47 citations · 317 across the 48 of their papers we have counts for
29 papers · 1 filter
On optimal bases for multiscale PDEs and Bayesian homogenization
Shi Chen, Zhiyan Ding, Qin Li +1
PDE solutions are numerically represented by basis functions. Classical methods employ pre-defined bases that encode minimum desired PDE properties, which naturally cause redundant…
Second-order diffusion limit for the phonon transport equation-asymptotics and numerics
Anjali Nair, Qin Li, Weiran Sun
We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive t…
High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville
Shi Chen, Zhiyan Ding, Qin Li +1
We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equation and the radiative transfer equation (RTE) as physical models, in the high-freq…
Low-rank approximation for multiscale PDEs
Ke Chen, Shi Chen, Qin Li +2
Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing mult…
A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks
Shi Chen, Zhiyan Ding, Qin Li +1
Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We des…
Bridging and Improving Theoretical and Computational Electric Impedance Tomography via Data Completion
Tan Bui-Thanh, Qin Li, Leonardo Zepeda-Núñez
In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data…
Manifold Learning and Nonlinear Homogenization
Shi Chen, Qin Li, Jianfeng Lu +1
We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tang…
Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds
Zhiyan Ding, Qin Li, Jianfeng Lu
Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (t…
Classical limit for the varying-mass Schrödinger equation with random inhomogeneities
Shi Chen, Qin Li, Xu Yang
The varying-mass Schrödinger equation (VMSE) has been successfully applied to model electronic properties of semiconductor hetero-structures, for example, quantum dots and quantum…
Schwarz iteration method for elliptic equation with rough media based on random sampling
Ke Chen, Qin Li, Stephen J. Wright
We propose a computationally efficient Schwarz method for elliptic equations with rough media. A random sampling strategy is used to find low-rank approximations of all local solut…
Ensemble Kalman Sampler: mean-field limit and convergence analysis
Zhiyan Ding, Qin Li
Ensemble Kalman Sampler (EKS) is a method to find approximately samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unk…
Ensemble Kalman Inversion: mean-field limit and convergence analysis
Zhiyan Ding, Qin Li
Ensemble Kalman Inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems. It samples particles from a prior distribution, and introduces a motion to move…
Dynamical low-rank integrator for the linear Boltzmann equation: error analysis in the diffusion limit
Zhiyan Ding, Lukas Einkemmer, Qin Li
Dynamical low-rank algorithms are a class of numerical methods that compute low-rank approximations of dynamical systems. This is accomplished by projecting the dynamics onto a low…
A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient
Ke Chen, Qin Li, Jianfeng Lu +1
Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an…
Generalized multiscale finite element method for the steady state linear Boltzmann equation
Eric Chung, Yalchin Efendiev, Yanbo Li +1
The Boltzmann equation, as a model equation in statistical mechanics, is used to describe the statistical behavior of a large number of particles driven by the same physics laws. D…
A Numerical method for coupling the BGK model and Euler equation through the linearized Knudsen layer
Hongxu Chen, Qin Li, Jianfeng Lu
The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is sm…
Random Sampling and Efficient Algorithms for Multiscale PDEs
Ke Chen, Qin Li, Jianfeng Lu +1
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In con…
A new numerical approach to inverse transport equation with error analysis
Qin Li, Ruiwen Shu, Li Wang
The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover the optica…
Sensitivity analysis of Burgers' equation with shocks
Qin Li, Jian-Guo Liu, Ruiwen Shu
Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accurac…
Stability of Stationary Inverse Transport Equation in Diffusion Scaling
Ke Chen, Qin Li, Li Wang
We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains…
Uniform regularity for linear kinetic equations with random input based on hypocoercivity
Qin Li, Li Wang
In this paper we study the effect of randomness in kinetic equations that preserve mass. Our focus is in proving the analyticity of the solution with respect to the randomness, whi…
An asymptotic preserving method for transport equations with oscillatory scattering coefficients
Qin Li, Jianfeng Lu
We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as Knudsen number…
Exploring the locally low dimensional structure in solving random elliptic PDEs
Thomas Y. Hou, Qin Li, Pengchuan Zhang
We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to sim…
Implicit Asymptotic Preserving Method for Linear Transport Equations
Qin Li, Li Wang
The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from…