UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.
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Item Analyzing Inverse Design Problems from a Topological Perspective(2024) Chen, Qiuyi; Fuge, Mark; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Inverse design (ID) problems are inverse problems that aim to rapidly retrieve the subset of valid designs having the desired performances and properties under the given conditions. In practice, this can be solved by training generative models to approximate and sample the posterior distributions of designs. However, little has been done to understand their mechanisms and limitations from a theoretical perspective. This dissertation leverages theoretical tools from general and differential topology to answer these three questions of inverse design: what does a set of valid designs look like? How helpful are the data-driven generative models for retrieving the desired designs from this set? What topological properties affect the subset of desired designs? The dissertation proceeds by dismantling inverse (design) problems into two major subjects: that is, the representing and probing of a given set of valid designs (or data), and the retrieval of the desired designs (or data) from this given set. It draws inspiration from topology and geometry to investigate them and makes the main contributions below: 1. Chapter 3 details a novel representation learning method called Least Volume, which has properties similar to nonlinear PCA for representing datasets. It can minimize the representation's dimension automatically and, as shown in Chapter 4, conducts contrastive learning when applied to labeled datasets. 2. Two conditional generative models are developed to generate performant 2-D airfoils and 3-D heat sinks in Chapter 5 and 6 respectively. They can produce realistic designs to warm-start further optimization, with the relevant chapters detailing their acceleration effects. 3. Lastly, Chapter 7 describes how to use Least volume to solve high-dimensional inverse problems efficiently. Specifically, using examples from physic system identification, the chapter uncovers the correlation between the inverse problem's uncertainty and its intrinsic dimensions.Item Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems(2016) Zhong, Ming; Tadmor, Eitan; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)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.