We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold T. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of T, nonequilibrium phase transitions take place in the N→∞ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant.

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains, with a focus on multiscale diffusions. Although diffusion is often studied in terms of continuum observables, it is a consequence of the motion of individual particles. Incorporating interactions between constituents and geometry often runs into complications, since interactions typically act on multiple length scales. We address this issue by studying different types of multiscale models and by applying them to a variety of scenarios known for their inherent complexity.

Our contributions can be grouped in two parts. In the first part, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct finite element schemes with desirable approximation properties that resolve the microscopic domain variations and support parallel execution. In the second part of the thesis, we investigate certain interacting particle systems and their links to families of partial differential equations. In this spirit, we analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within an interacting particle system in an extension of the classical Ehrenfest model.

Our overall focus is two-fold. On the one hand, we increase the theoretical understanding of multiscale models by providing modeling, analysis and simulation of specific two-scale couplings. On the other hand, we design computational frameworks and tailored implementations to improve the application of multiscale modeling to complex scenarios and large-scale systems. In this way, our contributions aim to expand the capacity of mathematical modeling to numerically approximate the rich and complex physical world.

We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains. Our contributions can be grouped in two parts. First, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct convergent and efficient finite element schemes that resolve the microscopic domain variations. Second, we investigate certain interacting particle systems and their links to a family of partial differential equations. We analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We also construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within a particle system extending the classical Ehrenfest model.

Our focus is two-fold: we increase the theoretical understanding of certain two-scale couplings, while on the other hand, we develop computational multiscale frameworks for a variety of scenarios known for their inherent complexity.

High level traffic characteristics have the potential to be useful for inference of various host characteristics. This work proposes the novel Flow-Discretize Order (FDO) approach for describing session characteristics in an intuitive manner, while also retaining flow ordering information. The FDO approach allows for flexible construction of flow descriptors, by using different flow properties and applying appropriate discretization. The individual flow descriptors are concatenated to form session descriptors. By utilizing string distance metrics, such as the Damerau-Levenshtein distance (DLD), it is possible to perform both unsupervised and supervised learning on the FDO session descriptors. Here, we utilize FDO as a tool for OS and browser identification coupled to a particular user activity, in this case watching YouTube videos. The variable-length nature of FDO session descriptors precludes learning methods expecting fixed dimensionality from being used. However, experiments show that excellent performance are provided by methods operating on distances such as hierarchical Ward for the unsupervised case, and k-NN for the supervised case. The supervised learning evaluation shows that over 99% accuracy can be achieved for both operating system and browser identification based on video session characteristics. The FDO framework also provides multiple promising avenues for further research and improvements such as improved methods for discretization boundary placement, more elaborate feature selection approaches, and more fine-grained DLD weights. 

In this paper, we study the numerical approximation of a coupled system of elliptic–parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulation results illustrating the expected behaviour of the system.

Npoint particles move within a billiard table made of two circular cavities connected by a straight channel. The usual billiard dynamics is modified so that it remains deterministic, phase space volumes preserving and time reversal invariant. Particles move in straight lines and are elastically reflected at the boundary of the table, as usual, but those in a channel that are moving away from a cavity invert their motion (rebound), if their number exceeds a given thresholdT. When the geometrical parameters of the billiard table are fixed, this mechanism gives rise to non-equilibrium phase transitions in the largeNlimit: lettingT/Ndecrease, the homogeneous particle distribution abruptly turns into a stationary inhomogeneous one. The equivalence with a modified Ehrenfest two urn model, motivated by the ergodicity of the billiard with no rebound, allows us to obtain analytical results that accurately describe the numerical billiard simulation results. Thus, a stochastic exactly solvable model that exhibits non-equilibrium phase transitions is also introduced.

We study the upscaling of a system of many interacting particles through a heterogenous thin elongated obstacle as modeled via a two-dimensional diffusion problem with a one-directional nonlinear convective drift. Assuming that the obstacle can be described well by a thin composite strip with periodically placed microstructures, we aim at deriving the upscaled model equations as well as the effective transport coefficients for suitable scalings in terms of both the inherent thickness at the strip and the typical length scales of the microscopic heterogeneities. Aiming at computable scenarios, we consider that the heterogeneity of the strip is made of an array of periodically arranged impenetrable solid rectangles and identify two scaling regimes what concerns the small asymptotics parameter for the upscaling procedure: the characteristic size of the microstructure is either significantly smaller than the thickness of the thin obstacle or it is of the same order of magnitude. We scale up the diffusion-polynomial drift model and list computable formulas for the effective diffusion and drift tensorial coefficients for both scaling regimes. Our upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We illustrate numerically the concentration profile of the chemical species passing through the upscaled strip in the finite thickness regime and point out that trapping of concentration inside the strip is likely to occur in at least two conceptually different transport situations: (i) full diffusion/dispersion matrix and nonlinear horizontal drift, and (ii) diagonal diffusion matrix and oblique nonlinear drift.

We develop and implement a numerical model to simulate the effect of photocatalytic asphalt on reducing the concentration of nitrogen monoxide (NO) due to the presence of heavy traffic in an urban environment. The contributions in this paper are threefold: we model and simulate the spread and breakdown of pollution in an urban environment, we provide a parameter estimation process that can be used to find missing parameters, and finally, we train and compare this simulation with different data sets. We analyse the results and provide an outlook on further research.

We study the evacuation dynamics of a crowd evacuating from a complex geometry in the presence of a fire as well as of a slowly spreading smoke curtain.The crowd is composed of two kinds of individuals: those who know the layout of the building, and those who do not and rely exclusively on potentially informed neighbors to identify a path towards the exit. We aim to capture the effect the knowledge of the environment has on the interaction between evacuees and their residence time in the presence of fire and evolving smoke. Our approach is genuinely multiscale—we employ a two-scale model that is able to distinguish between compressible and incompressible pedestrian flow regimes and allows for micro and macro pedestrian dynamics. Simulations illustrate the expected qualitative behavior of the model. We finish with observations on how mixing evacuees with different levels of knowledge impacts important evacuation aspects.

We study a two-scale coupled system consisting of a macroscopic elliptic equation and a microscopic parabolic equation. This system models the interplay between a gas and liquid close to equilibrium within a porous medium with distributed microstructures. We use formal homogenization arguments to derive the target system. We start by proving well-posedness and inverse estimates for the two-scale system. We follow up by proposing a Galerkin scheme which is continuous in time and discrete in space, for which we obtain well-posedness, a priori error estimates and convergence rates. Finally, we propose a numerical error reduction strategy by refining the grid based on residual error estimators.