52 research outputs found
Point cloud discretization of Fokker-Planck operators for committor functions
The committor functions provide useful information to the understanding of
transitions of a stochastic system between disjoint regions in phase space. In
this work, we develop a point cloud discretization for Fokker-Planck operators
to numerically calculate the committor function, with the assumption that the
transition occurs on an intrinsically low-dimensional manifold in the ambient
potentially high dimensional configurational space of the stochastic system.
Numerical examples on model systems validate the effectiveness of the proposed
method.Comment: 17 pages, 11 figure
Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
Let be a connected, closed, orientable Riemannian surface and denote
by the -th eigenvalue of the Laplace-Beltrami operator on
. In this paper, we consider the mapping .
We propose a computational method for finding the conformal spectrum
, which is defined by the eigenvalue optimization problem
of maximizing for fixed as varies within a conformal
class of fixed volume . We also propose a
computational method for the problem where is additionally allowed to vary
over surfaces with fixed genus, . This is known as the topological
spectrum for genus and denoted by . Our
computations support a conjecture of N. Nadirashvili (2002) that
, attained by a sequence of surfaces degenerating to
a union of identical round spheres. Furthermore, based on our computations,
we conjecture that ,
attained by a sequence of surfaces degenerating into a union of an equilateral
flat torus and identical round spheres. The values are compared to
several surfaces where the Laplace-Beltrami eigenvalues are well-known,
including spheres, flat tori, and embedded tori. In particular, we show that
among flat tori of volume one, the -th Laplace-Beltrami eigenvalue has a
local maximum with value . Several properties are also studied
computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure
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