We propose a 2-dimensional cellular automaton model to simulate pedestrian traffic. It is a v max = 1 model with exclusion statistics and parallel dynamics. Long-range interactions between the pedestrians are mediated by a so called floor field which modifies the transition rates to neighbouring cells. This field, which can be discrete or continuous, is subject to diffusion and decay. Furthermore it can be modified by the motion of the pedestrians. Therefore the model uses an idea similar to chemotaxis, but with pedestrians following a virtual rather than a chemical trace. Our main goal is to show that the introduction of such a floor field is sufficient to model collective effects and self-organization encountered in pedestrian dynamics, e.g. lane formation in counterflow through a large corridor. As an application we also present simulations of the evacuation of a large room with reduced visibility, e.g. due to failure of lights or smoke.
We present scalable algorithms for parallel adaptive mesh refinement and coarsening (AMR), partitioning, and 2:1 balancing on computational domains composed of multiple connected two-dimensional quadtrees or three-dimensional octrees, referred to as a forest of octrees. By distributing the union of octants from all octrees in parallel, we combine the high scalability proven previously for adaptive single-octree algorithms with the geometric flexibility that can be achieved by arbitrarily connected hexahedral macromeshes, in which each macroelement is the root of an adapted octree. A key concept of our approach is an encoding scheme of the interoctree connectivity that permits arbitrary relative orientations between octrees. Based on this encoding we develop interoctree transformations of octants. These form the basis for high-level parallel octree algorithms, which are designed to interact with an application code such as a numerical solver for partial differential equations. We have implemented and tested these algorithms in the p4est software library. We demonstrate the parallel scalability of p4est on its own and in combination with two geophysics codes. Using p4est we generate and adapt multioctree meshes with up to 5.13 × 10 11 octants on as many as 220,320 CPU cores and execute the 2:1 balance algorithm in less than 10 seconds per million octants per process.
We address the solution of large-scale statistical inverse problems in the framework of Bayesian inference. The Markov chain Monte Carlo (MCMC) method is the most popular approach for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. MCMC methods face two central difficulties when applied to large-scale inverse problems: first, the forward models (typically in the form of partial differential equations) that map uncertain parameters to observable quantities make the evaluation of the probability density at any point in parameter space very expensive; and second, the high-dimensional parameter spaces that arise upon discretization of infinite-dimensional parameter fields make the exploration of the probability density function prohibitive. The challenge for MCMC methods is to construct proposal functions that simultaneously provide a good approximation of the target density while being inexpensive to manipulate. Here we present a so-called Stochastic Newton method in which MCMC is accelerated by constructing and sampling from a proposal density that builds a local Gaussian approximation based on local gradient and Hessian (of the log posterior) information. Thus, the method exploits tools (adjoint-based gradients and Hessians) that have been instrumental for fast (often mesh-independent) solution of deterministic inverse problems. Hessian manipulations (inverse, square root) are made tractable by a low-rank approximation that exploits the compact nature of the data misfit operator. This is analogous to a reduced model of the parameter-to-observable map. The method is applied to the Bayesian solution of an inverse medium problem governed by 1D seismic wave propagation. We compare the Stochastic Newton method with a reference black box MCMC method as well as a gradient-based Langevin MCMC method, and observe at least two orders of magnitude improvement in convergence for problems with up to 65 parameters. Numerical evidence suggests that a 1025 parameter problem converges at the same rate as the 65 parameter problem.
Plate tectonics is regulated by driving and resisting forces concentrated at plate boundaries, but observationally constrained high-resolution models of global mantle flow remain a computational challenge. We capitalized on advances in adaptive mesh refinement algorithms on parallel computers to simulate global mantle flow by incorporating plate motions, with individual plate margins resolved down to a scale of 1 kilometer. Back-arc extension and slab rollback are emergent consequences of slab descent in the upper mantle. Cold thermal anomalies within the lower mantle couple into oceanic plates through narrow high-viscosity slabs, altering the velocity of oceanic plates. Viscous dissipation within the bending lithosphere at trenches amounts to approximately 5 to 20% of the total dissipation through the entire lithosphere and mantle.
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