On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function

Physical demand but not dexterity is associated with motor flexibility during rapid reaching in healthy young adults

Healthy humans are able to place light and heavy objects in small and large target locations with remarkable accuracy. Here we examine how dexterity demand and physical demand affect flexibility in joint coordination and end-effector kinematics when healthy young adults perform an upper extremity reaching task. We manipulated dexterity demand by changing target size and physical demand by increasing external resistance to reaching. Uncontrolled manifold analysis was used to decompose variability in joint coordination patterns into variability stabilizing the end-effector and variability de-stabilizing the end-effector during reaching. Our results demonstrate a proportional increase in stabilizing and de-stabilizing variability without a change in the ratio of the two variability components as physical demands increase. We interpret this finding in the context of previous studies showing that sensorimotor noise increases with increasing physical demands. We propose that the larger de-stabilizing variability as a function of physical demand originated from larger sensorimotor noise in the neuromuscular system. The larger stabilizing variability with larger physical demands is a strategy employed by the neuromuscular system to counter the de-stabilizing variability so that performance stability is maintained. Our findings have practical implications for improving the effectiveness of movement therapy in a wide range of patient groups, maintaining upper extremity function in old adults, and for maximizing athletic performance

Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: two approaches

In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at $x_1=0$ is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space~$\mathbb{R}^N$. The second one is based on the theory of dynamical systems

Multiscale simulations of porous media flows in flow-based coordinate system

In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

We study the preconditioning of Markov chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite volume methods. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen--Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than 10 times if MCMC simulations are preconditioned using coarse-scale models