Inverse Scattering and Tomography, Delay Differential Equations in Cyber-Physical Systems

Siamak RabieniaHaratbar, Purdue University

Abstract

This dissertation concludes three mathematical works in Inverse Problems as well as an engineering work in Cyber-Physical Systems. Our mathematical works are in the area of Inverse Problems and Scattering Theory where the main focuses are on Support theorem of Vectorial Light-ray transform in Minkowski Spaces, Dynamical X-ray Tomography, and Inverse Scattering of the biharmonic operator. For the first project, we prove a support theorem for vector fields whose integral lines vanish on an open set of light-like lines. In this work, we employ the method of microlocal analysis and Pseudo Differential Operators. We illustrate the application of our results for the inverse recovery of the hyperbolic Dirichlet-to-Neumann map through various examples. The second project is motivated by an inverse problem arising from medical imaging where we investigate a dynamic operator, A , integrating over a family of level curves when the object changes between the measurements. Microlocal analysis is used to determine which singularities can be recovered by the data-set. We prove that not all singularities can be recovered, depending on the particular movement of the object compare to the X-ray source. We then find sufficient conditions under which the reconstruction is possible. We also show that one can establish stability estimates and injectivity results under the Visibility, the Local and Semi-global Bolker conditions. We illustrate the implementation of our results in Fan-beam geometry. In the final project, we consider a perturbed biharmonic operator and study the inverse scattering problem for this operator by investigating the recovery process of the magnetic field A and the potential field V. Using the high-frequency asymptotic of the scattering amplitude of the biharmonic operator, we prove the unique recovery of curl A and V − 12∇ · A. By investigating the near-field scattering, we show that the high-frequency asymptotic expansion up to an error O(λ −4 ) (where λ is the frequency or the spectral parameter) recovers the same above quantities but does not provide any additional information about the magnetic and the potential fields. We also establish stability estimates for curl A and V − 12∇·A. Our engineering work is in the area of Cyber-Physical systems where we study Real-time hybrid simulation (RTHS). By the use of RTHS, which is an efficient technique that investigate cyber-physical system in a cost-effective way, we introduce powerful indicators to examine the structural behavior and seismic resilience of a structure through various settings.

Degree

Ph.D.

Advisors

Stefanov, Purdue University.

Subject Area

Medical imaging|Engineering|Mathematics|Medicine

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