Numerical Solution of Riemann–Hilbert Problems: Painlevé II

Numerical Solution of Riemann–Hilbert Problems: Painlevé II: "

Abstract

We describe a new, spectrally accurate method for solving matrix-valued Riemann–Hilbert problems numerically. The effectiveness
of this approach is demonstrated by computing solutions to the homogeneous Painlevé II equation. This can be used to relate
initial conditions with asymptotic behavior.

• Content Type Journal Article
• DOI 10.1007/s10208-010-9079-8
• Authors
  
  
  • Sheehan Olver, Oxford University Mathematical Institute, 24–29 St Giles’, Oxford, UK
  
  

• Journal Foundations of Computational Mathematics
• Online ISSN 1615-3383
• Print ISSN 1615-3375

"

Fast Matrix-Vector Multiplication in the Sparse-Grid Galerkin Method

Fast Matrix-Vector Multiplication in the Sparse-Grid Galerkin Method: "

Abstract

Sparse grid discretization of higher dimensional partial differential equations is a means to break the curse of dimensionality.
For classical sparse grids based on the one-dimensional hierarchical basis, a sophisticated algorithm has been devised to
calculate the application of a vector to the Galerkin matrix in linear complexity, despite the fact that the matrix is not
sparse. However more general sparse grid constructions have been recently introduced, e.g. based on multilevel finite elements,
where the specified algorithms only have a log-linear scaling. This article extends the idea of the linear scaling algorithm
to more general sparse grid spaces. This is achieved by abstracting the algorithm given in (Balder and Zenger, SIAM J. Sci.
Comput. 17:631, 1996) from specific bases, thereby identifying the prerequisites for performing the algorithm. In this way one can easily adapt
the algorithm to specific discretizations, leading for example to an optimal linear scaling algorithm in the case of multilevel
finite element frames.

• Content Type Journal Article
• DOI 10.1007/s10915-010-9438-2
• Authors
  
  
  • Andreas Zeiser, Sekretariat MA 3-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
  
  

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

"

Symmetric parareal algorithms for Hamiltonian systems. (arXiv:1011.6222v1 [math.NA])

Symmetric parareal algorithms for Hamiltonian systems. (arXiv:1011.6222v1 [math.NA]): "

The parareal in time algorithm allows to efficiently use parallel computing
for the simulation of time-dependent problems. It is based on a decomposition
of the time interval into subintervals, and on a predictor-corrector strategy,
where the propagations over each subinterval for the corrector stage are
concurrently performed on the processors. In this article, we are concerned
with the long time integration of Hamiltonian systems. Geometric,
structure-preserving integrators are preferably employed for such systems
because they show interesting numerical properties, in particular excellent
preservation of the total energy of the system. Using a symmetrization
procedure and/or a (possibly also symmetric) projection step, we introduce here
several variants of the original plain parareal in time algorithm [Lions, Maday
and Turinici 2001, Baffico, Bernard, Maday, Turinici and Zerah 2002, Bal and
Maday 2002] that are better adapted to the Hamiltonian context. These variants
are compatible with the geometric structure of the exact dynamics, and are easy
to implement. Numerical tests on several model systems illustrate the
remarkable properties of the proposed parareal integrators over long
integration times. Some formal elements of understanding are also provided.

"