Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential

The Term Structures of Equity and Interest Rates

This paper proposes a dynamic risk-based model capable of jointly explaining the term structure of interest rates, returns on the aggregate market, and the risk and return characteristics of value and growth stocks. Both the term structure of interest rates and returns on value and growth stocks convey information about how the representative investor values cash flows of different maturities. We model how the representative investor perceives risks of these cash flows by specifying a parsimonious stochastic discount factor for the economy. Shocks to dividend growth, the real interest rate, and expected inflation are priced, but shocks to the price of risk are not. Given reasonable assumptions for dividends and inflation, we show that the model can simultaneously account for the behavior of aggregate stock returns, an upward-sloping yield curve, the failure of the expectations hypothesis, and the poor performance of the capital asset pricing model

Why Is Long-Horizon Equity Less Risky? A Duration-Based Explanation of the Value Premium

We propose a dynamic risk-based model that captures the value premium. Firms are modeled as long-lived assets distinguished by the timing of cash flows. The stochastic discount factor is specified so that shocks to aggregate dividends are priced, but shocks to the discount rate are not. The model implies that growth firms covary more with the discount rate than do value firms, which covary more with cash flows. When calibrated to explain aggregate stock market behavior, the model accounts for the observed value premium, the high Sharpe ratios on value firms, and the poor performance of the CAPM

The Declining Equity Premium: What Role Does Macroeconomic Risk Play?

Aggregate stock prices, relative to virtually any indicator of fundamental value, soared to unprecedented levels in the 1990s. Even today, after the market declines since 2000, they remain well above historical norms. Why? We consider one particular explanation: a fall in macroeconomic risk, or the volatility of the aggregate economy. Empirically, we find a strong correlation between low-frequency movements in macroeconomic volatility and low-frequency movements in the stock market. To model this phenomenon, we estimate a two-state regime switching model for the volatility and mean of consumption growth, and find evidence of a shift to substantially lower consumption volatility at the beginning of the 1990s. We then use these estimates from postwar data to calibrate a rational asset pricing model with regime switches in both the mean and standard deviation of consumption growth. Plausible parameterizations of the model are found to account for a significant portion of the run-up in asset valuation ratios observed in the late 1990s

Analysis of Videotape Evidence in Police Misconduct Cases

Many evidentiary issues arise with respect to the admission of videotape evidence and computer generated simulations at trial, and the authors of this Article address these issues as they arise in police misconduct cases. Professor Schwartz provides insight into and analysis of the evidentiary principles that govern the use of video and computer simulation evidence at trial in cases where police misconduct is at issue. His discussion first addresses the issues that concern the admissibility of videotape evidence, then discusses the role of a videotape on summary judgment, and lastly, analyzes evidentiary issues with respect to computer generated simulations

Lunar basalt chronology, mantle differentiation and implications for determining the age of the Moon

Despite more than 40 years of studying Apollo samples, the age and early evolution of the Moon remain contentious. Following the formation of the Moon in the aftermath of a giant impact, the resulting Lunar Magma Ocean (LMO) is predicted to have generated major geochemically distinct silicate reservoirs, including the sources of lunar basalts. Samples of these basalts, therefore, provide a unique opportunity to characterize these reservoirs. However, the precise timing and extent of geochemical fractionation is poorly constrained, not least due to the difficulty in determining accurate ages and initial Pb isotopic compositions of lunar basalts. Application of an in situ ion microprobe approach to Pb isotope analysis has allowed us to obtain precise crystallization ages from six lunar basalts, typically with an uncertainty of about ±10Ma, as well as constrain their initial Pb-isotopic compositions. This has enabled construction of a two-stage model for the Pb-isotopic evolution of lunar silicate reservoirs, which necessitates the prolonged existence of high-μ reservoirs in order to explain the very radiogenic compositions of the samples. Further, once firm constraints on U and Pb partitioning behaviour are established, this model has the potential to help distinguish between conflicting estimates for the age of the Moon. Nonetheless, we are able to constrain the timing of a lunar mantle reservoir differentiation event at 4376±18Ma, which is consistent with that derived from the Sm–Nd and Lu–Hf isotopic systems, and is interpreted as an average estimate of the time at which the high-μ urKREEP reservoir was established and the Ferroan Anorthosite (FAN) suite was formed