Date/Time : 03 Mar 2025 16:00
Venue : S16 #03-06
Speaker : Li Zhuoyuan, Peking University

Data assimilation, as a key technique for merging physical models with observational to estimate state variables, has been widely explored in the fields of weather forecasting, oceanography, and ecology. However, conventional assimilation frameworks, such as variational methods (3D/4DVar) and sequential methods (EnKF), often struggle with the high dimensionality and pronounced nonlinearity inherent in both physical and observational models. In this talk, I will introduce our recent work on leveraging deep learning to address these challenges, focusing on data-driven reduced-order modeling and deep generative modeling.

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Date/Time : 13 Feb 2025 16:00
Venue : S17 #05-11 (SR5)
Speaker : Xiong Yunfeng, Beijing Normal University

Human behaviors have non-negligible impacts on spread of contagious disease. For instance, large-scale gathering and high mobility of population could lead to accelerated disease transmission, while public behavioral changes in response to pandemics may effectively reduce contacts and suppress the peak of the outbreak. In order to understand how spatial characteristics like population mobility and clustering interplay with epidemic outbreaks, we formulate an interacting particle system via an agent-based biased random walk model on a two-dimensional lattice. The “popularity” and “awareness” variables are taken into consideration to capture human natural and preventive behavioral factors, which are assumed to guide and bias agent movement in a combined way. Numerical study of the particle system and its continuum limit, a class of Keller-Segel PDEs with logarithm sensitivity, reveals that the spatial heterogeneity, like social influence locality and spatial clustering induced by self-aggregation, potentially suppresses the contacts between agents and consequently flats the epidemic curve. Surprisedly, disease responses might not necessarily reduce the susceptibility of informed individuals and even aggravate disease outbreak if each individual responds independently upon their awareness. The disease control is achieved effectively only if there are coordinated public-health interventions and public compliance to these measures. Our model may help evaluate a variety of public-health policies.

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Date/Time : 12 Feb 2025 16:30
Venue : S17 #06-11 (SR6)
Speaker : Cai Yongyong, Beijing Normal University

We introduce two conservative Crank-Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrodinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. Both Crank-Nicolson finite difference methods enjoy the second order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.

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Date/Time : 10 Feb 2025 16:00
Venue : S16 #03-06
Speaker : Zhang Lei, Shanghai Jiao Tong University

Quantum computing has garnered considerable interest in the past few years owing to its potential to offer up to exponential acceleration compared to classical computational methods. In this talk, I will first introduce the Schrödingerisation method by Jin, Liu and Yu [1], followed by its concrete quantum circuit implementation [2], with examples of heat equation and Maxwell equation [3]. I will further demonstrate applications to linear multiscale PDEs [4] and multiscale Hamilton-Jacobi equations [5].

1. Shi Jin, Nana Liu and Yue Yu, Quantum Simulation of Partial Differential Equations via Schrödingerization,Phys. Rev. Lett. 133, 230602, 2024.
2. Junpeng Hu, Shi Jin, Nana Liu, Lei Zhang, Quantum Circuits for partial differential equations via Schrödingerisation, Quantum, 8, 1563, 2024.
3. Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, Lei Zhang, Schrödingerization based Quantum Circuits for Maxwell’s Equation with time-dependent source terms, arXiv:2411.10999, 2024.
4. Junpeng Hu, Shi Jin, Lei Zhang, Quantum algorithms for multiscale partial differential equations, (SIAM) Multiscale Model. Simulation, 22(3), 1030-1067, 2024.
5. Kezhen Wang, Lei Zhang, Quantum Algorithms for Multiscale Hamilton-Jacobi Equation, preprint.

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Date/Time : 03 Feb 2025 16:30
Venue : S16 #03-04
Speaker : Lin Bo, Huairou Laboratory

In this talk, I will introduce a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging technique to solve the drift-diffusion model for semiconductor devices. The proposed scheme combines the robustness of the Scharfetter-Gummel method, which is a classical discretization for this model, with the high-order accuracy and hp-flexibility by the HDG scheme. Gradient-based indicators are introduced to utilize the hp-adaptivity. Numerical examples suggest that the proposed scheme does not exhibit oscillations even when applied to heavily doped and sharp PN-junctions.

