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Kinetic equations are ubiquitous in physics and engineering. In most settings, they appear in the inverse form -- parameters in the equations are unknown, and measurements are taken to infer for their values. The group has been focusing on studying the inverse theory for these kinetic equations and have cumulated some interesting results. In particular, we study
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Inverse problem in the semi-classical limit, and high frequency waves
Inverse problem in the semi-classical limit, and high frequency waves
In the high frequency limit, Helmholtz and Schroedinger equations become the Newton's second law (Liouville equation). While inverse Helmholtz and inverse Schroedinger are ill-conditioned, inverse Liouville is a very nice problem. How does the problem gain its conditioning while the frequency increases? The plot on the left shows the reconstruction of the media using high and low frequencies (Helmholtz).
High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville, S. Chen, Z. Ding, Q. Li and L. Zepeda-Nunez. SIAM-Imaging, Vol 16 (1), 2023
Computation of optimal beams in weak turbulence. Q.~Li, A.~Nair and S.~Stechmann, accepted, Optics Continuum, 2022
Scintillation minimization versus intensity maximization in optimal beams, A. Nair, Q. Li and S. Stechmann, Optics Letters, Volume 48, issue 5
Estimating the time-evolving refractivity of a turbulent medium using optical beam measurements: a data assimilation approach, A. Nair, Q. Li and S. Stechmann, accepted, JOSA A, 2024
Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models, Borong Zhang, Martín Guerra, Qin Li, Leonardo Zepeda-Núñez, submitted,
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Plasma simulation related inversion
Plasma simulation related inversion
Fusion energy holds immense promise as a source of clean, safe, and sustainable energy. Vlasov-Poisson equation is regarded as an accurate model for plasma dynamics. As such, Vlasov-Poisson constrained optimization can potentially bring a new perspective for fusion energy generation.
Plot: top row: without external potential; second row: parameter scan for controlling plasma; last row: two instances of control results.
Suppressing Instability in a Vlasov-Poisson System by an External Electric Field Through Constrained Optimization. L. Einkemmer, Q. Li, L. Wang and Y. Yang, JCP 498(1), 2023, https://www.sciencedirect.com/science/article/pii/S002199912300757X
Reconstruction of the doping profile in Vlasov-Poisson, Inverse Problems, 40(11), R-Y, Lai, Q. Li and W. Sun, https://iopscience.iop.org/article/10.1088/1361-6420/ad7c78
Control of Instability in a Vlasov-Poisson System Through an External Electric Field, Lukas Einkemmer, Qin Li, Clément Mouhot, Yukun Yue, submitted, JCP, https://arxiv.org/abs/2407.15008
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Inverse problem from phonon transport
Inverse problem from phonon transport
Phonon transport equation is a microscopic description for heat conductance. By shining energy into the material and measuring the temperature change, one can recover the heat conductance.
The plots on the right show: 1. set up of the experiments; 2. reconstruction by running SGD; 3. evolution of the phonon transport equation.
Unique reconstruction of the heat-reflection indices at solid interfaces. W. Sun and Q. Li accepted, SIAM J. Math. Anal., 2021, vol 54 (5)
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Inverse problems from math biology
Inverse problems from math biology
Chemotaxis equation is a classical equation used to describe bacteria motion (run-and-tumble model). It is the kinetic version of macroscopic description such as the Keller-Segel model. In the lab, however, one only has access to pictures that count the density -- a macroscopic quantity. Can we use these macroscopic measurements to infer microscopic quantities such as transition kernel? Turns out the answer is yes if measurements are conducted in a smart way.
Multi-scale PDE inverse problem in bacterial movement. K.~Hellmuth, C.~Klingenberg, and Q.~Li, accepted, proceeding of HYP2022 · XVIII International Conference on Hyperbolic Problems: Theory, Numerics, Applications, 2023
Numerical reconstruction of the kinetic chemotaxis kernel from macroscopic measurements, wellposedness and illposedness, K. Hellmuth, C. Klingenberg, Q. Li and M. Tang, accepted, SIAP
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Multiscale Inverse Problems
Multiscale Inverse Problems
Generally speaking, inverse kinetic equations are usually well-conditioned, but their corresponding inverse fluid limits are ill-conditioned. How to explain such transition? We take the perspective of the Knudsen number and trace the conditioning's dependence on Kn.
