Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented

Chimpanzee faces under the magnifying glass: emerging methods reveal cross-species similarities and individuality

Independently, we created descriptive systems to characterize chimpanzee facial behavior, responding to a common need to have an objective, standardized coding system to ask questions about primate facial behaviors. Even with slightly different systems, we arrive at similar outcomes, with convergent conclusions about chimpanzee facial mobility. This convergence is a validation of the importance of the approach, and provides support for the future use of a facial action coding system for chimpanzees,ChimpFACS. Chimpanzees share many facial behaviors with those of humans. Therefore, processes and mechanisms that explain individual differences in facial activity can be compared with the use of a standardized systems such asChimpFACSandFACS. In this chapter we describe our independent methodological approaches, comparing how we arrived at our facial coding categories. We present some Action Descriptors (ADs) from Gaspar’s initial studies, especially focusing on an ethogram of chimpanzee and bonobo facial behavior, based on studies conducted between 1997 and 2004 at three chimpanzee colonies (The Detroit Zoo; Cleveland Metroparks Zoo; and Burger’s Zoo) and two bonobo colonies (The Columbus Zoo and Aquarium; The Milwaukee County Zoo). We discuss the potential significance of arising issues, the minor qualitative species differences that were found, and the larger quantitative differences in particular facial behaviors observed between species, e.g., bonobos expressed more movements containing particular action units (Brow Lowerer, Lip Raiser, Lip Corner Puller) compared with chimpanzees. The substantial interindividual variation in facial behavior within each species was most striking. Considering individual differences and the impact of development, we highlight the flexibility in facial activity of chimpanzees. We discuss the meaning of facial behaviors in nonhuman primates, addressing specifically individual attributes of Social Attraction, facial expressivity, and the connection of facial behavior to emotion. We do not rule out the communicative function of facial behavior, in which case an individual’s properties of facial behavior are seen as influencing his or her social life, but provide strong arguments in support of the role of facial behavior in the expression of internal states

The future of spaceborne altimetry. Oceans and climate change: A long-term strategy

The ocean circulation and polar ice sheet volumes provide important memory and control functions in the global climate. Their long term variations are unknown and need to be understood before meaningful appraisals of climate change can be made. Satellite altimetry is the only method for providing global information on the ocean circulation and ice sheet volume. A robust altimeter measurement program is planned which will initiate global observations of the ocean circulation and polar ice sheets. In order to provide useful data about the climate, these measurements must be continued with unbroken coverage into the next century. Herein, past results of the role of the ocean in the climate system is summarized, near term goals are outlined, and requirements and options are presented for future altimeter missions. There are three basic scientific objectives for the program: ocean circulation; polar ice sheets; and mean sea level change. The greatest scientific benefit will be achieved with a series of dedicated high precision altimeter spacecraft, for which the choice of orbit parameters and system accuracy are unencumbered by requirements of companion instruments

On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method