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Date/Time : 24 Jan 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Wu Xinming, Fudan University

In this talk, we propose a finite element method for solving the coupled model of classical drift-diffusion equations and Schrödinger-Poisson equations in simulating a resonant tunneling diode (RTD). The coupling coefficient $\Theta$ and the quantum electron density $n^{\rm qu}$ are defined by weighted integrals of $|\psi_p|^2$ where $\psi_p$ is the wave function of Schrödinger equation for each momentum $p\in\mathbb{R}$. We first prove a growth rate of $|\psi_p|$ as $|p|\to +\infty$. This enables us to truncate the momentum space $\mathbb{R}$ to a bounded interval $[-P,P]$, which corresponds to an energy interval $[0,E_P]$. Based on a posteriori error estimate for computing $\Theta$ and $n^{\rm qu}$, we propose an energy-adaptive algorithm. The algorithm reduces the number of nodes greatly by capturing resonance peaks accurately and adjusting the partition of $[0,E_P]$ adaptively. Numerical experiment for an RTD shows that, based on energy-adaptive grids, the Gummel method for solving the coupled model is robust and can converge in several iterative steps.

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Date/Time : 21 Jan 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Yunpeng Shi, University of California, Davis

In this talk, we explore two classical image processing tasks motivated by cryo-electron microscopy imaging: tomographic image denoising and rigid image registration. Both tasks inherently involve operations of 2D rotations, where leveraging specific transforms can significantly enhance the speed and robustness of the algorithms.

In the first part of the talk, I will introduce an unsupervised image denoiser based on estimating the covariance matrix of clean images. A key insight is that if the image manifold is invariant under global in-plane rotations, this symmetry can be exploited to accelerate computations and reduce dimensionality. I will discuss recent advances in fast expansion into steerable bases that allow us to efficiently utilize this rotational symmetry, leading to a thousand fold improvement in the speed of covariance estimation over existing methods. This technique has been successfully applied to joint deconvolution and denoising of large-scale, real-world cryo-EM images.

In the second part, I will present a fast algorithm for aligning images using optimal transport. Our method leverages the sliced Wasserstein distance by computing the 1D Wasserstein distance between radial line projections of input images. By applying a special transform to the frequency domain, we achieve efficient alignment of two L by L images in O(L^2 log L) operations — matching the complexity of alignment using Euclidean distance. I will demonstrate the robustness of our method to translations and deformations.

Finally, I will comment on the practical limitations of both methods and discuss open questions that remain, highlighting areas for future exploration.

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Date/Time : 12 Dec 2024 16:30
Venue : S17 #04-04 (SR3)
Speaker : Huang Jizu, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

In this talk, we propose a novel accuracy-preserved tensor-train (APTT) method to solve the Boltzmann-BGK equation. We use the Crank-Nicolson Leap Frog (CNLF) scheme for temporal discretization, based on which the collision term can be evaluated only once at each time step. A second-order scheme based on the upwind rule is employed for spatial discretization to match the accuracy of the CNLF scheme. At each time step, the linear system is constructed with the tensor-train (TT) format, where the matrix, the right-hand side, and the collision term involved in the linear system are all represented using the low-rank TT format. Based on such a representation, an efficient and effective iterative solver can be implemented for solving the linear system, which can reduce two orders of magnitude in terms of both time and memory costs compared with classical methods. We present a theoretical analysis to show that the proposed APTT method can maintain accuracy and an extensive set of 3D3V Boltzmann-BGK test examples to demonstrate its effectiveness.

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Date/Time : 12 Nov 2024 16:00
Venue : S16 #03-06
Speaker : Cao Yu, Shanghai Jiao Tong University

Quantum dynamical systems are essential models for understanding the time evolution of our nature. The mathematically simplest models in this context are quantum semigroup dynamics. When a system is closed (as special instances), its dynamics are described by Schrödinger’s equation or equivalently, the von Neumann equation. There is growing interest in simulating the time-dependent Schrödinger’s equation using quantum devices due to its applications in quantum adiabatic computing, quantum control theory, and solving partial differential equations (PDEs). We will explore Sambe-Howland’s clock as a unifying framework for addressing time-dependent problems through time-independent techniques. This approach is compatible with both analog and digital quantum computing and can address some unresolved questions in recent studies. On the other hand, in open systems, semigroup dynamics are described by the Lindblad equation, which models the interaction between the system and its environment. In this talk, we will also discuss a family of structure-preserving algorithms for simulating Lindblad equations, which have close connections to designing stochastic and quantum algorithms.

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