An Iterative Minimization Formulation for Saddle Point Search
冷竞（复旦大学博士）
时间: 4月17日2：00——2：45 地点X2511
We proposes and
analyzes an iterative minimization formulation for searching index-1 saddle
points of an energy function. We give a general and rigorous description of
eigenvector following methodology in this iterative scheme by considering an
auxiliary optimization problem at each iteration
in which the new objective function is locally defined near the current guess.
We prove that this scheme has a quadratic local convergence rate in terms of
number of iterations, in comparison to the linear rate of the gentlest ascent
dynamics (E and Zhou, nonlinearity, vol 24, p1831, 2011) and many other
existing methods. We also propose the generalization of the new methodology for
saddle points of higher index and for constrained energy functions on manifold.
Preliminary numerical results on the nature of this iterative minimization
formulation are presented.
On the Two-Dimensional Muskat Problem with Monotone Large Initial Data
邓凡（复旦大学博士）
时间: 4月17日3：00——3：45 地点X2511
We consider the
evolution of two incompressible, immiscible fluids with different densities in
porous media, known as the Muskat problem, which in two dimensions is analogous
to the Hele-Shaw cell. We establish, for a class of large and monotone initial
data, the global existence of weak solutions. The proof is based on a local
well-posedness result for the initial data with certain specific asymptotics at
spatial infinity and a new maximum principle for the first derivative of the
graph function.
