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.

READY FOR KINDERGARTEN: A TRAINING PROGRAM DESIGNED TO ENCOURAGE PARENT-CHILD CONVERSATION DURING THE PRESCHOOL YEARS

Many children in the United States begin kindergarten unprepared to converse in the academic language surrounding instruction, putting them at greater risk for later language and reading difficulties. Importantly, correlational research has shown there are certain experiences prior to kindergarten that foster the oral language skills needed to understand and produce academic language. The focus of this dissertation was on increasing one of these experiences: parent-child conversations about abstract and non-present concepts, known as decontextualized language (DL). Decontextualized language involves talking about non-present concepts such as events that happened in the past or future, or abstract discussions such as providing explanations or defining unknown words. As caregivers’ decontextualized language input to children aged three to five is consistently correlated with kindergarten oral language skills and later reading achievement, it is surprising no experimental research has been done to establish this relation causally. The study described in this dissertation filled this literature gap by designing, implementing, and evaluating a decontextualized language training program for parents of 4-year-old children (N=30). After obtaining an initial measure of decontextualized language, parents were randomly assigned to a control condition or training condition, the latter of which educated parents about decontextualized language and why it is important. All parents then audio-recorded four mealtime conversations over the next month, which were transcribed and reliably coded for decontextualized language. Results indicated that trained parents boosted their DL from roughly 17 percent of their total utterances at baseline to approximately 50 percent by the mid-point of the study, and remained at these boosted levels throughout the duration of the study. Children’s DL was also boosted by similar margins, but no improvement in children’s oral language skills was observed, measured prior to, and one month following training. Further, exploratory analyses pointed to parents’ initial use of DL and their theories of the malleability of intelligence (i.e., growth mindsets) as moderators of training gains. Altogether, these findings are a first step in establishing DL as a viable strategy for giving children the oral language skills needed to begin kindergarten ready to succeed in the classroom.