Tuesday, January 26, 2021

2021-02-01 / 15:30 ~ 16:30

by 홍영준()

Deep neural networks have achieved state-of-the-art performance in a variety of fields. The exponential growth of machine learning models and the extreme success of deep learning have seen application across a multitude of disciplines. Recent works observe that a class of widely used neural networks can be viewed as the Euler method of numerical discretization. From the numerical discretization perspective, Total Variation Diminishing (TVD) Runge-Kutta methods are more advanced techniques than the explicit Euler method that produce both accurate and stable solutions. Motivated by the TVD property and a generalized Runge-Kutta method, we proposed new networks which improve robustness against adversarial attacks. If time permits, we explore a deep learning methodology that can be applied to the data-driven discovery of numerical PDEs.

2021-02-01 / 13:00 ~ 14:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 최우철(성균관대)

Distributed convex optimization has received a lot of interest from many researchers since it is widely used in various applications, containing wireless network sensor and machine learning. Recently, A. S. Berahas et al (2018) introduced a variant of the distributed gradient descent called the Near DGD+ which combines nested communications and gradient descent steps. They proved that this scheme finds the optimum point using a constant step size when the target function is strongly convex and smooth function. In the first part, we show that the scheme attains O(1/t) convergence rate for convex and smooth function. In addition we obtain a convergence result of the scheme for quasi-strong convex function. In the second part, we use the idea of Near DGD+ to design a variant of the push-sum gradient method on directed graph. This talk is based on a joint work with Doheon Kim and Seok-bae Yun.

2021-02-01 / 11:10 ~ 12:10

학과 세미나/콜로퀴엄 - PDE 세미나:

by (서울대)

In this talk, we introduce continuous-time deterministic optimal control problems with entropy regularization. Applying the dynamic programming principle, we derive a novel class of Hamilton-Jacobi-Bellman (HJB) equations and prove that the optimal value function of the maximum entropy control problem corresponds to the unique viscosity solution of the HJB equation. Our maximum entropy formulation is shown to enhance the regularity of the viscosity solution and to be asymptotically consistent as the effect of entropy regularization diminishes. A salient feature of the HJB equations is computational tractability. Generalized Hopf-Lax formulas can be used to solve the HJB equations in a tractable grid-free manner without the need for numerically optimizing the Hamiltonian. We further show that the optimal control is uniquely characterized as Gaussian in the case of control affine systems and that, for linear-quadratic problems, the HJB equation is reduced to a Riccati equation, which can be used to obtain an explicit expression of the optimal control. Lastly, we discuss how to extend our results to continuous-time model-free reinforcement learning by taking an adaptive dynamic programming approach.

2021-02-01 / 10:00 ~ 11:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by (카톨릭대)

We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. This is a part of herding problem, which considers the relationship between shepherd dogs and sheep. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region. Numerical simulations of such models quickly become unfeasible for a large number of interacting agents, as the number of interactions grows $O(N^2)$ for $N$ agents. For reducing the computational cost to $O(N)$, we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. In this approximated dynamics, the corresponding optimal control can be computed efficiently using a classical gradient descent. The resulting control is not optimal for the original system, but for a reduced RBM model. We therefore adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. This leads to a semi-feedback control strategy, where the control is applied only for a short time interval to the original system, and then compute the optimal control for the next time interval with the state of the (controlled) original dynamics.

2021-01-29 / 15:30 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 석진명(경기대)

Since a ground breaking work by Cazenave-Lions in 1982, showing uniqueness (up to symmetries) of variationally constructed solutions to Hamiltonian PDEs has played an indispensable role for verifying their orbital stability. In this talk, we discuss how to obtain the uniqueness of a family of binary star solutions to the Euler-Poisson equations, variationally constructed by McCann in 2006. Main methodology is based on perturbation arguments crucially relying on the exact asymptotic behaviors of solutions.

2021-01-29 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - 대수기하학: Birational Covers and the Bloch-Beilinson filtration.

by Pablo Pelaez(Universidad Nacional Autonoma de Mexico)

In the context of Voevodsky’s triangulated category of motives, we will describe a tower of triangulated functors which induce a finite filtration on the Chow groups. For smooth projective varieties, this finite filtration is a good candidate for the (still conjectural) Bloch-Beilinson filtration.

2021-01-26 / 16:00 ~ 17:00

학과 세미나/콜로퀴엄 - PDE 세미나:

by 김기현(카이스트)

We consider the Cauchy problem of the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. It is $L^2$-critical, has the pseudoconformal symmetry, and admits a soliton $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transform to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the high equivariance case $m \geq 1$, the pseudoconformal blow-up for smooth finite energy solutions in fact occurs in a codimension one sense; it is stable under a codimension one perturbation, but also exhibits an instability mechanism. In the radial case $m=0$, however, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions, but their blow-up rates differ from the pseudoconformal rate by a power of logarithm. We will explore these interesting blow-up dynamics (with more focus on the latter) via modulation analysis. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.

Events for the 취소된 행사 포함