Automatic differentiation of numerical integration algorithms

Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

     

Nonlinear eigenvalue problems for quasilinear equations in unbounded domains

We prove the existence of a solution of the nonlinear equation in IR^N and in exterior domains, respectively. We concentrate to the case when p ≥ N and the nonlinearity f(x, · ) is “superlinear” and “subcritical”.     

Flamelet-Modeling of Inverse Rich Diffusion Flames

An experimental and numerical investigation of a confined laminar inverse diffusion flame (IDF) with pure oxygen as oxidizer and carbon dioxide diluted methane as fuel with a global stoichiometry of partial oxidation processes (equivalence ratio of 2.5) is presented. The present burner setup allows studying both the flame and the post-flame zone in a simplified geometry considering typical operating conditions as found in large-scale gasifiers. This partial oxidation flame setup is characterized by very high temperatures close to the stoichiometric oxidation zone due to oxy-fuel combustion, whereas lower temperatures and slow endothermic post-flame conversion reactions with long residence times are found in the fuel rich post-flame region. The scope of this paper is to investigate different modeling approaches suitable for both regimes by comparing the simulation results to detailed experimental data. Planar OH laser-induced fluorescence (OH-LIF) was performed for measuring the hydroxyl radical in the reaction zone and the results are compared to CFD calculations. Based on this comparison, the necessary level of detail of diffusion flux modeling, which includes Soret and Dufour effects, is analyzed and established. Finally, steady and unsteady non-premixed flamelet approaches based on a single mixture fraction are used in order to study their applicability for both the oxidation and post-flame zone. Significantly different time scales are obtained using different flamelet paths. Their influence on the results is investigated in the steady flamelet and the Lagrangian flamelet approach.     

Numerical modeling of chlorine concentration in water storage tanks

In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed. Copyright © 2015 John Wiley & Sons, Ltd.     

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a?1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.     

Convergent sequences of sparse graphs: A large deviations approach

Models based on sparse graphs are of interest to many communities: they appear as basic models in combinatorics, probability theory, optimization, statistical physics, information theory, and more applied fields of social sciences and economics. Different notions of similarity (and hence convergence) of sparse graphs are of interest in different communities. In probability theory and combinatorics, the notion of Benjamini‐Schramm convergence, also known as left‐convergence, is used quite frequently. Statistical physicists are interested in the the existence of the thermodynamic limit of free energies, which leads naturally to the notion of right‐convergence. Combinatorial optimization problems naturally lead to so‐called partition convergence, which relates to the convergence of optimal values of a variety of constraint satisfaction problems. The relationship between these different notions of similarity and convergence is, however, poorly understood. In this paper we introduce a new notion of convergence of sparse graphs, which we call Large Deviations or LD‐convergence, and which is based on the theory of large deviations. The notion is introduced by “decorating” the nodes of the graph with random uniform i.i.d. weights and constructing corresponding random measures on and . A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on . The corresponding large deviations rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object. We then establish that LD‐convergence implies the other three notions of convergence discussed above, and at the same time establish several previously unknown relationships between the other notions of convergence. In particular, we show that partition‐convergence does not imply left‐ or right‐convergence, and that right‐convergence does not imply partition‐convergence. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 52–89, 2017     

Estimating the efficient price from the order flow: A Brownian Cox process approach

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price, etc.). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price from the order flow.     

Liouville numbers, Rajchman measures, and small Cantor sets

We show that the set of Liouville numbers carries a positive measure whose Fourier transform vanishes at infinity. The proof is based on a new construction of a Cantor set of Hausdorff dimension zero supporting such a measure.