Applied Math
PhD Seminar

2021 Spring

The seminar of this semester is organized by Bichen Lu and Xu Huang, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan.

Past Presentations

2021-03-11 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Matrix completion is about recovering a matrix from its partial revealed entries, and it can often be achieved by exploiting the inherent simplicity or low dimensional structure of the target matrix. For instance, a typical notion of matrix simplicity is low rank. In this talk we will study matrix completion based on another low dimensional structure, namely the low rank Hankel structure in the Fourier domain. It is shown that matrices with this structure can be exactly recovered by solving a convex optimization program provided the sampling complexity is nearly optimal.

2021-03-18 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We present a fully discrete convergent finite difference scheme for the Q-tensor flow of liquid crystals based on the energy-stable semi-discrete scheme by Zhao, Yang, Gong, and Wang (Comput. Methods Appl. Mech. Engrg. 2017). We prove stability properties of the scheme and show convergence to weak solutions of the Q-tensor flow equations. We demonstrate the performance of the scheme in numerical simulations.

2021-03-25 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The well-known Constantin-Lax-Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities. De Gregorio modified the CLM model by adding a convective term, which is known important for fluid dynamics. We present two results on the De Gregorio model, based on a joint work with Prof. Z. Lei and J. Liu. The first one is the global well-posedness of such a model for general initial data with non-negative (or non-positive) vorticity. The second one is an exponential stability result of ground states, which is similar to the recent significant work of Jia et al. (Ration Mech Anal, 231:1269–1304, 2019).

2021-04-01 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The 2D eigenvalue problem (2D EVP) is a class of the 2-parameter eigenvalue problems and dates back to the work of Blum and Chang in 1970s. 2D EVP seeks real scalars $\lambda, \mu$, and a corresponding vector $x$ satisfying the following equations \begin{align*} Ax & = \lambda x + \mu Cx,\\ x^HCx & =0, \\ x^Hx & =1, \end{align*} where $A$ and $C$ are Hermitian and $C$ is indefinite. We will briefly introduce its applications and fundamental theory, including its relation to eigenvalue optimization, variational characterization and number of solutions. We will introduce a 2D Rayleigh Quotient method to solve it, which is very suitable for large scale problems. Examples are given to demonstrate the efficiency of the algorithm compared to other eigenvalue optimization methods. Generalizations, e.g., to kDEVP, will also be covered.

2021-04-08 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, the Cauchy problem for the Moore-Gibson-Thompson (MGT) equations will be introduced, which can describe the propagation of sound (nonlinear acoustics) in thermoviscous fluids. Concerning the linearized MGT equations, some qualitative properties of solutions will be shown, including sharp decay estimates, asymptotic profiles, large-time approximations, and singular limits with respect to the thermal relaxation tending to zero. Then, global (in time) existence of small data solution or blow-up of solutions for the semilinear MGT equations and Jordan-MGT equations will be presented. Particularly, the blow-up phenomena for the semilinear MGT equations in the conservative case and the dissipative case are quite different. This talk is based on joint works with Ryo Ikehata and Alessandro Palmieri.

2021-04-15 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand As a phase space language, the Wigner quantum dynamics bears a close analogy to classical mechanics and its numerical resolution has been drawing growing attention in the past few decades, especially in studying nanoscale semiconductors, quantum many-body systems and quantum tomography. However, the high dimensionality and oscillatory nature of the Wigner function give rise to a formidable challenge in both computation and data storage. In this talk, we will discuss our recent progress in the stochastic algorithm for the time-dependent Wigner equation. We would like to share our experience on how to establish the mathematical framework of the stochastic algorithm for the partial differential equation, how to find out the fundamental numerical sign problem that limits the efficiency of the existing algorithms, how to borrow the basic idea from harmonic analysis, combinatorics, number theory and high-dimensional statistical learning to overcome the notorious sign problem, as well as how to combine all these ingredients to make reliable Wigner simulations in 6-D phase space. These works are joint with Prof. Sihong Shao.

2021-04-22 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The seismic inverse problem is one of the most important problems in geosciences, such as the earthquake location and the seismic tomography. Seismic signals are recorded by stations at the surface to determine the earthquake information and the structure of the earth’s interior. From the mathematical point of view, the seismic inverse problem can be formulated as a PDE constrained nonlinear optimization problem. Compared to the traditional L2 norm which suffers from the cycle skipping problem, the objective function based on the optimal transport metric holds better convexity property and high resistance to the noise. Thus, more accurate and robust inversion results can be obtained by introducing the optimal transport theory to seismic inverse problems.

