48,559 research outputs found
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
Formality and Star Products
These notes, based on the mini-course given at the PQR2003 Euroschool held in
Brussels in 2003, aim to review Kontsevich's formality theorem together with
his formula for the star product on a given Poisson manifold. A brief
introduction to the employed mathematical tools and physical motivations is
also given.Comment: 49 pages, 9 figures; proceedings of the PQR2003 Euroschool. Version 2
has minor correction
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
Pacemakers in large arrays of oscillators with nonlocal coupling
We model pacemaker effects of an algebraically localized heterogeneity in a 1
dimensional array of oscillators with nonlocal coupling. We assume the
oscillators obey simple phase dynamics and that the array is large enough so
that it can be approximated by a continuous nonlocal evolution equation. We
concentrate on the case of heterogeneities with positive average and show that
steady solutions to the nonlocal problem exist. In particular, we show that
these heterogeneities act as a wave source, sending out waves in the far field.
This effect is not possible in 3 dimensional systems, such as the complex
Ginzburg-Landau equation, where the wavenumber of weak sources decays at
infinity. To obtain our results we use a series of isomorphisms to relate the
nonlocal problem to the viscous eikonal equation. We then use Fredholm
properties of the Laplace operator in Kondratiev spaces to obtain solutions to
the eikonal equation, and by extension to the nonlocal problem.Comment: 26 page
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