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

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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.