Bayesian inverse problem for linear and nonlinear RTE, instability in the fluid regime. Q. Li, K. Newton and L. Wang, Computation, 10(2), 2022
Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting. K. Hellmuth, C. Klingenberg, Q. Li and M. Tang, Computation, 9(11), 2021
On diffusive scaling in acousto-optic imaging. F. Chung, R-Y. Lai and Q. Li. Inverse Problems. 36(8), 2019
Parameter Reconstruction for general transport equation. R. Lai and Q. Li. SIAM J. Math. Anal., 52(3), 2020.
Diffusive optical tomography in the Bayesian framework. K. Newton, Q. Li and A. Stuart. SIAM-Multiscale Model. Simul., 18(2), 2020
Applications of Kinetic Tools to Inverse Transport Problems. Q. Li and W. Sun. Inverse Problems., 36(3), 2020
Inverse problems for the stationary transport equation in the diffusion scaling. R-Y Lai, Q. Li, and G. Uhlmann. , SIAM J. Appl. Math., 79(6), 2019
Two-level Markov chain Monte Carlo methods for the inverse radiative transfer equation. K. Newton, and Q. Li. Entropy, 21(3), 291, 2019.
Stability of inverse transport equation in diffusion scaling and Fokker-Planck limit. K. Chen, Q. Li and L. Wang. SIAM J. Appl. Math., 78(5), 2018.
Stability of stationary inverse transport equation in diffusion scaling. K. Chen, Q. Li and L. Wang. Inverse Problems, 34, 2017.
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General theory in inverse problems and UQ
General theory in inverse problems and UQ
We develop general theory for inverse problems and UQ.
Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation. Q. Li, L. Wang and Y. Yang, accepted, SIAM J. Numer. Anal.
Inference of interaction kernels in mean-field models of opinion dynamics. W.~Chu, Q.~Li and M.~Porter, accepted, SIAM-AP, 2024
Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients, C.~Zheng, J.~Qiu, Q.~Li, X.~Zhong, 94(68), Journal of Scientific Computing, 2023
Solving the wide-band inverse scattering problem via equivariant neural networks. Q.~Li, L.~Zepada Nunez and B.~Zhang, Journal of Computational and Applied Math, 451(1)
Bridging and Improving Theoretical and Computational Electric Impedance Tomography via Data Completion, T. Bui-Thanh, Q. Li, L. Zepeda-Nunez, SISC, 44(3), 2022
Reconstruction of the emission coefficient in the nonlinear radiative transfer equation. C. Klingenberg, R-Y. Lai, and Q. Li. SIAM J. Appl. Math., 81(1), 2021
Structured random sketching for PDE inverse problems. K. Chen, Q. Li, K. Newton and S. Wright. SIAM J. Matrix Anal. Appl., 41(4), 2020
Parameter Reconstruction for general transport equation. R. Lai and Q. Li. SIAM J. Math. Anal., 52(3), 2020.
Randomized sampling for basis functions construction in generalized finite element methods. K. Chen, Q. Li, J. Lu and S. Wright. SIAM-Multiscale Model. Simul., 18(2), 2020
Applications of Kinetic Tools to Inverse Transport Problems. Q. Li and W. Sun. Inverse Problems., 36(3), 2020
Understanding and Predicting Nonlinear Turbulent Dynamical Systems with Information Theory. N. Chen, X. Hou, Q. Li and Y. Li. Atmosphere, 10(5), 248, 2019.
On quantifying uncertainties for the linearized BGK kinetic equation. C. Klingenberg, Q. Li and M. Pirner. Proc. 16th Int’l Conf. on Hyperbolic Problems, 2017.
A new numerical approach to inverse transport equation with error analysis. Q. Li, R. Shu and L. Wang. SIAM J. Numer. Anal., 56(6), 2019.
Galerkin method for stationary radiative transfer equations with uncertain coefficients. X. Zhong and Q. Li. J. Sci. Comput., Feb, 2018.
Online learning in optical tomography: a stochastic approach. K. Chen, Q. Li and J. Liu. Inverse Problems, 34(7), 2018.
Uniform regularity for linear kinetic equations with random input based on hypocoercivity. Q. Li and L. Wang. SIAM/ASA J. on Uncertainty Quantification 5, 2017
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