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.

Benefits and Effects of Interactive Gaming Exercise as Compared to Traditional Exercise Techniques: An Investigation of the Relationship Between Physical Activity Levels and Mode of Activity in College Students

Gemstone Team F.I.T.N.E.S.S. (Fun Interactive Techniques for New Exercise and Sport Styles)

The Design and Implementation of an Educational Computer Game and Its Study as a Motivational Tool for Middle School ESOL Students

Gemstone Team ILL (Interactive Language Learning)

Crime in College Park: understanding crime levels, perceptions, and environmental design in an off-campus student-occupied neighborhood

Gemstone Team Crime Prevention and Perception

Effects of Web-Based Self-Reporting: College Students’ Self-Efficacy Regarding Fruit and Vegetable Intake

This study evaluated the effect of an online diet-tracking tool on college students’ self-efficacy regarding fruit and vegetable intake. A convenience sample of students completed online self-efficacy surveys before and after a six-week intervention in which they tracked dietary intake with an online tool. Group one (n=22 fall, n=43 spring) accessed a tracking tool without nutrition tips; group two (n=20 fall, n=33 spring) accessed the tool and weekly nutrition tips. The control group (n=36 fall, n=60 spring) had access to neither. Each semester there were significant changes in self-efficacy from pre- to post-test for men and for women when experimental groups were combined (p<0.05 for all); however, these changes were inconsistent. Qualitative data showed that participants responded well to the simplicity of the tool, the immediacy of feedback, and the customized database containing foods available on campus. Future models should improve user engagement by increasing convenience, potentially by automation.