New developments in inverse theory for differential equation networks: from trees to general graphs

这个项目旨在发展微分方程网络的逆理论，也称为量子图。网络结构在许多科学和工程问题中起着基础性的作用。传统上，这些模型被用来描述桥梁、空间结构、天线、输电线路杆、钢网钢筋和其他典型的土木工程对象。最近，这些模型被用于在更小的尺度上模拟现象，包括分层材料，如陶瓷或金属泡沫，渗滤网络和碳和石墨烯纳米管，以及石墨烯带。虽然微分方程网络的逆理论对于所有上述应用都很重要，但是它还没有得到充分的发展。该项目将发展一般微分方程网络或量子图的逆理论，重点是波，热，薛定谔，狄拉克和其他微分方程的逆谱和动力学问题，这些问题对物理和工程应用很重要。至少两个博士学位学生们将积极参与该项目，并将就此专题开设一门新的研究生课程。量子图的逆问题以前几乎只在树上研究过，也就是说，没有圈的图。该项目将考虑具有周期的量子图，目标是更广泛适用的微分方程网络。沿着方程的未知系数，可以恢复图的拓扑结构及其几何参数。所提出的方法来解决反问题的量子图与圈将联合收割机频谱和动力学方法相结合。它将使用逆理论中的边界控制方法，该方法利用了动力系统的可控性和可识别性之间的深层联系，以及最近开发的用于解决树上逆问题的剥叶方法。扩展这些方法的微分方程的一般图形将需要新的发展的边界控制和剥叶方法，以及结果和方法的组合图论，偏微分方程，谱分析，和非调和傅立叶级数。相应的偏微分方程的可控性的结果，一般图的周期也将得到。将开发用于解决一般微分方程网络的控制和逆问题的数值算法，并将进行广泛的数值实验以验证其有效性。该项目由数学科学部的应用数学项目和促进竞争性研究的既定项目（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project intends to develop inverse theory for differential equation networks, also called quantum graphs. Network-like structures play a fundamental role in many problems of science and engineering. Classically, these models were used to describe bridges, space structures, antennas, transmission-line posts, steel-grid reinforcements and other typical objects of civil engineering. More recently, these models are being used to model phenomena on a much smaller scale, including hierarchical materials like ceramic or metallic foams, percolation networks and carbon and graphene nano-tubes, and graphene ribbons. While inverse theory of differential equation networks is important for all aforementioned applications, it has not, however, been sufficiently developed. This project will develop the inverse theory of general differential equation networks, or, quantum graphs, with a focus on inverse spectral and dynamical problems for the wave, heat, Schrodinger, Dirac and other differential equations which are important for applications in physics and engineering. At least two Ph.D. students will take an active part in the project, and a new graduate course on this topic will be developed. Inverse problems for quantum graphs have previously been studied almost exclusively on trees, that is, graphs without cycles. This project will consider quantum graphs with cycles, targeting more broadly applicable differential equation networks. Along with unknown coefficients of the equations, the topology of the graph and its geometrical parameters will be recovered. The proposed approach to solving inverse problems for quantum graphs with cycles will combine spectral and dynamical methods. It will use the boundary control method in inverse theory, which exploits deep connections between controllability and identifiability of dynamical systems and a recently developed leaf-peeling method for solving inverse problems on trees. Extension of these methods to differential equations on general graphs will require new developments of the boundary control and leaf peeling methods as well as results and methods of combinatorial graph theory, partial differential equations, spectral analysis, and nonharmonic Fourier series. Controllability results for the corresponding partial differential equations on general graphs with cycles will be also obtained. Numerical algorithms for solving control and inverse problems for general differential equation networks will be developed, and extensive numerical experiments will be performed to validate their effectiveness. This project is jointly funded by the Applied Mathematics Program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.