Convergence of a split-step Hermite method for the Gross-Pitaevskii equation

Convergence of a split-step Hermite method for the Gross-Pitaevskii equation: "

An error analysis is given for a discretization of the Gross–Pitaevskii equation by Strang splitting in time and Hermite collocation in space. A second-order error bound in L² for the semidiscretization error of the Strang splitting in time is proven under suitable regularity assumptions on the exact solution. For the semidiscretization in space, high-order convergence is shown, depending on the regularity of the exact solution. The analyses of the semidiscretizations in time and space are finally combined into an error analysis of the fully discrete method.

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Towards an Efficient Tile Matrix Inversion of Symmetric Positive Definite Matrices on Multicore Architectures. (arXiv:1002.4057v1 [cs.MS])

Towards an Efficient Tile Matrix Inversion of Symmetric Positive Definite Matrices on Multicore Architectures. (arXiv:1002.4057v1 [cs.MS]): "

The algorithms in the current sequential numerical linear algebra libraries
(e.g. LAPACK) do not parallelize well on multicore architectures. A new family
of algorithms, the tile algorithms, has recently been introduced. Previous
research has shown that it is possible to write efficient and scalable tile
algorithms for performing a Cholesky factorization, a (pseudo) LU
factorization, and a QR factorization. In this extended abstract, we attack the
problem of the computation of the inverse of a symmetric positive definite
matrix. We observe that, using a dynamic task scheduler, it is relatively
painless to translate existing LAPACK code to obtain a ready-to-be-executed
tile algorithm. However we demonstrate that non trivial compiler techniques
(array renaming, loop reversal and pipelining) need then to be applied to
further increase the parallelism of our application. We present preliminary
experimental results.

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A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems

A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems: "

Abstract

We study a tailored finite point method (TFPM) for solving the convection-diffusion-reaction equation. The solution basis
functions for the TFPM are constructed for a 5 point, 7 point and 9 point stencil. Some truncation error calculations are
given. Numerical tests are given on problems containing a boundary or interior layer. The tests compare TFPM with several
versions of a Petrov-Galerkin finite element schemes, and suggest that TFPM gives a superior resolution of the layers.

• Content Type Journal Article
• DOI 10.1007/s10915-010-9354-5
• Authors
  
  
  • Yintzer Shih, National Chung Hsing University Department of Applied Mathematics 250 Kuo-Kuang Road Taichung 40227 Taiwan
  • R. Bruce Kellogg, University of South Carolina Department of Mathematics Columbia SC 29208 USA
  • Peishan Tsai, National Chung Hsing University Department of Applied Mathematics 250 Kuo-Kuang Road Taichung 40227 Taiwan
  
  

• Journal Journal of Scientific Computing
• Online ISSN 1573-7691
• Print ISSN 0885-7474

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Spatial discretization of restricted group algebras. (arXiv:1002.4104v1 [math.OA])

Spatial discretization of restricted group algebras. (arXiv:1002.4104v1 [math.OA]): "

We consider spatial discretizations by the finite section method of the
restricted group algebra of a finitely generated discrete group, which is
represented as a concrete operator algebra via its left-regular representation.
Special emphasis is paid to the quasicommutator ideal of the algebra generated
by the finite sections sequences and to the stability of sequences in that
algebra. For both problems, the sequence of the discrete boundaries plays an
essential role. Finally, for commutative groups and for free non-commutative
groups, the algebras of the finite sections sequences are shown to be fractal.

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An iterative scheme for solving equations with locally $\sigma$-inverse monotone operators. (arXiv:1002.4165v1 [math.NA])

An iterative scheme for solving equations with locally $\sigma$-inverse monotone operators. (arXiv:1002.4165v1 [math.NA]): "

An iterative scheme for solving ill-posed nonlinear equations with locally
$\sigma$-inverse monotone operators is studied in this paper. A stopping rule
of discrepancy type is proposed. The existence of $u_{n_\delta}$ satisfying the
proposed stopping rule is proved. The convergence of this element to the
minimal-norm solution is justified mathematically.

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