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Chunlong Sun

Publications and source records attributed to Chunlong Sun.

7 recordsLinked to original sources
arXiv2025

Quantitative estimates for a nonlinear inverse source problem in a coupled diffusion equations with uncertain measurements

This work considers a nonlinear inverse source problem in a coupled diffusion equation from the terminal observation. Theoretically, under some conditions on problem data, we build the uniqueness theorem for this inverse problem and show two Lipschitz-type stability results in $L^2$ and $(H^1(\cdot))^*$ norms, respectively. However, in practice, we could only observe the measurements at discrete sensors, which contain the noise. Hence, this work further investigates the recovery of the unknown source from the discrete noisy measurements. We propose a stable inversion scheme and provide probabilistic convergence estimates between the reconstructions and exact solution in two cases: convergence respect to expectation and convergence with an exponential tail. We provide several numerical experiments to illustrate and complement our theoretical analysis.

math.NA↗
arXiv2025

Inverse source problem of sub-diffusion of variable exponent

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the analytical extensibility of the solutions and the weak unique continuation principle are proved, which results in the uniqueness of the inverse space-dependent source problem from local internal observation. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

math.NA↗
arXiv2024

Conditional well-posedness and data-driven method for identifying the dynamic source in a coupled diffusion system from one single boundary measurement

This work considers the inverse dynamic source problem arising from the time-domain fluorescence diffuse optical tomography (FDOT). We recover the dynamic distributions of fluorophores in biological tissue by the one single boundary measurement in finite time domain. We build the uniqueness theorem of this inverse problem. After that, we introduce a weighted norm and establish the conditional stability of Lipschitz type for the inverse problem by this weighted norm. The numerical inversions are considered under the framework of the deep neural networks (DNNs). We establish the generalization error estimates rigorously derived from Lipschitz conditional stability of inverse problem. Finally, we propose the reconstruction algorithms and give several numerical examples illustrating the performance of the proposed inversion schemes.

math.NA↗
arXiv2023

Simultaneous uniqueness and numerical inversion for an inverse problem in the time-domain diffuse optical tomography with fluorescence

In this work, an inverse problem on the determination of multiple coefficients arising from the time-domain diffuse optical tomography with fluorescence (DOT-FDOT) is investigated. We simultaneously recover the distribution of background absorption coefficient, photon diffusion coefficient as well as the fluorescence absorption in biological tissue by the time-dependent boundary measurements. We build the uniqueness theorem of this multiple coefficients simultaneous inverse problem. After that, the numerical inversions are considered. We introduce an accelerated Landweber iterative algorithm and give several numerical examples illustrating the performance of the proposed inversion schemes.

math.NA↗
arXiv2021

Uniqueness and numerical inversion in the time-domain fluorescence diffuse optical tomography

This work considers the time-domain fluorescence diffuse optical tomography (FDOT). We recover the distribution of fluorophores in biological tissue by the boundary measurements. With the Laplace transform and the knowledge of complex analysis, we build the uniqueness theorem of this inverse problem. After that, the numerical reconstructions are considered. We introduce a non-iterative inversion strategy by peak detection and an iterative inversion algorithm under the framework of regularizing scheme, then give several numerical examples in three-dimensional space illustrating the performance of the proposed inversion schemes.

math.AP↗
arXiv2019

Expression of the peak time for time-domain boundary measurements in diffuse light

Light propagation through diffusive media can be described by the diffusion equation in a space-time domain. Further, fluorescence can be described by a system of coupled diffusion equations. This paper analyzes time-domain measurements, which measure the temporal point-spread function (TPSF), at a boundary of such diffusive media with a given source and detector. We focus on the temporal position of the TPSF maximum, which we refer to as the peak time. Although some unique properties of solutions of this system have been numerically studied, we give a mathematical analysis of peak time, providing proof of the existence, uniqueness, and the explicit expression of the peak time. We clearly show the relationship between the peak time and the object position in a medium.

cs.CE↗