An Iterative Minimization Formulation for Saddle Point Search

冷競(jìng)（復(fù)旦大學(xué)博士）

時(shí)間: 4月17日2：00——2：45   地點(diǎn)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

鄧凡（復(fù)旦大學(xué)博士）

時(shí)間: 4月17日3：00——3：45    地點(diǎn)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.