A mathematical program is an optimization problem expressed as an objective function of multiple variables subject to set of constraints. When the optimization problem has specific structure, the problem class usually has a special name. A linear program is the optimization of a linear objective function subject to linear constraints. An integer program is a linear program where some of the variables must take only integer values. A semidefinite program is a linear program where the variables are arranged in a matrix and for all feasible solutions, this matrix must be positive semidefinite. There are general-purpose solvers for each of these classes of mathematical program. There are usually many ways to express a problem as a correct, say, linear program. However, equivalent formulations can have significantly different practical tractability. In this poster, we present new formulations for two classic discrete optimization problems, maximum cut (max cut) and the graphical traveling salesman problem (GTSP), that are significantly stronger, and hence more computationally tractable, than any previous formulations of their class. Both partially answer longstanding open theoretical questions in polyhedral combinatorics.
Singlet oxygen generators are multiphase flow chemical reactors used to generate energetic oxygen to be used as a fuel for chemical oxygen iodine lasers. In this paper, a theoretical model of the generator is presented along with its solutions over ranges of parameter space and oxygen maximizing optimizations. The singlet oxygen generator (SOG) is a low-pressure, multiphase flow chemical reactor that is used to produce molecular oxygen in an electronically excited state, i.e. singlet delta oxygen. The primary product of the reactor, the energetic oxygen, is used in a stage immediately succeeding the SOG to dissociate and energize iodine. The gas mixture including the iodine is accelerated to a supersonic speed and lased. Thus the SOG is the fuel generator for the chemical oxygen iodine laser (COIL). The COIL has important application for both military purposes--it was developed by the US Air Force in the 1970s--and, as the infrared beam is readily absorbed by metals, industrial cutting and drilling. The SOG appears in various configurations, but the one in focus here is a crossflow droplet generator SOG. A gas consisting of molecular chlorine and a diluent, usually helium, is pumped through a roughly rectangular channel. An aqueous solution of hydrogen peroxide and potassium hydroxide is pumped through small holes into the channel and perpendicular to the direction of the gas flow. So doing causes the solution to become aerosolized. Dissociation of the potassium hydroxide draws a proton from the hydrogen peroxide generating an HO{sub 2} radical in the liquid. Chlorine diffuses into the liquid and reacts with the HO{sub 2} ion producing the singlet delta oxygen; some of the oxygen diffuses back into the gas phase. The focus of this work is to generate a predictive multiphase flow model of the SOG in order to optimize its design. The equations solved are the so-called Eulerian-Eulerian form of the multiphase flow Navier-Stokes equations wherein one set of the equations represents the gas phase and another equation set of size m represents the liquid phase. In this case, m is representative of the division of the liquid phase into distinct representations of the various droplet sizes distributed in the reactor. A stabilized Galerkin formulation is used to solve the equation set on a computer. The set of equations is large. There are five equations representing the gas phase: continuity, vector momentum, heat. There are 5m representing the liquid phase: number density, vector momentum, heat. Four mass transfer equations represent the gas phase constituents and there are m advection diffusion equations representing the HO{sub 2} ion concentration in the liquid phase. Thus we are taking advantage of and developing algorithms to harness the power of large parallel computing architectures to solve the steady-state form of these equations numerous times so as to explore the large parameter space of the equations via continuation methods and to maximize the generation of singlet delta oxygen via optimization methods. Presented here will be the set of equations that are solved and the methods we are using to solve them. Solutions of the equations will be presented along with solution paths representing varying aerosol loading-the ratio of liquid to gas mass flow rates-and simple optimizations centered around maximizing the oxygen production and minimizing the amount of entrained liquid in the gas exit stream. Gas-entrained liquid is important to minimize as it can destroy the lenses and mirrors present in the lasing cavity.
We consider how to distribute sparse matrices among processes to reduce communication costs in parallel sparse matrix computations, specifically, sparse matrix-vector multiplication. Our main contributions are: (i) an exact graph model for communication with general (two-dimensional) matrix distribution, and (ii) a recursive partitioning algorithm based on nested dissection (substructuring). We show that the communication volume is closely linked to vertex separators. We have implemented our algorithm using hypergraph partitioning software to enable a fair comparison with existing methods. We present numerical results for sparse matrices from several application areas, with up to 9 million nonzeros. The results show that our new approach is superior to traditional 1d partitioning and comparable to a current leading partitioning method, the finegrain hypergraph method, in terms of communication volume. Our nested dissection method has two advantages over the fine-grain method: it is faster to compute, and the resulting distribution requires fewer communication messages.
A loose two-way coupling of SNL's Presto v2.8 and CTH v8.1 analysis code has been developed to support the analysis of explosive loading of structures. Presto is a Lagrangian, three-dimensional explicit, transient dynamics code in the SIERRA mechanics suite for the analysis of structures subjected to impact-like loads. CTH is a hydro code for modeling complex multi-dimensional, multi-material problems that are characterized by large deformations and/or strong shocks. A fundamental assumption in this loose coupling is that the compliance of the structure modeled with Presto is significantly smaller than the compliance of the surrounding medium (e.g. air) modeled with CTH. A current limitation of the coupled code is that the interaction between CTH and thin structures modeled in Presto (e.g. shells) is not supported. Research is in progress to relax this thin-structure limitation.