Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case

The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments

Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost functional with an additional penalty on the variance of the state. The expressions for the gradient and Hessian corresponding to either problem contain expected value operators. Due to the large number of uncertainties considered in our model, we suggest to evaluate these expectations using a multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that this results in the gradient and Hessian corresponding to the MLMC estimator of the original cost functional. Furthermore, we show that the use of certain correlated samples yields a reduction in the total number of samples required. Two optimization methods are investigated: the nonlinear conjugate gradient method and the Newton method. For both, a specific algorithm is provided that dynamically decides which and how many samples should be taken in each iteration. The cost of the optimization up to some specified tolerance $\tau$ is shown to be proportional to the cost of a gradient evaluation with requested root mean square error $\tau$. The algorithms are tested on a model elliptic diffusion problem with lognormal diffusion coefficient. An additional nonlinear term is also considered.Comment: This work was presented at the IMG 2016 conference (Dec 5 - Dec 9, 2016), at the Copper Mountain conference (Mar 26 - Mar 30, 2017), and at the FrontUQ conference (Sept 5 - Sept 8, 2017

Carrie Jo Law v. Robert Frank Law and M. Lynne Larson : Brief of Appellant

Appeal from Order of Dismissal of Complaint in Intervention Fourth District Court for Millard County Honorable John Wahlquist Judge Pro Te

A Review of the Associations of Alcohol Consumption with Race/Ethnicity, Religion, and Socioeconomic Status

This literature review explores the relationship between alcohol use and different demographic variables, specifically race/ethnicity, religion, and socioeconomic status. The aims of the paper are: 1) define alcohol use and discuss the physical and mental health implications of alcohol use and 2) explore the associations of alcohol use with religiosity, race/ethnicity, and socioeconomic status. Thirty four articles that directly address the correlates of race/ethnicity, religion, SES, health, and alcohol consumption were reviewed after extensive literature search. Studies indicate that religious individuals consume less alcohol than non-religious individuals. European Americans and Native Americans have the highest rates of alcohol consumption. Socioeconomic status proves to be the most inconsistent of the demographic variables studied; results vary about the effects of socioeconomic status on alcohol use. Gaps in current literature, the need for consistency in vocabulary, and focus for future studies are discussed