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.

Observed Social Problem Solving and Friendship Quality in Children with a Traumatic Brain Injury

Children who have experienced a traumatic brain injury (TBI) are at risk for a variety of maladaptive cognitive, behavioral and social outcomes (Yeates et al., 2007). Research involving the social problem solving (SPS) abilities of children with TBI indicates a preference for lower level strategies when compared to children who have experienced an orthopedic injury (OI; Hanten et al., 2008, 2011). Research on SPS in non-injured populations has highlighted the significance of the identity of the social partner (Rubin et al., 2006). Within the pediatric TBI literature few studies have utilized friends as the social partner in SPS contexts, and fewer have used in-vivo SPS assessments. The current study aimed to build on existing research of SPS in children with TBI by utilizing an observational coding scheme to capture in-vivo problem solving behaviors between children with TBI and a best friend. The current study included children with TBI (n = 41), children with OI (n = 43), and a non-injured typically developing group (n = 41). All participants were observed completing a task with a friend and completed a measure of friendship quality. SPS was assessed using an observational coding scheme that captured SPS goals, strategies, and outcomes. It was expected children with TBI would produce fewer successes, fewer direct strategies, and more avoidant strategies. ANOVAs tested for group differences in SPS successes, direct strategies and avoidant strategies. Analyses were run to see if positive or negative friendship quality moderated the relation between group type and SPS behaviors. Group differences were found between the TBI and non-injured group in the SPS direct strategy of commands. No group differences were found for other SPS outcome variables of interest. Moderation analyses partially supported study hypotheses regarding the effect of friendship quality as a moderator variable. Additional analyses examined SPS goal-strategy sequencing and grouped SPS goals into high cost and low cost categories. Results showed a trend supporting the hypothesis that children with TBI had fewer SPS successes, especially with high cost goals, compared to the other two groups. Findings were discussed highlighting the moderation results involving children with severe TBI.