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1. Active subspace-based dimension reduction for chemical kinetics applications with epistemic uncertainty

   Vohra, M. (March 2019, Combustion and flame)

   We focus on an efficient approach for quantification of uncertainty in complex chemical reaction networks with a large number of uncertain parameters and input conditions. Parameter dimension reduction is accomplished by computing an active subspace that predominantly captures the variability in the quantity of interest (QoI). In the present work, we compute the active subspace for a H2/O2 mechanism that involves 19 chemical reactions, using an efficient iterative strategy. The active subspace is first computed for a 19-parameter problem wherein only the uncertainty in the pre-exponents of the individual reaction rates us considered. This is followed by the analysis of a 36-dimensional case wherein the activation energies and initial conditions are also considered uncertain. In both cases, a 1-dimensional active subspace is observed to capture the uncertainty in the QoI, which indicates enormous potential for efficient statistical analysis of complex chemical systems. In addition, we explore links between active subspaces and global sensitivity analysis, and exploit these links for identification of key contributors to the variability in the model response. 

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2. Saddle point least squares discretization for convection-diffusion

   https://doi.org/10.1080/00036811.2023.2291511  

   Bacuta, Constantin; Hayes, Daniel; O'Grady, Tyler (August 2024, Applicable Analysis)

   We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models. 

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3. Counting and Sampling Orientations on Chordal Graphs

   https://doi.org/10.1007/978-3-030-96731-4_29  

   Bezakova, Ivona; Sun, Wenbo (January 2022, International Conference and Workshops on Algorithms and Computation (WALCOM))

   We study counting problems for several types of orientations of chordal graphs: source-sink-free orientations, sink-free orientations, acyclic orientations, and bipolar orientations, and, for the latter two, we also present linear-time uniform samplers. Counting sink-free, acyclic, or bipolar orientations are known to be #P-complete for general graphs, motivating our study on a restricted, yet well-studied, graph class. Our main focus is source-sink-free orientations, a natural restricted version of sink-free orientations related to strong orientations, which we introduce in this work. These orientations are intriguing, since despite their similarity, currently known FPRAS and sampling techniques (such as Markov chains or sink-popping) that apply to sink-free orientations do not seem to apply to source-sink-free orientations. We present fast polynomialtime algorithms counting these orientations on chordal graphs. Our approach combines dynamic programming with inclusion-exclusion (going two levels deep for source-sink-free orientations and one level for sinkfree orientations) throughout the computation. Dynamic programming counting algorithms can be typically used to produce a uniformly random sample. However, due to the negative terms of the inclusion-exclusion, the typical approach to obtain a polynomial-time sampling algorithm does not apply in our case. Obtaining such an almost uniform sampling algorithm for source-sink-free orientations in chordal graphs remains an open problem. Little is known about counting or sampling of acyclic or bipolar orientations, even on restricted graph classes. We design efficient (linear-time) exact uniform sampling algorithms for these orientations on chordal graphs. These algorithms are a byproduct of our counting algorithms, but unlike in other works that provide dynamic-programming-based samplers, we produce a random orientation without computing the corresponding count, which leads to a faster running time than the counting algorithm (since it avoids manipulation of large integers). 

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4. Adaptive Discretization for Episodic Reinforcement Learning in Metric Spaces

   https://doi.org/10.1145/3410048.3410059  

   Sinclair, Sean R.; Banerjee, Siddhartha; Lee Yu, Christina (July 2020, ACM SIGMETRICS Performance Evaluation Review)

   null (Ed.)

   We present an efficient algorithm for model-free episodic reinforcement learning on large (potentially continuous) state-action spaces. Our algorithm is based on a novel Q-learning policy with adaptive data-driven discretization. The central idea is to maintain a finer partition of the state-action space in regions which are frequently visited in historical trajectories, and have higher payoff estimates. We demonstrate how our adaptive partitions take advantage of the shape of the optimal Q-function and the joint space, without sacrificing the worst-case performance. In particular, we recover the regret guarantees of prior algorithms for continuous state-action spaces, which additionally require either an optimal discretization as input, and/or access to a simulation oracle. Moreover, experiments demonstrate how our algorithm automatically adapts to the underlying structure of the problem, resulting in much better performance compared both to heuristics and Q-learning with uniform discretization. 

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5. Endothermic and exothermic chemically reacting plumes

   https://doi.org/10.1017/S0022112008002917  

   CONROY, DEVIN T.; SMITH, STEFAN G. (October 2008, Journal of Fluid Mechanics)

   null (Ed.)

   We develop a model for a turbulent plume in an unbounded ambient that takes into account a general exothermic or endothermic chemical reaction. These reactions can have an important effect on the plume dynamics since the entrainment rate, which scales with the vertical velocity, will be a function of the heat release or absorption. Specifically, we examine a second-order non-reversible reaction, where one species is present in the plume from a pure source and the other is in the environment. For uniform ambient density and species fields the reaction has an important effect on the deviation from pure plume behaviour as defined by the source parameter Γ. In the case of an exothermic reaction the density difference between the plume and the reference density increases and the plume is ‘lazy’, whereas for an endothermic reaction this difference decreases and the plume is more jet-like. Furthermore, for chemical and density-stratified environments, the reaction will have an important effect on the buoyancy flux because the entrainment rate will not necessarily decrease with distance from the source, as in traditional models. As a result, the maximum rise height of the plume for exothermic reactions may actually decrease with reaction rate if this occurs in a region of high ambient density. In addition, we investigate non-Boussinesq effects, which are important when the heat of reaction is large enough. 

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