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.

Language Outcomes of the Play and Language for Autistic Youngsters (PLAY) Project Home Consultation model—An Extended Analysis

The current study is a post-hoc analysis of data from the original randomized control trial of the Play and Language for Autistic Youngsters (PLAY) Home Consultation program, a parent-mediated, DIR/Floortime based early intervention program for children with ASD (Solomon, Van Egeren, Mahone, Huber, & Zimmerman, 2014). We examined 22 children from the original RCT who received the PLAY program. Children were split into two groups (high and lower functioning) based on the ADOS module administered prior to intervention. Fifteen-minute parent-child video sessions were coded through the use of CHILDES transcription software. Child and maternal language, communicative behaviors, and communicative functions were assessed in the natural language samples both pre- and post-intervention. Results demonstrated significant improvements in both child and maternal behaviors following intervention. There was a significant increase in child verbal and non-verbal initiations and verbal responses in whole group analysis. Total number of utterances, word production, and grammatical complexity all significantly improved when viewed across the whole group of participants; however, lexical growth did not reach significance. Changes in child communicative function were especially noteworthy, and demonstrated a significant increase in social interaction and a significant decrease in non-interactive behaviors. Further, mothers demonstrated an increase in responsiveness to the child’s conversational bids, increased ability to follow the child’s lead, and a decrease in directiveness. When separated for analyses within groups, trends emerged for child and maternal variables, suggesting greater gains in use of communicative function in both high and low groups over changes in linguistic structure. Additional analysis also revealed a significant inverse relationship between maternal responsiveness and child non-interactive behaviors; as mothers became more responsive, children’s non-engagement was decreased. Such changes further suggest that changes in learned skills following PLAY parent training may result in improvements in child social interaction and language abilities.