191,467 research outputs found
Discretization methods for homogeneous fragmentations
Homogeneous fragmentations describe the evolution of a unit mass that breaks
down randomly into pieces as time passes. They can be thought of as continuous
time analogs of a certain type of branching random walks, which suggests the
use of time-discretization to shift known results from the theory of branching
random walks to the fragmentation setting. In particular, this yields
interesting information about the asymptotic behaviour of fragmentations.
On the other hand, homogeneous fragmentations can also be investigated using
a powerful technique of discretization of space due to Kingman, namely, the
theory of exchangeable partitions of . Spatial discretization is especially
well-suited to develop directly for continuous times the conceptual method of
probability tilting of Lyons, Pemantle and Peres.Comment: 21 page
Polynomial mechanics and optimal control
We describe a new algorithm for trajectory optimization of mechanical
systems. Our method combines pseudo-spectral methods for function approximation
with variational discretization schemes that exactly preserve conserved
mechanical quantities such as momentum. We thus obtain a global discretization
of the Lagrange-d'Alembert variational principle using pseudo-spectral methods.
Our proposed scheme inherits the numerical convergence characteristics of
spectral methods, yet preserves momentum-conservation and symplecticity after
discretization. We compare this algorithm against two other established methods
for two examples of underactuated mechanical systems; minimum-effort swing-up
of a two-link and a three-link acrobot.Comment: Final version to EC
Robust Discretization of Flow in Fractured Porous Media
Flow in fractured porous media represents a challenge for discretization
methods due to the disparate scales and complex geometry. Herein we propose a
new discretization, based on the mixed finite element method and mortar
methods. Our formulation is novel in that it employs the normal fluxes as the
mortar variable within the mixed finite element framework, resulting in a
formulation that couples the flow in the fractures with the surrounding domain
with a strong notion of mass conservation. The proposed discretization handles
complex, non-matching grids, and allows for fracture intersections and
termination in a natural way, as well as spatially varying apertures. The
discretization is applicable to both two and three spatial dimensions. A priori
analysis shows the method to be optimally convergent with respect to the chosen
mixed finite element spaces, which is sustained by numerical examples
Generalized semi-infinite programming: Numerical aspects
Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization and an exchange method are derived under fairly general assumptions. The question under which conditions GSIP represents a convex problem is answered
Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization
Schwarz methods are attractive parallel solvers for large scale linear
systems obtained when partial differential equations are discretized. For
hybridizable discontinuous Galerkin (HDG) methods, this is a relatively new
field of research, because HDG methods impose continuity across elements using
a Robin condition, while classical Schwarz solvers use Dirichlet transmission
conditions. Robin conditions are used in optimized Schwarz methods to get
faster convergence compared to classical Schwarz methods, and this even without
overlap, when the Robin parameter is well chosen. We present in this paper a
rigorous convergence analysis of Schwarz methods for the concrete case of
hybridizable interior penalty (IPH) method. We show that the penalization
parameter needed for convergence of IPH leads to slow convergence of the
classical additive Schwarz method, and propose a modified solver which leads to
much faster convergence. Our analysis is entirely at the discrete level, and
thus holds for arbitrary interfaces between two subdomains. We then generalize
the method to the case of many subdomains, including cross points, and obtain a
new class of preconditioners for Krylov subspace methods which exhibit better
convergence properties than the classical additive Schwarz preconditioner. We
illustrate our results with numerical experiments.Comment: 25 pages, 5 figures, 3 tables, accepted for publication in SINU
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