Gary Froyland

The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems

In a given dynamical system there are essentially two different types of information which could be of practical interest: on the one hand there is the need to describe the behavior of single trajectories in detail. This information is helpful for the analysis of transient behavior and also in the investigation of geometric properties of dynamical systems. On the other hand, if the underlying invariant set is generated by complicated dynamics then the computation of single trajectories may give misleading results. In this case there still exists important set related information covering both topological and statistical aspects of the underlying dynamical behavior. Within the DFG-Schwerpunkt we have focussed on the development of set oriented methods for the numerical approximation of invariant sets (e.g. invariant manifolds, global attractors, chain recurrent sets) (natural) invariant measures almost invariant sets

Origin, dynamics and evolution of ocean garbage patches from observed surface drifters

Much of the debris in the near-surface ocean collects in so-called garbage patches where, due to convergence of the surface flow, the debris is trapped for decades to millennia. Until now, studies modelling the pathways of surface marine debris have not included release from coasts or factored in the possibilities that release concentrations vary with region or that pathways may include seasonal cycles. Here, we use observational data from the Global Drifter Program in a particle-trajectory tracer approach that includes the seasonal cycle to study the fate of marine debris in the open ocean from coastal regions around the world on interannual to centennial timescales. We find that six major garbage patches emerge, one in each of the five subtropical basins and one previously unreported patch in the Barents Sea. The evolution of each of the six patches is markedly different. With the exception of the North Pacific, all patches are much more dispersive than expected from linear ocean circulation theory, suggesting that on centennial timescales the different basins are much better connected than previously thought and that inter-ocean exchanges play a large role in the spreading of marine debris. This study suggests that, over multi-millennial timescales, a significant amount of the debris released outside of the North Atlantic will eventually end up in the North Pacific patch, the main attractor of global marine debris.

LP-based disaggregation approaches to solving the open pit mining production scheduling problem with block processing selectivity

Given a discretisation of an orebody as a block model, the open pit mining production scheduling problem (OPMPSP) consists of finding the sequence in which the blocks should be removed from the pit, over the lifetime of the mine, such that the net present value (NPV) of the operation is maximised. In practice, due to the large number of blocks and precedence constraints linking them, blocks are typically aggregated to form larger scheduling units. We aim to solve the OPMPSP, formulated as a mixed integer programme (MIP), so that aggregates are used to schedule the mining process, while individual blocks are used for processing decisions. We propose an iterative disaggregation method that refines the aggregates (with respect to processing) up to the point where the refined aggregates defined for processing produce the same optimal solution for the linear programming (LP) relaxation of the MIP as the optimal solution of the LP relaxation with individual block processing. We propose several strategies of creating refined aggregates for the MIP processing, using duality results and exploiting the problem structure. These refined aggregates allow the solution of very large problems in reasonable time with very high solution quality in terms of NPV.

Transport in time-dependent dynamical systems: Finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.

Extracting Dynamical Behavior via Markov Models

Statistical properties of chaotic dynamical systems are difficult to estimate reliably. Using long trajectories as data sets sometimes produces misleading results. It has been recognized for some time that statistical properties are often stable under the addition of a small amount of noise. Rather than analyzing the dynamical system directly, we slightly perturb it to create a Markov model. The analogous statistical properties of the Markov model often have “closed forms” and are easily computed numerically. The Markov construction is observed to provide extremely robust estimates and has the theoretical advantage of allowing one to prove convergence in the noise → 0 limit and produce rigorous error bounds for statistical quantities. We review the latest results and techniques in this area.

Optimizing the landside operation of a container terminal

This paper concerns the problem of operating a landside container exchange area that is serviced by multiple semi-automated rail mounted gantry cranes (RMGs) that are moving on a single bi-directional traveling lane. Such a facility is being built by Patrick Corporation at the Port Botany terminal in Sydney. The gantry cranes are a scarce resource and handle the bulk of container movements. Thus, they require a sophisticated analysis to achieve near optimal utilization. We present a three-stage algorithm to manage the container exchange facility, including the scheduling of cranes, the control of associated short-term container stacking, and the allocation of delivery locations for trucks and other container transporters. The key components of our approach are a time scale decomposition, whereby an integer program controls decisions across a long time horizon to produce a balanced plan that is fed to a series of short time scale online subproblems, and a highly efficient space-time divisioning of short-term storage areas. A computational evaluation shows that our heuristic can find effective solutions for the planning problem; on real-world data it yields a solution at most 8% above a lower bound on optimal RMG utilization.

A strengthened formulation and cutting planes for the open pit mine production scheduling problem

We present an integer programming formulation for the open pit mine production scheduling problem. We strengthen this formulation by adding inequalities derived by combining the precedence and production constraints. The addition of these inequalities decreases the computational requirements to obtain the optimal integer solution, in many cases by a significant margin.

An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems

Abstract The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, are currently the subject of intense study. In the autonomous setting, the use of transfer operators (Perron–Frobenius operators) to identify invariant and almost-invariant sets has been particularly successful. In the nonautonomous (time-dependent) setting, coherent sets, a time-parameterised family of minimally dispersive sets, are a natural extension of almost-invariant sets. The present work introduces a new analytic transfer operator construction that enables the calculation of finite-time coherent sets (sets are that minimally dispersive over a finite time interval). This new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude. The construction can also be applied to general Markov processes on continuous state space.

Approximating physical invariant measures of mixing dynamical systems in higher dimensions

Invariant measures of higher dimensional transformations are hard to calculate. We present new results on the estimation of absolutely continous invariant measures of mixing transformations , including a new method of proof of Ulams conjecture. The method involves constructing nite matrix approximations to the Perron-Frobenius operator from increasingly ner partitions of the state space X. We show that at a nite stage, our approximations are close to a special operator which would yield a correct answer. The exponential mixing property guarantees that the system is suuciently insensitive to any approximation errors, showing our computed invariant density is close to the true invariant density. Our method has the advantages of having very relaxed conditions on the partitions, being applicable to higher dimensional systems, and potentially applicable to a wide class of maps.

Ulam's method for random interval maps

We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulams finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory 17 177-86). We extend Ulams construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogous convergence result. In addition, we obtain a convergence rate for our approximations to the unique ACIM, and provide rigorous bounds for the L1 error of the Ulam approximation.