Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material. Accepted for publication in Journal of Fluid Mechanic

Interest rate models with Markov chains

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Fluctuations in ballistic transport from Euler hydrodynamics

We propose a general formalism, within large deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the "extended fluctuation relations" of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of "freeness" much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.Comment: 27+22 pages, one figur

Middleware-based Database Replication: The Gaps between Theory and Practice

The need for high availability and performance in data management systems has been fueling a long running interest in database replication from both academia and industry. However, academic groups often attack replication problems in isolation, overlooking the need for completeness in their solutions, while commercial teams take a holistic approach that often misses opportunities for fundamental innovation. This has created over time a gap between academic research and industrial practice. This paper aims to characterize the gap along three axes: performance, availability, and administration. We build on our own experience developing and deploying replication systems in commercial and academic settings, as well as on a large body of prior related work. We sift through representative examples from the last decade of open-source, academic, and commercial database replication systems and combine this material with case studies from real systems deployed at Fortune 500 customers. We propose two agendas, one for academic research and one for industrial R&D, which we believe can bridge the gap within 5-10 years. This way, we hope to both motivate and help researchers in making the theory and practice of middleware-based database replication more relevant to each other.Comment: 14 pages. Appears in Proc. ACM SIGMOD International Conference on Management of Data, Vancouver, Canada, June 200

Non-Equilibrium Steady States for Networks of Oscillators

Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity

An FPGA Based Implementation of the Exact Stochastic Simulation Algorithm

Mathematical and statistical modeling of biological systems is a desired goal for many years. Many biochemical models are often evaluated using a deterministic approach, which uses differential equations to describe the chemical interactions. However, such an approach is inaccurate for small species populations as it neglects the discrete representation of population values, presents the possibility of negative populations, and does not represent the stochastic nature of biochemical systems. The Stochastic Simulation Algorithm (SSA) developed by Gillespie is able to properly account for these inherent noise fluctuations. Due to the stochastic nature of the Monte Carlo simulations, large numbers of simulations must be run in order to get accurate statistics for the species populations and reactions. However, the algorithm tends to be computationally heavy and leads to long simulation runtimes for large systems. Therefore, this thesis explores implementing the SSA on a Field Programmable Gate Array (FPGA) to improve performance. Employing the Field programmable Gate Arrays exploits the parallelism present in the SSA, providing speedup over the software implementations that execute sequentially. In contrast to prior work that requires re-construction and re-synthesis of the design to simulate a new biochemical system, this work explores the use of reconfigurable hardware in implementing a generic biochemical simulator