污染溯源反问题的快速重构算法与理论

Inverse source problems and their fast numerical methods are one of the most attractive fields of wide investigation in the community of inverse problems. Important applications arise in environmental and emergent alert and monitoring such as chemical raw materials leakage, source identification of water pollution, fires and diffusion of toxic gases in MTR and buildings. Such problems are highly ill-posed in itself, as well as difficulties such as the measurement of temporal and spatial limitations, the inaccuracy of the measurement data, and real-time requirements of reconstruction. These challenges has seriously hindered the reconstruction process from efficient and effective solution. This proposal is concerned with the inverse problem of reconstructing contaminant source, based on the convection-diffusion model, and using space-time finite element methods, sparse regularization and sensitivity analysis theory. Systematic investigations will be carried out on feasible, fast and stable numerical methods, including: (a) semi-Lagrange finite element discretization; (b) construction of efficient multigrid methods; (c) space-time adaptive methods; (d) sparse regularization of space-time point sources; (e) choice rule of the regularization parameter, etc. The ultimate goal is to overcome the limitations of instability of space-time measurement data, low efficiency of traditional reconstruction methods, and large latency and low accuracy of the resolution. The propsed project explores the fundamental research on algorithmic and theoretical aspects of efficient numerical methods in the inverse source reconstruction problems, which enables relevant institutions for real-time monitoring and precaution plans for emergencies like pollutions.

污染溯源问题及其快速数值方法是目前反问题研究的热点问题之一，在环保（化工原料泄漏、水污染的溯源），以及突发灾害（地铁及建筑物火灾、毒气扩散）的预警与监测等方面有着重要的实际应用，但由于此类问题具有高度不适定性，加之测量的时空局限性、测量数据的非精确性、以及重构的实时性等难点，严重制约了相关研究的进展。本项目将以对流扩散模型为研究对象，采用稀疏正则化与伴随敏感性分析理论，对污染溯源反问题的时空有限元重构，系统探讨可行、快速、稳定的数值方法，包括a.半拉格朗日时空有限元离散格式；b.多重网格方法；c.基于时空的自适应方法；d.时空点源的稀疏正则化；e.正则化参数选择；以期克服时空测量的局限带来的不稳定性，以及传统方法重构速度慢和解析度过粗而导致的时延和精度低问题。项目为污染溯源的快速数值方法的研究做出基础性的探索，可以为相关机构针对污染溯源与监测等提供科学的决策依据。

溯源及其可视化成像问题及其快速数值方法是偏微分方程数值解中的一类重要问题,在环保，以及突发灾害的预警与监测，空间与地下探测等方面有着重要的实际应用，针对此类问题具有高度不适定性，加之测量的时空局限性、测量数据的非精确性、以及重构成像的实时性等难点，本项目以对流扩散和散射模型为研究对象，采用自适应技巧、伴随敏感性分析、多尺度成像理论，对污染及散射问题中的溯源及其可视化成像问题的时空重构，系统探讨可行、快速、稳定的数值方法，包括a. 溯源的唯一性理论研究；b.自适应溯源方法及其收敛性分析；c.多尺度溯源成像方法及其示性分析；克服了传统方法重构速度慢和解析度过粗而导致的时延和精度低问题。项目为溯源及其可视化成像的快速数值方法的研究做出基础性的探索，可以为相关机构针对污染溯源与监测等提供科学的决策依据。