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.

Aging African American Women and the Impact of Positive Psychological Factors on Quality of Life

Aging African-American women are disproportionately affected by negative health outcomes and mortality. Life stress has strong associations with these health outcomes. The purpose of this research was to understand how aging African American women manage stress. Specifically, the effects of coping, optimism, resilience, and religiousness as it relates to quality of life were examined. This cross-sectional exploratory study used a self-administered questionnaire and examined quality of life in 182 African-American women who were 65 years of age or older living in senior residential centers in Baltimore using convenience sampling. The age range for these women was 65 to 94 years with a mean of 71.8 years (SD = 5.6). The majority (53.1%) of participants completed high school, with 23 percent (N = 42) obtaining college degrees and 19 percent (N = 35) holding advanced degrees. Nearly 58 percent of participants were widowed and 81 percent were retired. In addition to demographics, the questionnaire included the following reliable and valid survey instruments: The Brief Cope Scale (Carver, Scheier, & Weintraub, 1989), Optimism Questionnaire (Scheier, Carver, & Bridges, 1994), Resilience Survey (Wagnild & Young, 1987), Religiousness Assessment (Koenig, 1997), and Quality of Life Questionnaire (Cummins, 1996). Results revealed that the positive psychological factors examined were positively associated with and significant predictors of quality of life. The bivariate correlations indicated that of the six coping dimensions measured in this study, planning (r=.68) was the most positively associated with quality of life. Optimism (r=.33), resilience (=.48), and religiousness (r=.30) were also significantly correlated with quality of life. In the linear regression model, again the coping dimension of planning was the best predictor of quality of life (beta = .75, p <.001). Optimism (beta = .31, p <.001), resilience (beta = .34, p, .001) and religiousness (beta = .17, p <.01) were also significant predictors of quality of life. It appears as if positive psychology plays an important role in improving quality of life among aging African-American women.