An extension of the projected gradient method to a Banach space setting with application in structural topology optimization

For the minimization of a nonlinear cost functional $j$ under convex constraints the relaxed projected gradient process $\varphi_{k+1} = \varphi_{k} + \alpha_k(P_H(\varphi_{k}-\lambda_k \nabla_H j(\varphi_{k}))-\varphi_{k})$ is a well known method. The analysis is classically performed in a Hilbert space $H$. We generalize this method to functionals $j$ which are differentiable in a Banach space. Thus it is possible to perform e.g. an $L^2$ gradient method if $j$ is only differentiable in $L^\infty$. We show global convergence using Armijo backtracking in $\alpha_k$ and allow the inner product and the scaling $\lambda_k$ to change in every iteration. As application we present a structural topology optimization problem based on a phase field model, where the reduced cost functional $j$ is differentiable in $H^1\cap L^\infty$. The presented numerical results using the $H^1$ inner product and a pointwise chosen metric including second order information show the expected mesh independency in the iteration numbers. The latter yields an additional, drastic decrease in iteration numbers as well as in computation time. Moreover we present numerical results using a BFGS update of the $H^1$ inner product for further optimization problems based on phase field models

Collective Motion of Predictive Swarms

Theoretical models of populations and swarms typically start with the assumption that the motion of agents is governed by the local stimuli. However, an intelligent agent, with some understanding of the laws that govern its habitat, can anticipate the future, and make predictions to gather resources more efficiently. Here we study a specific model of this kind, where agents aim to maximize their consumption of a diffusing resource, by attempting to predict the future of a resource field and the actions of other agents. Once the agents make a prediction, they are attracted to move towards regions that have, and will have, denser resources. We find that the further the agents attempt to see into the future, the more their attempts at prediction fail, and the less resources they consume. We also study the case where predictive agents compete against non-predictive agents and find the predictors perform better than the non-predictors only when their relative numbers are very small. We conclude that predictivity pays off either when the predictors do not see too far into the future or the number of predictors is small.Comment: 16 pages, 7 figure

A NOTE ON HOUSING WEALTH AND PRIVATE CONSUMPTION

This paper analyses the relationship between house prices and private consumption of the US economy. Based on Granger's causality test, we ask whether this relationship is driven by causality or whether it is merely an ambiguous connection. Based on latest quartely data, our results show that there is indeed a causal relationship with changes in house prices affeting private consumption, fundamentally supporting economic theory. Considering existing research, it is therefore suggested that the US economy is at the outset of a severe economic downturn, confirming pessimist's expectations.Wealth effect; House prices; Consumption; Granger causality

Development of a microbalance suitable for space application

The tapered element oscillating microbalance (TEOM), an ultrasensitive mass measurement device which is suitable for both particulate and vapor deposition measurements is described. The device can be used in contamination measurements, surface reaction studies, particulate monitoring systems or any microweighing activity where either laboratory or field monitoring capability is desired. The active element of the TEOM consists of a tube or reed constructed of a material with high mechanical quality factor and having a special taper. The element is firmly mounted at the wide end while the other end supports a substrate surface which can be composed of virtually any material. The tapered element with the substrate at the free (narrow) end is set into oscillation in a clamped free mode. A feedback system maintains the oscillation whose natural frequency will change in relation to the mass deposited on the substrate

Ice in space: An experimental and theoretical investigation

Basic knowledge is provided on the behavior of ice and ice particles under a wide variety of conditions including those of interplanetary space. This information and, in particular, the lifetime of ice particles as a function of solar distance is an absolute requirement for a proper interpretation of photometric profiles in comets. Because fundamental properties of ice and ice particles are developed in this report, the applicability of this information extends beyond the realm of comets into any area where volatile particles exist, be it in space or in the earth's atmosphere