Algebraic multigrid within defect correction for the linearized Euler equations

Given the continued difficulty of developing geometric multigrid methods that provide robust convergence for unstructured discretizations of compressible flow problems in aerodynamics, we turn to algebraic multigrid (AMG) as an alternative with the potential to automatically deal with arbitrary sources of stiffness on unstructured grids. We show here that AMG methods are able to solve linear problems associated with first-order discretizations of the compressible Euler equations extremely rapidly. In order to solve the linear problems resulting from second-order discretizations that are of practical interest, we employ AMG applied to the first-order system within a defect correction iteration. It is demonstrated on two- and three-dimensional test cases in a range of flow regimes (sub-, trans- and supersonic) that the described method converges rapidly and robustly. Copyright © 2009 John Wiley & Sons, Ltd.

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Original source : http://dx.doi.org/10.1002%2Fnla.687...

Boundary Measures for Geometric Inference

Abstract  We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric
inference. The main contribution of this work is the proof of a quantitative stability theorem for boundary measures using
tools of convex analysis and geometric measure theory. As a corollary we obtain a stability result for Federer’s curvature
measures of a compact set, showing that they can be reliably estimated from point-cloud approximations.

• Content Type Journal Article
• DOI 10.1007/s10208-009-9056-2
• Authors
  
  
  • Frédéric Chazal, INRIA Saclay Geometrica Orsay France
  • David Cohen-Steiner, INRIA Sophia-Antipolis Geometrica Sophia-Antipolis France
  • Quentin Mérigot, INRIA Sophia-Antipolis Geometrica Sophia-Antipolis France
  
  

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

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Original source : http://www.springerlink.com/content/t1834k344266k2...

Solving the Poisson equation on small aspect ratio domains using unstructured meshes. (arXiv:0912.1976v1 [math.NA])

We discuss the ill conditioning of the matrix for the discretised Poisson
equation in the small aspect ratio limit, and motivate this problem in the
context of nonhydrostatic ocean modelling. Efficient iterative solvers for the
Poisson equation in small aspect ratio domains are crucial for the successful
development of nonhydrostatic ocean models on unstructured meshes. We introduce
a new multigrid preconditioner for the Poisson problem which can be used with
finite element discretisations on general unstructured meshes; this
preconditioner is motivated by the fact that the Poisson problem has a
condition number which is independent of aspect ratio when Dirichlet boundary
conditions are imposed on the top surface of the domain. This leads to the
first level in an algebraic multigrid solver (which can be extended by further
conventional algebraic multigrid stages), and an additive smoother. We
illustrate the method with numerical tests on unstructured meshes, which show
that the preconditioner makes a dramatic improvement on a more standard
multigrid preconditioner approach, and also show that the additive smoother
produces better results than standard SOR smoothing. This new solver method
makes it feasible to run nonhydrostatic unstructured mesh ocean models in small
aspect ratio domains.

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Original source : http://arxiv.org/abs/0912.1976...

Mathematics of Photoacoustic and Thermoacoustic Tomography. (arXiv:0912.2022v1 [math.AP])

This is the manuscript of the chapter for a planned Handbook of Mathematical
Methods in Imaging that surveys the mathematical models, problems, and
algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT
and PAT represent probably the most developed of the several novel ``hybrid''
methods of medical imaging. These new modalities combine different physical
types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a
way that the resolution and contrast of the resulting method are much higher
than those achievable using only acoustic or electromagnetic measurements.

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Original source : http://arxiv.org/abs/0912.2022...

Convergence of numerical schemes for interaction equations of short and long waves. (arXiv:0912.2027v1 [math.NA])

We study numerical approximations of systems of partial differential
equations modeling the interaction of short and long waves. The short waves are
modeled by a nonlinear Schr"odinger equation which is coupled to another
equation modeling the long waves. Here, we consider the case where the long
wave equation is either a hyperbolic conservation law or a Korteweg--de Vries
equation. In the former case, we prove the strong convergence of a
Lax--Friedrichs type scheme towards the unique entropy solution of the problem,
while in the latter case we prove convergence of a finite difference scheme
towards the global solution of the problem.

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Original source : http://arxiv.org/abs/0912.2027...

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. (arXiv:0912.1968v1 [math.NA])

In this article we compare the mean-square stability properties of the
Theta-Maruyama and Theta-Milstein method that are used to solve stochastic
differential equations. As a simple extension of the standard geometric
Brownian motion as a test equation for the linear stability analysis, we
consider a scalar linear test equation with several multiplicative noise terms.
This test equation allows to begin investigating the influence of
multi-dimensional noise on the stability behaviour of the methods while the
analysis is still tractable. Our findings include: (i) the stability condition
for the Theta-Milstein method and thus, for some choices of Theta, the
conditions on the step-size, are much more restrictive than those for the
Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein
method explicitly depends on the noise terms. Further, we investigate the
effect of introducing (partially) implicitness in the diffusion approximation
terms of Milstein-type methods, thus obtaining the possibility to control the
stability properties of these methods with a further method parameter Sigma.
Numerical examples illustrate the results and provide a comparison of the
stability behaviour of the different methods.

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Original source : http://arxiv.org/abs/0912.1968...