An accelerated first-order method with complexity analysis for solving cubic regularization subproblems

We propose a first-order method to solve the cubic regularization subproblem (CRS) based on a novel reformulation. The reformulation is a constrained convex optimization problem whose feasible region admits an easily computable projection. Our reformulation requires computing the minimum eigenvalue of the Hessian. To avoid the expensive computation of the exact minimum eigenvalue, we develop a surrogate problem to the reformulation where the exact minimum eigenvalue is replaced with an approximate one. We then apply first-order methods such as the Nesterov's accelerated projected gradient method (APG) and projected Barzilai-Borwein method to solve the surrogate problem. As our main theoretical contribution, we show that when an $\epsilon$-approximate minimum eigenvalue is computed by the Lanczos method and the surrogate problem is approximately solved by APG, our approach returns an $\epsilon$-approximate solution to CRS in $\tilde O(\epsilon^{-1/2})$ matrix-vector multiplications (where $\tilde O(\cdot)$ hides the logarithmic factors). Numerical experiments show that our methods are comparable to and outperform the Krylov subspace method in the easy and hard cases, respectively. We further implement our methods as subproblem solvers of adaptive cubic regularization methods, and numerical results show that our algorithms are comparable to the state-of-the-art algorithms

Chinese students’ adjustment to studying in UK Higher Education: Academic self-efficacy and psychological well-being

With the internationalization of higher education and the rapid economic and social development of China the number of Chinese international students pursuing higher education in the UK has expanded considerably over the past several decades. These sojourners face a variety of challenges, both academically and psychologically, in their adjustment to this new cultural environment. This longitudinal research explored the academic self-efficacy and psychological well-being of Chinese international students and the relationships between these two variables over time, during their adaptation to UK higher education. In addition to Chinese international students in UK universities, data were also obtained from Chinese university students in China to be used as a comparison sample to better understand the general academic and psychological status of Chinese students studying in their home country. Data was collected through a quantitatively driven mixed methods design utilizing questionnaires and semi-structured in-person interviews. The questionnaire included brief measurements of academic self-efficacy, academic stress level, personality, and flourishing scale. Findings show academic performance, academic stress, academic support, and English language proficiency contribute greatly to students’ academic self-efficacy. Interactions with host nationals, social difficulty, academic stress, discrimination, and personality were proven to be predictive of students’ psychological well-being. Findings also revealed that students’ academic self-efficacy and psychological well-being are positively correlated with each other across time. These findings will be useful for faculty, staff, and even future international students to enable them to better understand the adjustment difficulties faced and to offer programming and support to facilitate this process

Femtosecond UV Transient Absorption Study of Coumarin Dimers

The phenomena of photo-induced chemical reactions can exhibit kinetics as short as a few femtoseconds. The development of femtosecond UV pulses enables the investigation of such ultrafast phenomena in more experimentally challenging, UV absorbing systems. This thesis describes, chronologically, development of a UV transient absorption (TA) system and its application to various coumarin-based molecular systems. The experimental section details the TA system comprising of three major components: two tunable UV excitation sources giving a total excitation range of 250-350 nm, supercontinuum generation enabling 240-700 nm probe range, and multichannel detection with greater than 5 ×10-5 OD sensitivity. The potential of the TA setup was tested on the study of ultrafast relaxation in 7-hydroxy coumarin. Following successful tests, the photo-induced cleavage reaction of coumarin dimer was studied in order to understand the ultrafast dynamics. The experiments utilized 280 nm excitation and broadband (300-650 nm) probing. The results revealed fast cleavage occurring through short-lived nonradiative (<200 fs) singlet states. Two branched kinetic models were developed to describe the monomer formation and dimer relaxation dynamics, identify intermediate states, and determine the quantum yields. The anti-hh displayed the highest cleavage efficiency of ~20 %. Finally, variation in the quantum yields between isomers was rationalized using a sequential bond cleavage mechanism over a concerted, “pericyclic-style” ring cleavage