4,149 research outputs found
Eigenvalue computations in the context of data-sparse approximations of integral operators
In this work, we consider the numerical solution of a large eigenvalue problem resulting from a finite rank discretization of an integral operator. We are interested in computing a few eigenpairs, with an iterative method, so a matrix representation that allows for fast matrix-vector products is required. Hierarchical matrices are appropriate for this setting, and also provide cheap LU decompositions required in the spectral transformation technique. We illustrate the use of freely available software tools to address the problem, in particular SLEPc for the eigensolvers and HLib for the construction of H-matrices. The numerical tests are performed using an astrophysics application. Results show the benefits of the data-sparse representation compared to standard storage schemes, in terms of computational cost as well as memory requirements.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion under projects TIN2009-07519, TIN2012-32846 and AIC10-D-000600 and by Fundacao para a Ciencia e a Tecnologia - FCT under project FCT/MICINN proc 441.00.Román Moltó, JE.; Vasconcelos, PB.; Nunes, AL. (2013). Eigenvalue computations in the context of data-sparse approximations of integral operators. Journal of Computational and Applied Mathematics. 237(1):171-181. doi:10.1016/j.cam.2012.07.021S171181237
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
NLO Renormalization in the Hamiltonian Truncation
Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical
technique for solving strongly coupled QFTs, in which the full Hilbert space is
truncated to a finite-dimensional low-energy subspace. The accuracy of the
method is limited only by the available computational resources. The
renormalization program improves the accuracy by carefully integrating out the
high-energy states, instead of truncating them away. In this paper we develop
the most accurate ever variant of Hamiltonian Truncation, which implements
renormalization at the cubic order in the interaction strength. The novel idea
is to interpret the renormalization procedure as a result of integrating out
exactly a certain class of high-energy "tail states". We demonstrate the power
of the method with high-accuracy computations in the strongly coupled
two-dimensional quartic scalar theory, and benchmark it against other existing
approaches. Our work will also be useful for the future goal of extending
Hamiltonian Truncation to higher spacetime dimensions.Comment: 28pp + appendices, detailed version of arXiv:1706.0612
Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum
The computation of eigenvalues of large-scale matrices arising from finite
element discretizations has gained significant interest in the last decade.
Here we present a new algorithm based on slicing the spectrum that takes
advantage of the rank structure of resolvent matrices in order to compute m
eigenvalues of the generalized symmetric eigenvalue problem in operations, where is a small constant
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