Abstract : High-dimensional problems appear in various scientific areas and suffers from the notorious “curse of dimensionality” issue. This mini-symposium brings together the scientists working on quantum chemistry, classical and quantum kinetic theory and geophysics, etc., to share and exchanges their ideas in solving high-dimensional problems. Our mini-symposium will highlight the recent developments in both deterministic and particle-based methods. It is expected to elucidate the following questions: 1. Why do we need to solve high-dimensional problems? 2. What are common difficulties? 3. How to alleviate the curse of dimensionality by either novel mathematical methodologies or sophisticated usage of high-performance computing environment?
[04360] An efficient stochastic particle method for high-dimensional nonlinear PDEs
Format : Talk at Waseda University
Author(s) :
Sihong Shao (Peking University)
Abstract : We introduce a stochastic particle method (SPM) to solve high-dimensional nonlinear PDEs in the weak sense. The weak formulation is a time-dependent high-dimensional integral, and different test functions can bring us various information about the solution. To determine the dynamics of the particle system, we linearize the nonlinear terms using the previous time step solutions and establish a relationship of weak formulation between adjacent time steps via the Lawson-Euler scheme. The resulting stochastic particles follow the behavior of the solution in an adaptive manner, therefore mitigating curse of dimensionality to a certain extent. Numerical experiments on the 6-D Allen-Cahn equation and 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the accuracy and efficiency of SPM. This work is joint with Zhengyang Lei and Yunfeng Xiong.
[04066] Overcoming the dynamical sign problem via adaptive particle annihilation
Format : Talk at Waseda University
Author(s) :
Yunfeng Xiong (Beijing Normal University)
Abstract : The dynamical sign problem poses a fundamental obstacle to particle-based simulations in high dimensional space. To resolve it, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE). SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each cluster. Combining SPADE with the stationary phase approximation, we attempt to simulate the Wigner dynamics in 6-D and 12-D phase space.
[02525] A splitting Hamiltonian Monte Carlo method for efficient sampling
Format : Talk at Waseda University
Author(s) :
Lei Li (Shanghai Jiao Tong University)
Lin Liu (Shanghai Jiao Tong University)
Yuzhou Peng (Shanghai Jiao Tong University)
Abstract : In this talk, I will introduce a splitting Hamiltonian Monte Carlo algorithm, which can be computationally efficient when combined with the random mini-batch strategy. By splitting the potential energy into numerically nonstiff and stiff parts, one makes a proposal using the nonstiff part, followed by a Metropolis rejection step using the stiff part that is often easy to compute. The splitting allows efficient sampling from systems with singular potentials and/or multiple potential barriers. We also use random batch strategies to reduce the computational cost in generating the proposals for problems arising from many-body systems and Bayesian inference, and estimate both the strong and the weak errors in the Hamiltonian induced by the random batch approximation.
[02255] Ergodicity and sharp error estimate of Stochastic Gradient Langevin Dynamics
Format : Talk at Waseda University
Author(s) :
Yuliang Wang (Shanghai Jiao Tong University)
Abstract : We establish a sharp error estimate for the Stochastic Gradient Langevin Dynamics (SGLD). Under mild assumptions, we obtain a uniform-in-time $O(\eta^2)$ bound for the KL-divergence between SGLD and the Langevin diffusion, where $\eta$ is the step size. Based on this, we are able to obtain an $O(\eta)$ bound for its sampling error in terms of Wasserstein or total variation distances. Moreover, we prove the geometric ergodicity of SGLD algorithm under $W_1$ distance without global convexity.
[04069] Multi-level Monte Carlo methods in stochastic density functional theory
Format : Talk at Waseda University
Author(s) :
Huajie Chen (Beijing Normal University)
Abstract : The stochastic density functional theory (sDFT) has become an attractive approach in electronic structure calculations. The computational complexity of Hamiltonian diagonalization is replaced by introducing a set of random orbitals leading to sub-linear scaling of evaluating the ground-state observables. This work investigates the convergence and acceleration of the self-consistent field (SCF) iterations for sDFT in the presence of statistical error. We also study some variance reduction schemes by multi-level Monte Carlo methods that can accelerate the SCF convergence.
[04185] Inchworm Monte Carlo Method for Spin Chain Models in Open Quantum Systems
Format : Talk at Waseda University
Author(s) :
Zhenning Cai (National University of Singapore)
Abstract : We consider open quantum systems where the quantum system is coupled to a harmonic bath. When the coupling is weak, we can mimic Feynman's methodology to represent the dynamics as the sum of infinite integrals represented by diagrams. In this talk, we will discuss an efficient diagrammatic approach, known as the inchworm Monte Carlo method, to compute the observables in open quantum systems. Applications to spin chain models will be considered in the numerical tests.
[04220] A short-memory operator splitting scheme for constant-Q viscoelastic wave equation
Format : Talk at Waseda University
Author(s) :
Yunfeng Xiong (Beijing Normal University)
Xu Guo (Shandong University)
Abstract : We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. An operator splitting scheme is introduced to solve the resulting set of equations, where the auxiliary dynamics can be solved exactly, so that it gets rid of the numerical stiffness and discretization errors. Numerical experiments on both 1-D diffusive wave equation and 2-D constant-Q P-and S-wave equations are presented to validate the accuracy and efficiency of the proposed scheme.
[04363] Solving Boltzmann equation with neural sparse representation
Format : Talk at Waseda University
Author(s) :
Zhengyi Li (Peking Univeristy)
Yanli Wang (Beijing Computational Science Research Center)
Hongsheng Liu (Huawei Technologies Co. Ltd)
Zidong Wang (Huawei Technologies Co. Ltd)
Bin Dong (Beijing International Center for Mathematical Research & Center for Machine Learning Research, Peking University)
Abstract : We consider the neural sparse representation to solve Boltzmann equation. The different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision model, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution.
