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- ItemThe space of bounded variation with infinite-dimensional codomain(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Patterson, Robert I.A.; Renger, D.R. MichielWe study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.
- ItemNon-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, Alexander; Patterson, Robert I.A.; Peletier, Mark A.; Renger, D.R. MichielWe study stochastic interacting particle systems that model chemical reaction networks on the microscopic scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we also study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations enables us to derive a non-linear relation between thermodynamic fluxes and free energy driving force.
- ItemCell size error in stochastic particle methods for coagulation equations with advection(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Patterson, Robert I.A.; Wagner, WolfgangThe paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.
- ItemA kinetic equation for the distribution of interaction clusters in rarefied gases(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Patterson, Robert I.A.; Simonella, Sergio; Wagner, WolfgangWe consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.
- ItemConvergence of stochastic particle systems undergoing advection and coagulation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Patterson, Robert I.A.The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction.
- ItemSimulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bartsch, Clemens; John, Volker; Patterson, Robert I.A.A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.
- ItemStochastic weighted particle methods for population balance equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Patterson, Robert I.A.; Kraft, Markus; Wagner, WolfgangA class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.
- ItemPercolation for D2D networks on street systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Cali, Elie; En-Najjari, Taoufik; Gafur, Nila Novita; Hirsch, Christian; Jahnel, Benedikt; Patterson, Robert I.A.We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold.
- ItemLarge deviations of reaction fluxes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Patterson, Robert I.A.; Renger, D.R. MichielWe study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration.
- ItemDynamical large deviations of countable reaction networks under a weak reversibility condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Patterson, Robert I.A.; Renger, D.R. MichielA dynamic large deviations principle for a countable reaction network including coagulation-fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.