Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

Lin, Guang; Ou, Na; Zhang, Zecheng; Zhang, Zhidong

doi:10.1088/1361-6420/ad2904

Title: Restoring the discontinuous heat equation source using sparse boundary data and dynamic sensors

Journal Article · 29 February 2024 · Inverse Problems

DOI: https://doi.org/10.1088/1361-6420/ad2904 · OSTI ID:2317694

; Ou, Na; Zhang, Zecheng;

Abstract This study focuses on addressing the inverse source problem associated with the parabolic equation. We rely on sparse boundary flux data as our measurements, which are acquired from a restricted section of the boundary. While it has been established that utilizing sparse boundary flux data can enable source recovery, the presence of a limited number of observation sensors poses a challenge for accurately tracing the inverse quantity of interest. To overcome this limitation, we introduce a sampling algorithm grounded in Langevin dynamics that incorporates dynamic sensors to capture the flux information. Furthermore, we propose and discuss two distinct dynamic sensor migration strategies. Remarkably, our findings demonstrate that even with only two observation sensors at our disposal, it remains feasible to successfully reconstruct the high-dimensional unknown parameters.

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Sponsoring Organization:: USDOE Office of Science (SC), Fusion Energy Sciences (FES); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0024583

OSTI ID:: 2317694

Alternate ID(s):: OSTI ID: 2301752

Journal Information:: Inverse Problems, Journal Name: Inverse Problems Journal Issue: 4 Vol. 40; ISSN 0266-5611

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United Kingdom

Language:: English

References (22)

Mcmc Methods for Diffusion Bridges Beskos, Alexandros; Roberts, Gareth; Stuart, Andrew Stochastics and Dynamics, Vol. 08, Issue 03 https://doi.org/10.1142/S0219493708002378	journal	September 2008
Inverse Source Problems Isakov, Victor https://doi.org/10.1090/surv/034	book	January 1990
Langevin Diffusions and Metropolis-Hastings Algorithms Roberts, G. O. Methodology and Computing in Applied Probability, Vol. 4, Issue 4, p. 337-357 https://doi.org/10.1023/A:1023562417138	journal	December 2002
Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements Lin, Guang; Zhang, Zecheng; Zhang, Zhidong Inverse Problems, Vol. 38, Issue 12 https://doi.org/10.1088/1361-6420/ac99f9	journal	November 2022
Multi-variance replica exchange SGMCMC for inverse and forward problems via Bayesian PINN Lin, Guang; Wang, Yating; Zhang, Zecheng Journal of Computational Physics, Vol. 460 https://doi.org/10.1016/j.jcp.2022.111173	journal	July 2022
Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data Li, Zhiyuan; Zhang, Zhidong Inverse Problems, Vol. 36, Issue 11 https://doi.org/10.1088/1361-6420/abbc5d	journal	October 2020
Inverse problems: A Bayesian perspective Stuart, A. M. Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000061	journal	May 2010
An inverse source problem for the heat equation and the enclosure method Ikehata, Masaru Inverse Problems, Vol. 23, Issue 1 https://doi.org/10.1088/0266-5611/23/1/009	journal	December 2006
Recovering an unknown source in a fractional diffusion problem Rundell, William; Zhang, Zhidong Journal of Computational Physics, Vol. 368 https://doi.org/10.1016/j.jcp.2018.04.046	journal	September 2018
The uniqueness of inverse problems for a fractional equation with a single measurement Kian, Yavar; Li, Zhiyuan; Liu, Yikan Mathematische Annalen, Vol. 380, Issue 3-4 https://doi.org/10.1007/s00208-020-02027-z	journal	July 2020
User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient Dalalyan, Arnak S.; Karagulyan, Avetik Stochastic Processes and their Applications, Vol. 129, Issue 12 https://doi.org/10.1016/j.spa.2019.02.016	journal	December 2019
B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD Lin, Guang; Moya, Christian; Zhang, Zecheng Journal of Computational Physics, Vol. 473 https://doi.org/10.1016/j.jcp.2022.111713	journal	January 2023
MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster Cotter, S. L.; Roberts, G. O.; Stuart, A. M. Statistical Science, Vol. 28, Issue 3 https://doi.org/10.1214/13-STS421	journal	August 2013
Exponential Convergence of Langevin Distributions and Their Discrete Approximations Roberts, Gareth O.; Tweedie, Richard L. Bernoulli, Vol. 2, Issue 4 https://doi.org/10.2307/3318418	journal	December 1996
Identification of a discontinuous source in the heat equation Hettlich, F.; Rundell, W. Inverse Problems, Vol. 17, Issue 5 https://doi.org/10.1088/0266-5611/17/5/315	journal	September 2001
Stability for inverse source problems by Carleman estimates Huang, X.; Yu Imanuvilov, O.; Yamamoto, M. Inverse Problems, Vol. 36, Issue 12 https://doi.org/10.1088/1361-6420/aba892	journal	December 2020
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models Efendiev, Y.; Hou, T.; Luo, W. SIAM Journal on Scientific Computing, Vol. 28, Issue 2 https://doi.org/10.1137/050628568	journal	January 2006
On the Identification of Source Term in the Heat Equation from Sparse Data Rundell, William; Zhang, Zhidong SIAM Journal on Mathematical Analysis, Vol. 52, Issue 2 https://doi.org/10.1137/19M1279915	journal	January 2020
Inverse problems for heat equation and space–time fractional diffusion equation with one measurement Helin, Tapio; Lassas, Matti; Ylinen, Lauri Journal of Differential Equations, Vol. 269, Issue 9 https://doi.org/10.1016/j.jde.2020.05.022	journal	October 2020
On an adaptive preconditioned Crank–Nicolson MCMC algorithm for infinite dimensional Bayesian inference Hu, Zixi; Yao, Zhewei; Li, Jinglai Journal of Computational Physics, Vol. 332 https://doi.org/10.1016/j.jcp.2016.11.024	journal	March 2017
An inverse source problem for parabolic equations with local measurements Cheng, Jin; Liu, Jijun Applied Mathematics Letters, Vol. 103 https://doi.org/10.1016/j.aml.2020.106213	journal	May 2020
Reconstruction of a Space-Time-Dependent Source in Subdiffusion Models via a Perturbation Approach Jin, Bangti; Kian, Yavar; Zhou, Zhi SIAM Journal on Mathematical Analysis, Vol. 53, Issue 4 https://doi.org/10.1137/21M1397295	journal	January 2021

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