567 research outputs found
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Computational aspects of spectral invariants
The spectral theory of the Laplace operator has long been studied in connection with physics. It appears in the wave equation, the heat equation, Schroedinger's equation and in the expression of quantum effects such as the Casimir force. The Casimir effect can be studied in terms of spectral invariants computed entirely from the spectrum of the Laplace operator. It is these spectral invariants and their computation that are the object of study in the present work.
The objective of this thesis is to present a computational framework for the spectral zeta function and its derivative on a Euclidean domain in , with rigorous theoretical error bounds when this domain is polygonal. To obtain error bounds that remain practical in applications an improvement to existing heat trace estimates is necessary. Our main result is an original estimate and proof of a heat trace estimate for polygons that improves the one of van den Berg and Srisatkunarajah, using finite propagation speed of the corresponding wave kernel. We then use this heat trace estimate to obtain a rigorous error bound for computations. We will provide numerous examples of our computational framework being used to calculate for a variety of situations involving a polygonal domain, including examples involving cutouts and extrusions that are interesting in applications.
Our second result is the development a new eigenvalue solver for a planar polygonal domain using a partition of unity decomposition technique. Its advantages include multiple precision and ease of use, as well as reduced complexity compared to Finite Elemement Method. While we hoped that it would be able to contend with existing packages in terms of speed, our implementation was many times slower than MPSPack when dealing with the same problem (obtaining the first 5 digits of the principal eigenvalue of the regular unit hexagon).
Finally, we present a collection of numerical examples where we compute the spectral determinant and Casimir energy of various polygonal domains. We also use our numerical tools to investigate extremal properties of these spectral invariants. For example, we consider a square with a small square cut out of the interior, which is allowed to rotate freely about its center
Large Field Inflation from Axion Mixing
We study the general multi-axion systems, focusing on the possibility of
large field inflation driven by axions. We find that through axion mixing from
a non-diagonal metric on the moduli space and/or from St\"uckelberg coupling to
a U(1) gauge field, an effectively super-Planckian decay constant can be
generated without the need of "alignment" in the axion decay constants. We also
investigate the consistency conditions related to the gauge symmetries in the
multi-axion systems, such as vanishing gauge anomalies and the potential
presence of generalized Chern-Simons terms. Our scenario applies generally to
field theory models whose axion periodicities are intrinsically sub-Planckian,
but it is most naturally realized in string theory. The types of axion mixings
invoked in our scenario appear quite commonly in D-brane models, and we present
its implementation in type II superstring theory. Explicit stringy models
exhibiting all the characteristics of our ideas are constructed within the
frameworks of Type IIA intersecting D6-brane models on T6/OR and Type IIB
intersecting D7-brane models on Swiss-Cheese Calabi-Yau orientifolds.Comment: v2: references added, typos corrected; v1: 1+85 pages, 4 figure
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