202 research outputs found
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Nonlinear Geometric Optics Based Multiscale Stochastic Galerkin Methods for Highly Oscillatory Transport Equations with Random Inputs
We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG)
methods for a class of highly oscillatory transport equations that arise in
semiclassical modeling of non-adiabatic quantum dynamics. These models contain
uncertainties, particularly in coefficients that correspond to the potentials
of the molecular system. We first focus on a highly oscillatory scalar model
with random uncertainty. Our method is built upon the nonlinear geometrical
optics (NGO) based method, developed in \cite{NGO} for numerical approximations
of deterministic equations, which can obtain accurate pointwise solution even
without numerically resolving spatially and temporally the oscillations. With
the random uncertainty, we show that such a method has oscillatory higher order
derivatives in the random space, thus requires a frequency dependent
discretization in the random space. We modify this method by introducing a new
"time" variable based on the phase, which is shown to be non-oscillatory in the
random space, based on which we develop a gPC-SG method that can capture
oscillations with the frequency-independent time step, mesh size as well as the
degree of polynomial chaos. A similar approach is then extended to a
semiclassical surface hopping model system with a similar numerical conclusion.
Various numerical examples attest that these methods indeed capture accurately
the solution statistics {\em pointwisely} even though none of the numerical
parameters resolve the high frequencies of the solution.Comment: 35 page
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