Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

Development of a Real-Time Hierarchical 3D Path Planning Algorithm for Unmanned Aerial Vehicles

Unmanned aerial vehicles (UAVs) frequently operate in partially or entirely unknown environments. As the vehicle traverses the environment and detects new obstacles, rapid path replanning is essential to avoid collisions. This thesis presents a new algorithm called Hierarchical D* Lite (HD*), which combines the incremental algorithm D* Lite with a novel hierarchical path planning approach to replan paths sufficiently fast for real-time operation. Unlike current hierarchical planning algorithms, HD* does not require map corrections before planning a new path. Directional cost scale factors, path smoothing, and Catmull-Rom splines are used to ensure the resulting paths are feasible. HD* sacrifices optimality for real-time performance. Its computation time and path quality are dependent on the map size, obstacle density, sensor range, and any restrictions on planning time. For the most complex scenarios tested, HD* found paths within 10% of optimal in under 35 milliseconds.

ENGINEERING HIERARCHICAL MESO-/MICROPOROUS LAMELLAR ZEOLITES WITH VARIABLE TEXTURAL AND CATALYTIC PROPERTIES

Meso-/microporous zeolites combine the charactersitics of well-defined micropores of zeolite with efficient mass transfer consequences of mesopores to increase the efficiency of the catalysts in reactions involving bulky molecules. Different methods such as demetallation and templating have been explored for the synthesis of meso-/microporous zeolites. However, they all have limitations in production of meso-/microporous zeolites with tunable textural and catalytic properties using few synthesis steps. To address this challenge, a simple one-step dual template synthesis approach has been developed in this work to engineer lamellar meso-/microporous zeolites structures with tunable textural and catalytic properties. First, one-step dual template synthesis of meso-/microporous mordenite framework inverted (MFI) zeolite structures was investigated. Tetrapropyl ammonium hydroxide (TPAOH) and diquaternary ammonium surfactant ([C22H45-N+(CH3)2-C6H12-N+(CH3)2-C6H13]Br2, C22-6-6) were used as templates to produce micropores and mesopores, respectively. The variation in concentration ratios of dual templates and hydrothermal synthesis conditions resulted in production of multi-lamellar MFI and the hybrid lamellar-bulk MFI (HLBM) zeolite structures. The relationship between the morphology, porosity, acidity, and catalytic properties of these catalysts was systematically studied. Then, the validity of the proposed synthesis approach for production of other types of zeolites composites was examined by creating a meso-/microporous bulk polymorph A (BEA)-lamellar MFI (BBLM) composite. The resulted composite samples showed higher catalytic stability compared to their single component zeolites. The studies demonstrated the high potential of the one-step dual template synthesis procedure for engineering the textural and catalytic properties of the synthesized zeolites.