2021-04-29 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The proposal of this note is to study the back flow and blowup properties of solution to the geophysical boundary layer problem, which differs from the classical Prandtl boundary layer equations with a nonlocal integral term arising from the Coriolis force. Firstly, we show that the back flow point appears at the physical boundary in a finite time under certain constraint on the growth rate of the tangential velocity when both of the initial tangential velocity and the upstream velocity are monotonically increasing with respect to the normal variable of the boundary, even if the momentum of the outer flow is favorable for the classical Prandtl equations. Moreover, when the monotonicity condition is violated and the initial velocity and outflow velocity satisfy certain condition on a transversal plane, for any smooth solution decaying exponentially in the normal variable to the geophysical boundary layer problem, it is proved that its Sobolev norm  blows up in a finite time.

2021-05-06 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we consider blind super-resolution of point sources. As can be seen, this problem can be reformulated as a matrix recovery problem. By exploiting the low rank structure of the vectorized Hankel matrix associated with the target matrix, a convex approach called Vectorized Hankel Lift is proposed to exactly recover the target matrix with nearly optimal sampling complexity. Additionally, a new variant of the MUSIC method for line spectrum estimation arising from the framework for solving blind super-resolution may be of independent interest.

2021-05-13 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand When a constant supersonic Euler flow goes past a wedge or through a nozzle, a transonic shock may occur and we can easily calculate the state of the subsonic flow behind the shock. In this talk, I will discuss the stability of the background solution under a small perturbation of the boundary and an additional small exothermic reaction. This will be a nonlinear free boundary value problem with nonlinear boundary conditions for mixed type equations. To deal with these problems, some coordinate transformations and the fixed point theory will be introduced and we will face a linear elliptic equation with oblique derivative boundary conditions in a domain with angular points.

2021-05-20 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.

2021-05-27 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this talk we will present local and global well-posedness results for the 2D Peskin problem in critical spaces. Specifically, we will prove the local well-posedness for any initial data in satisfying the so-called well-stretched assumption. Then, we will show that when the initial string configuration is sufficiently close to an equilibrium in , global-in-time solution uniquely exists and it will converge to an equilibrium as . This is based on a joint work with Prof. Quoc-Hung Nguyen.

2021-06-03 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle. One of the key points is determining the position of the shock front. Compared with 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

2021-06-10 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We present a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography. The weak formulation of the PDE for the given inverse problem is leveraged, where the solution and the test function are parameterized as deep neural networks. Then, the weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of this approach. This presentation is based on the joint work with Gang Bao (Zhejiang U.), Xiaojing Ye (Georgia State U.) and Haomin Zhou (Georgia Tech.).

2021-06-17 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Elementary particles are either Bosons or Fermions. A gas of Bosons can be represented mathematically by a collection of interacting random loops. At very low temperatures, a gas of Bosons undergoes a phase transitions: the Bose-Einstein Condensate (BEC) appears. Despite the fundamental nature of the problem, a complete understanding of BEC is still at large. It had been conjectured by Richard Feynman that the BEC is represented by “infinite” loops. The recently developed theory of random interlacements provides a framework for a rigorous understanding of a canonical candidate of such paths. We will present the result of a recent preprint, showing this for the free and the mean-field gas. In the last part of the talk, we will examine the discontinuous phase transition for the Bose gas with Huang-Yang-Luttinger hard-core interaction. No knowledge of Bosons or random interlacements is required.

2021-06-24 16:10 - 17:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker–Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with external potentials, optimal transport problems, and aggregation-diffusion models. Based on the Energetic Variational Approach, a trajectory equation is first derived by using the balance between the maximal dissipation principle and least action principle. By a convex-splitting technique, we propose energy dissipating numerical schemes for the trajectory equation. Rigorous numerical analysis reveals that the nonlinear numerical schemes are uniquely solvable, naturally respect mass conservation and positivity at fully discrete level, and preserve steady states in an admissible convex set, where the discrete Jacobian of flow maps is positive. Under certain assumptions on smoothness and a positive Jacobian, the numerical schemes are shown to be second order accurate in space and first order accurate in time. Extensive numerical simulations are performed to demonstrate several valuable features of the proposed schemes. In addition to the preservation of physical structures, such as positivity, mass conservation, discrete energy dissipation, and steady states, numerical simulations further reveal that our numerical schemes are capable of solving degenerate cases of the Fokker–Planck equations effectively and robustly. It is shown that the developed numerical schemes have convergence order even in degenerate cases with the presence of solutions having compact support and can accurately and robustly compute the waiting time of free boundaries without any oscillation. The limitation of numerical schemes due to a singular Jacobian of the flow map is also discussed. This work is joint with Wenbin Chen, Chun Liu，Xingye Yue and Shenggao Zhou.