20,663 research outputs found
Unsolvability Cores in Classification Problems
Classification problems have been introduced by M. Ziegler as a
generalization of promise problems. In this paper we are concerned with
solvability and unsolvability questions with respect to a given set or language
family, especially with cores of unsolvability. We generalize the results about
unsolvability cores in promise problems to classification problems. Our main
results are a characterization of unsolvability cores via cohesiveness and
existence theorems for such cores in unsolvable classification problems. In
contrast to promise problems we have to strengthen the conditions to assert the
existence of such cores. In general unsolvable classification problems with
more than two components exist, which possess no cores, even if the set family
under consideration satisfies the assumptions which are necessary to prove the
existence of cores in unsolvable promise problems. But, if one of the
components is fixed we can use the results on unsolvability cores in promise
problems, to assert the existence of such cores in general. In this case we
speak of conditional classification problems and conditional cores. The
existence of conditional cores can be related to complexity cores. Using this
connection we can prove for language families, that conditional cores with
recursive components exist, provided that this family admits an uniform
solution for the word problem
An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems
In many scientific applications the solution of non-linear differential
equations are obtained through the set-up and solution of a number of
successive eigenproblems. These eigenproblems can be regarded as a sequence
whenever the solution of one problem fosters the initialization of the next. In
addition, in some eigenproblem sequences there is a connection between the
solutions of adjacent eigenproblems. Whenever it is possible to unravel the
existence of such a connection, the eigenproblem sequence is said to be
correlated. When facing with a sequence of correlated eigenproblems the current
strategy amounts to solving each eigenproblem in isolation. We propose a
alternative approach which exploits such correlation through the use of an
eigensolver based on subspace iteration and accelerated with Chebyshev
polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the
number of matrix-vector multiplications and parallelized using the Elemental
library framework. Numerical results show that ChFSI achieves excellent
scalability and is competitive with current dense linear algebra parallel
eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to
special issue of Concurrency and Computation: Practice and Experienc
Determining the Parameters of Massive Protostellar Clouds via Radiative Transfer Modeling
A one-dimensional method for reconstructing the structure of prestellar and
protostellar clouds is presented. The method is based on radiative transfer
computations and a comparison of theoretical and observed intensity
distributions at both millimeter and infrared wavelengths. The radiative
transfer of dust emission is modeled for specified parameters of the density
distribution, central star, and external background, and the theoretical
distribution of the dust temperature inside the cloud is determined. The
intensity distributions at millimeter and IR wavelengths are computed and
quantitatively compared with observational data. The best-fit model parameters
are determined using a genetic minimization algorithm, which makes it possible
to reveal the ranges of parameter degeneracy as well. The method is illustrated
by modeling the structure of the two infrared dark clouds IRDC-320.27+029 (P2)
and IRDC-321.73+005 (P2). The derived density and temperature distributions can
be used to model the chemical structure and spectral maps in molecular lines.Comment: Accepted for publication in Astronomy Report
Spectral tensor-train decomposition
The accurate approximation of high-dimensional functions is an essential task
in uncertainty quantification and many other fields. We propose a new function
approximation scheme based on a spectral extension of the tensor-train (TT)
decomposition. We first define a functional version of the TT decomposition and
analyze its properties. We obtain results on the convergence of the
decomposition, revealing links between the regularity of the function, the
dimension of the input space, and the TT ranks. We also show that the
regularity of the target function is preserved by the univariate functions
(i.e., the "cores") comprising the functional TT decomposition. This result
motivates an approximation scheme employing polynomial approximations of the
cores. For functions with appropriate regularity, the resulting
\textit{spectral tensor-train decomposition} combines the favorable
dimension-scaling of the TT decomposition with the spectral convergence rate of
polynomial approximations, yielding efficient and accurate surrogates for
high-dimensional functions. To construct these decompositions, we use the
sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of
tensors resulting from suitable discretizations of the target function. We
assess the performance of the method on a range of numerical examples: a
modifed set of Genz functions with dimension up to , and functions with
mixed Fourier modes or with local features. We observe significant improvements
in performance over an anisotropic adaptive Smolyak approach. The method is
also used to approximate the solution of an elliptic PDE with random input
data. The open source software and examples presented in this work are
available online.Comment: 33 pages, 19 figure
Ideal evolution of MHD turbulence when imposing Taylor-Green symmetries
We investigate the ideal and incompressible magnetohydrodynamic (MHD)
equations in three space dimensions for the development of potentially singular
structures. The methodology consists in implementing the four-fold symmetries
of the Taylor-Green vortex generalized to MHD, leading to substantial computer
time and memory savings at a given resolution; we also use a re-gridding method
that allows for lower-resolution runs at early times, with no loss of spectral
accuracy. One magnetic configuration is examined at an equivalent resolution of
points, and three different configurations on grids of
points. At the highest resolution, two different current and vorticity sheet
systems are found to collide, producing two successive accelerations in the
development of small scales. At the latest time, a convergence of magnetic
field lines to the location of maximum current is probably leading locally to a
strong bending and directional variability of such lines. A novel analytical
method, based on sharp analysis inequalities, is used to assess the validity of
the finite-time singularity scenario. This method allows one to rule out
spurious singularities by evaluating the rate at which the logarithmic
decrement of the analyticity-strip method goes to zero. The result is that the
finite-time singularity scenario cannot be ruled out, and the singularity time
could be somewhere between and More robust conclusions will
require higher resolution runs and grid-point interpolation measurements of
maximum current and vorticity.Comment: 18 pages, 13 figures, 2 tables; submitted to Physical Review
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