ncku-talk2024-05-01

Applied Mathematics Seminar

DATE 2024-05-01　15:10-16:00

PLACE 數學系館 3F 會議室

SPEAKER Professor Moody T. Chu（North Carolina State University）

TITLE On the Dynamics of the Maximin Flow

ABSTRACT In a complex system, such as the molecular dynamics, chemical kinetics, nucleation mechanism, or even the Lagrangian of a constrained convex programming problem, the presence of a saddle point often represents that a transition of events has occurred. Determining the locations of saddle points in the configuration space and the way they affect the transition provide critical information about the underlying complex system. This paper proposes a dynamical system approach to explore this problem. In addition to being capable of finding saddle points, the flow exhibits some intriguing behavior nearby a saddle point, which is demonstrated by graphic examples in various settings. Maximin flows also arise naturally from complex-valued differential equations over analytic vector fields due to the Cauchy-Riemann equations. The maximin flow can be cast as a gradient flow in the Krein space under indefinite inner product, whence the Lojasiewicz gradient inequality can be generalized. It is proved that a solution trajectory has finite arc length and, hence, converges to a singleton saddle point.

Applied Mathematics Seminar

DATE	2024-05-01　15:10-16:00

PLACE	數學系館 3F 會議室

SPEAKER	Professor Moody T. Chu（North Carolina State University）

TITLE	On the Dynamics of the Maximin Flow

ABSTRACT	In a complex system, such as the molecular dynamics, chemical kinetics, nucleation mechanism, or even the Lagrangian of a constrained convex programming problem, the presence of a saddle point often represents that a transition of events has occurred. Determining the locations of saddle points in the configuration space and the way they affect the transition provide critical information about the underlying complex system. This paper proposes a dynamical system approach to explore this problem. In addition to being capable of finding saddle points, the flow exhibits some intriguing behavior nearby a saddle point, which is demonstrated by graphic examples in various settings. Maximin flows also arise naturally from complex-valued differential equations over analytic vector fields due to the Cauchy-Riemann equations. The maximin flow can be cast as a gradient flow in the Krein space under indefinite inner product, whence the Lojasiewicz gradient inequality can be generalized. It is proved that a solution trajectory has finite arc length and, hence, converges to a singleton saddle point.