 174 and 220 in 1996. The SOHO/CELIAS experiment is one of only a few instrument suites located in the undisturbed solar wind, far away from planetary magnetospheres, that is able to measure a wide range of solar wind heavy ion species with relatively high counting statistics and fast measurement cadence. Our studies include both the investigation of long-term speed spectra in the form of accumulated count rates over the full measurement period and the analysis of short-term speed distributions, so-called 1D-reduced velocity distribution functions, that are recorded with the intrinsic cadence of the CELIAS/CTOF (Charge-Time-Of-Flight) sensor of about 5 minutes

Polynomial and rational approximation for electronic structure calculations

Atomic-scale simulation of matter has become an important research tool in physics, chemistry, material science and biology as it allows for insights which neither theoretical nor experimental investigation can provide. The most accurate of these simulations are based on the laws of quantum mechanics, in which case the main computational bottleneck becomes the evaluation of functions f(H) of a sparse matrix H (the Hamiltonian). One way to evaluate such matrix functions is through polynomial and rational approximation, the theory of which is reviewed in Chapter 2 of this thesis. It is well known that rational functions can approximate the relevant functions with much lower degrees than polynomials, but they are more challenging to use in practice since they require fast algorithms for evaluating rational functions r(H) of a matrix argument H. Such an algorithm has recently been proposed in the form of the Pole Expansion and Selected Inversion (PEXSI) scheme, which evaluates r(H) by writing r(x) = P k ck x−zk in partial-fraction-decomposed form and then employing advanced sparse factorisation techniques to evaluate only a small subset of the entries of the resolvents (H − z) −1 . This scheme scales better than cubically in the matrix dimension, but it is not a linear scaling algorithm in general. We overcome this limitation in Chapter 3 by devising a modified, linear-scaling PEXSI algorithm which exploits that most of the fill-in entries in the triangular factorisations computed by the PEXSI algorithm are negligibly small. Finally, Chapter 4 presents a novel algorithm for computing electric conductivities which requires evaluating a bivariate matrix function f(H, H). We show that the Chebyshev coefficients ck1k2 of the relevant function f(x1, x2) concentrate along the diagonal k1 ∼ k2 and that this allows us to approximate f(x1, x2) much more efficiently than one would expect based on a straightforward tensor-product extension of the one-dimensional arguments

Pulsar discoveries by volunteer distributed computing and the strongest continuous gravitational wave signal

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