14 research outputs found
Adaptive Localized Cayley Parametrization for Optimization over Stiefel Manifold
We present an adaptive parametrization strategy for optimization problems
over the Stiefel manifold by using generalized Cayley transforms to utilize
powerful Euclidean optimization algorithms efficiently. The generalized Cayley
transform can translate an open dense subset of the Stiefel manifold into a
vector space, and the open dense subset is determined according to a tunable
parameter called a center point. With the generalized Cayley transform, we
recently proposed the naive Cayley parametrization, which reformulates the
optimization problem over the Stiefel manifold as that over the vector space.
Although this reformulation enables us to transplant powerful Euclidean
optimization algorithms, their convergences may become slow by a poor choice of
center points. To avoid such a slow convergence, in this paper, we propose to
estimate adaptively 'good' center points so that the reformulated problem can
be solved faster. We also present a unified convergence analysis, regarding the
gradient, in cases where fairly standard Euclidean optimization algorithms are
employed in the proposed adaptive parametrization strategy. Numerical
experiments demonstrate that (i) the proposed strategy succeeds in escaping
from the slow convergence observed in the naive Cayley parametrization
strategy; (ii) the proposed strategy outperforms the standard strategy which
employs a retraction.Comment: 29 pages, 4 figures, 4 table
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes 路 Topp model 路 Reduced planar quartic Toppsystem 路 Singular point 路 Limit cycle 路 Hopf-saddle-node bifurcation 路 Perioddoubling bifurcation 路 Shilnikov homoclinic orbit 路 Chao