Date/Time : 28 Aug 2024 16:30
Venue : S17 #04-05 (SR2)
Speaker : Zhang Xiaolong, Nanyang Technological University & Hunan Normal University

The Logarithmic Schrödinger Equation (LogSE) exhibits rich dynamics and possesses unique properties compared to the standard Schrödinger equation with cubic nonlinearity. However, the logarithmic nonlinear term $f(u)=uln(∣u∣^2)$ presents significant challenges for both numerical solutions and error analysis, primarily due to low regularity and singularity. In this talk, we will characterize this low regularity within suitable fractional Sobolev spaces and derive the first set of error estimates for the time-splitting Fourier spectral methods with initial data of fractional-order regularity. These results are supported by recent regularity estimates for the LogSE from the PDE community and ample numerical evidence.(Joint work with Prof. Li-Lian Wang, NTU)

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Date/Time : 08 May 2024 16:30
Venue : S17 #04-05 (SR2)
Speaker : Junfu Yao, Johns Hopkins University

Minimal surfaces are known as critical points to the area functional. In this talk, I’ll focus on the question of finding minimal surfaces with prescribed behavior at the asymptotic infinity. This is called the asymptotic Plateau problem. I’ll briefly review previous results and present a min-max theory for finding solutions to this problem. As a corollary of the mountain-pass theorem, I’ll also discuss a topological uniqueness result.

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Date/Time : 20 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Jin Zeyu, Peking University

A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, we propose a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, we rigorously discuss the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory. Joint work with Ruo Li.

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Date/Time : 13 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Lin Bo, National University of Singapore

The committor function is a central object for quantifying the transitions between metastable states of dynamical systems. Recently, a number of computational methods based on deep neural networks have been developed for computing the high-dimensional committor function. The success of the methods relies on sampling adequate data for the transition, which still is a challenging task for complex systems at low temperatures. This talk will present a deep learning method with two novel adaptive sampling schemes. In the two schemes, the data are generated actively from a modified potential where the bias potential is constructed from the learned committor function. We theoretically show that the data in the two schemes are uniformly distributed along the transition tube. The efficiency of the method is illustrated on high-dimensional systems including the alanine dipeptide and a dimer system immersed in the water box.

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Date/Time : 06 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Li Yifei, National University of Singapore

Designing a numerical scheme that can preserve the geometric structure for anisotropic surface diffusion with an arbitrary anisotropic surface energy is a long-standing problem. In this talk, we propose and analyze a structure-preserving parametric finite element methods (SP-PFEM) for the evolution of a closed curve in 2D, which preserve two geometric structures – area conservation and energy dissipation – at the full-discretized level, The SP-PFEM innovates with a novel surface energy matrix and the Cahn-Hoffman ξ-vector, leading to a new geometric identity for dealing with the weighted mean curvature. This new identity allows our SP-PFEM to be easily extended to various geometric flows with anisotropic effects. Extensive numerical results demonstrate its efficiency, stability, and success in other geometric flows.

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Date/Time : 28 Feb 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Guo Ruchi, The Chinese University of Hong Kong

In physics and mathematics, a large class of PDE systems can be formulated as energy minimization problems subject to specific constraints. Lagrange multipliers are widely used for solving such problems, which however leads to min-max optimization problems known as saddle point systems. The development of fast solvers for saddle point systems, especially those nonlinear ones, presents considerable challenges in the sense that

(i) one has to consider the preconditioning in the primal and dual directions collectively;

(ii) the preconditioners have to evolve in iterations to accommodate the nonlinearity;

(iii) all the subsystems and subproblems cannot be solved exactly due to the high computational cost involved.

In this work, we develop an efficient transformed primal-dual (TPD) algorithm, and this is done by modifying the original gradient flow from the perspective of matrix factorization. We prove the global optimal linear convergence in terms of the condition numbers measured by variable metrics. To demonstrate the effectiveness of our approach, we apply the algorithm to a Darcy-Forchheimer model and a nonlinear ferromagnetic model, showing it is much more efficient than some traditional methods.

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Date/Time : 14 Feb 2024 16:30
Venue : S17 #04-06 (SR1)
Speaker : Liu Xin, Chinese Academy of Sciences

We propose constraint dissolving approaches for optimization problems over a class of Riemannian manifolds. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of the proposed approaches.

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Date/Time : 19 Jan 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Linfeng Zhang, AI for Science Institute, Beijing, and DP Technology

The rapid development of artificial intelligence is driving significant changes in the field of molecular and materials modeling, simulation, and design. In this report, we will explore the breakthroughs and progress in AI-assisted molecular modeling, with a focus on the shift from multiscale methods to large pre-trained models and the resulting software, data, and model ecosystem. In particular, we will investigate the possibility of a large atomic model that covers the entire periodic table and, based on this, envision the future of the materials genome.

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Date/Time : 17 Jan 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Sam Stechmann, University of Wisconsin–Madison

Sea ice floats on the surface of the Arctic Ocean, and plays an important role in Earth’s climate. Traditionally, sea ice has been modeled as a material, as a continuum, with various rheological models proposed. From a more detailed perspective, sea ice is a collection of many individual blocks of ice, called “floes.” In this talk, I will discuss approaches for floe-based (i.e., particle-based) models of sea ice, including multi-scale models to bridge the gap between floe-based and continuum-based models.

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