BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

On algebraic TVD-VOF methods for tracking material interfaces

We revisit simple algebraic VOF methods for advection of material interfaces based of the well established TVD paradigm. We show that greatly improved representation of contact discontinuities is obtained through use of a novel CFL-dependent limiter whereby the classical TVD bounds are exceeded. Perfectly crisp numerical interfaces are obtained with very limited numerical atomization (flotsam and jetsam) as compared to previous SLIC schemes. Comparison of the algorithm with accurate geometrical VOF shows larger error at given mesh resolution, but comparable efficiency when the reduced computational cost is accounted for

Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275

Algebraic coarsening in voter models with intermediate states

The introduction of intermediate states in the dynamics of the voter model modifies the ordering process and restores an effective surface tension. The logarithmic coarsening of the conventional voter model in two dimensions is eliminated in favour of an algebraic decay of the density of interfaces with time, compatible with Model A dynamics at low temperatures. This phenomenon is addressed by deriving Langevin equations for the dynamics of appropriately defined continuous fields. These equations are analyzed using field theoretical arguments and by means of a recently proposed numerical technique for the integration of stochastic equations with multiplicative noise. We find good agreement with lattice simulations of the microscopic model.Comment: 11 pages, 5 figures; minor typos correcte

JDFTx: software for joint density-functional theory

Density-functional theory (DFT) has revolutionized computational prediction of atomic-scale properties from first principles in physics, chemistry and materials science. Continuing development of new methods is necessary for accurate predictions of new classes of materials and properties, and for connecting to nano- and mesoscale properties using coarse-grained theories. JDFTx is a fully-featured open-source electronic DFT software designed specifically to facilitate rapid development of new theories, models and algorithms. Using an algebraic formulation as an abstraction layer, compact C++11 code automatically performs well on diverse hardware including GPUs. This code hosts the development of joint density-functional theory (JDFT) that combines electronic DFT with classical DFT and continuum models of liquids for first-principles calculations of solvated and electrochemical systems. In addition, the modular nature of the code makes it easy to extend and interface with, facilitating the development of multi-scale toolkits that connect to ab initio calculations, e.g. photo-excited carrier dynamics combining electron and phonon calculations with electromagnetic simulations.Comment: 9 pages, 3 figures, 2 code listing

Linguistics and some aspects of its underlying dynamics

In recent years, central components of a new approach to linguistics, the Minimalist Program (MP) have come closer to physics. Features of the Minimalist Program, such as the unconstrained nature of recursive Merge, the operation of the Labeling Algorithm that only operates at the interface of Narrow Syntax with the Conceptual-Intentional and the Sensory-Motor interfaces, the difference between pronounced and un-pronounced copies of elements in a sentence and the build-up of the Fibonacci sequence in the syntactic derivation of sentence structures, are directly accessible to representation in terms of algebraic formalism. Although in our scheme linguistic structures are classical ones, we find that an interesting and productive isomorphism can be established between the MP structure, algebraic structures and many-body field theory opening new avenues of inquiry on the dynamics underlying some central aspects of linguistics.Comment: 17 page