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This paper analyzes the convergence of a Petrov-Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

This work proposes an accelerated primal-dual dynamical system for affine constrained convex optimization and presents a class of primal-dual methods with nonergodic convergence rates. In continuous level, exponential decay of a novel Lyapunov function is established and in discrete level, implicit, semi-implicit and explicit numerical discretizations for the continuous model are considered sequentially and lead to new accelerated primal-dual methods for solving linearly constrained optimization problems. Special structures of the subproblems in those schemes are utilized to develop efficient inner solvers. In addition, nonergodic convergence rates in terms of primal-dual gap, primal objective residual and feasibility violation are proved via a tailored discrete Lyapunov function. Moreover, our method has also been applied to decentralized distributed optimization for fast and efficient solution.

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This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

Convergence analysis of accelerated first-order methods for convex optimization problems are developed from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient (NAG) flow, is derived from the connection between acceleration mechanism and A-stability of ODE solvers, and the exponential decay of a tailored Lyapunov function along with the solution trajectory is proved. Numerical discretizations of NAG flow are then considered and convergence rates are established via a discrete Lyapunov function. The proposed differential equation solver approach can not only cover existing accelerated methods, such as FISTA, Güler’s proximal algorithm and Nesterov’s accelerated gradient method, but also produce new algorithms for composite convex optimization that possess accelerated convergence rates. Both the convex and the strongly convex cases are handled in a unified way in our approach.

This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order γ (1 < γ < 2). We establish the stability of this method, and derive the optimal convergence in the H 1 (0, T ; L 2 (Ω))-norm and suboptimal convergence in the discrete L ∞ (0, T ; H 1 0 (Ω))-norm. Furthermore, we discuss the performance of this method in the case that the solution has singularity at t = 0, and show that optimal convergence rate with respect to the H 1 (0, T ; L 2 (Ω))norm can still be achieved by using graded grids in the time discretization. Finally, numerical experiments are performed to verify the theoretical results.

Objectives: The objective of this study was to quantify the risk of intracranial hypertension (ICH) with the intrauterine levonorgestrel (IUL) device Mirena ® . Methods: We used the United States Food and Drug Administration's Adverse Events Reporting System (FAERS) database to quantify a reporting odds ratio (ROR) for ICH and Mirena ® . We also conducted a retrospective cohort study using the IMS LifeLink ® database, comparing the risk of two oral contraceptives ethinyl estradiol (EE) with Mirena ® . A Bayesian sensitivity analysis was performed to account for the effect of body mass index (BMI). Results:The reported odds ratios (ORs) for ICH and papilledema with Mirena ® were 1.78 (95% confidence interval [CI] 1.41-2.25) and 1.50 (95% CI 1.10-2.05), respectively. In the cohort study, the OR for ICH and EE-norgestimate and EE-norethindrone compared with Mirena ® were 1.29 (95% CI 0.83-2.00) and 0.31 (95% CI 0.04-2.29), respectively. The presence of a strong confounder BMI did not affect the estimated OR (OR = 1.31, 95% CI 0.73-2.41 for EEnorgestimate; OR = 0.18, 95% CI 0.01-1.27 for EE-norethindrone). Conclusion: We found a higher than expected number of reports of ICH with Mirena ® in the FAERS database. We also found a similar risk of ICH with Mirena ® compared with the oral contraceptive EE-norgestimate. The higher risk of ICH with EE-norethindrone, another oral contraceptive should be further investigated.