Date/Time : 29 Apr 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Liu Jinpeng, Tsinghua University

Fault-tolerant quantum computers are expected to excel in simulating unitary dynamics, such as the dynamics of a quantum state under a Hamiltonian. Most applications in scientific and engineering computations involve non-unitary dynamics. First, we propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems. The LCHS method can achieve optimal cost in terms of state preparation. Second, we generalize the LCHS formula to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. We demonstrate the applicability of LCHS to a wide range of non-Hermitian dynamics, including linear parabolic PDEs and open quantum system dynamics. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

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Date/Time : 23 Apr 2025 16:30
Venue : S17 #04-05 (SR2)
Speaker : Li Yifei, University of Tübingen

In this talk, I introduce a Sobolev-like space of order $3/2$, denoted as $\widehat{H}^{3/2}$, for Lagrangian finite elements, especially for $C^0$ elements. It is motivated by the limitations of current stability analysis of the evolving surface finite element method (ESFEM), which relies exclusively on energy estimate frameworks. When establishing a PDE-based analysis framework for ESFEM, we encounter a fundamental regularity mismatch: the ESFEM employs $C^0$ elements, while PDE regularity theory requires $H^{3/2}$ regularity for solutions. Our $\widehat{H}^{3/2}$ space resolves this challenge by demonstrating compatibility with elliptic PDE regularity theory, trace inequalities, and inverse inequalities. Notably, we extend the domain deformation estimates in ESFEM to the $\widehat{H}^{3/2}$ setting, thereby providing a rigorous foundation for PDE-based convergence analysis of ESFEM. The talk concludes with applications of the $\widehat{H}^{3/2}$ theory to numerical analysis for a coupled bulk-surface problem.

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Date/Time : 26 Mar 2025 16:30
Venue : S17 #06-11 (SR6)
Speaker : Wang Chushan, National University of Singapore

The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singular potential and nonlinearity into the NLSE. Typical examples of such potential and nonlinearity include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, and the logarithmic nonlinearity. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of accurate, efficient, and structure-preserving schemes.

In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.

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Date/Time : 12 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Zhang Junyu, National University of Singapore

Mirror descent (MD) and its stochastic variants are well-understood first order methods in convex optimization under appropriate smoothness conditions and have been extended to general relative smooth and nonconvex problems in recent years. Despite being widely studied, there are still many issues in the existing research.

In this talk, we will first reveal the inappropriateness of the currently widely adopted optimality measures, which casts doubts about the validity of current complexity results, for both deterministic and stochastic MD. Second, due to the absence of Lipschitz continuity, most existing results on nonconvex stochastic mirror descent (SMD) do not allow popular acceleration techniques such as random shuffling and variance reduction, etc. To fix these issues, we propose a novel regularity condition that is widely satisfied by most of the popular kernels. Combined with the dual space perspective of mirror descent, a new optimality measure has been proposed and a Lipschitz-like gradient bound has been established. We elaborate how these concepts can help the analysis SMD by establishing new and improved complexity results for SMD with variance reduction.

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Date/Time : 05 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Anthony Chen, University of Michigan

Fast Summation refers to a family of techniques for the fast approximation of N-body sums. While traditionally fast summation has been applied to problems coming from astrophysics or electrodynamics, many problems in geophysical fluid dynamics can be rewritten as the computation of a spherical convolution, and when these integrals are discretized, the resulting problem is an N-body problem. In this talk, I discuss a novel spherical tree code/fast multipole method based on barycentric Lagrange interpolation, as well as applications to problems coming from geophysical fluid dynamics, including tidal modeling and vortex dynamics.

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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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