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Partitioning of rare earths in multiphase nuclear waste glass-ceramics
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Lunar highland rocks - Element partitioning among minerals. I - Electron microprobe analyses of Na, Mg, K and Fe in plagioclase; mg partitioning with orthopyroxene
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A Comparison of Partitioned and Non-Partitioned Matrix Solutions to Coupled Finite Element-Integral Equations
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The Pairwise Similarity Partitioning Algorithm: A Method for Unsupervised Partitioning of Geoscientific and Other Datasets Using Arbitrary Similarity Metrics
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Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations
We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Furthermore, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time.
Trace Element Partitioning Between Olivine and Melt in Lunar Basalts
Mineral/melt partition coefficients have been widely used to provide insights into magmatic processes. Olivine is one of the most abundant and important minerals in the lunar mantle and mare basalts. Yet, no systematic olivine/melt partitioning data are available for lunar conditions. We report trace element partition data between host mineral olivine and its melt inclusions in lunar basalts. Equilibrium is evaluated using the Fe-Mg exchange coefficient, leading to the choice of melt inclusion-host olivine pairs in lunar basalts 12040, 12009, 15016, 15647, and 74235. Partition coefficients of 21 elements (Li, Mg, Al, Ca, Ti, V, Cr, Mn, Fe, Co, Y, Zr, Nb, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) were measured. Except for Li, V, and Cr, these elements show no significant difference in olivine-melt partitioning compared to the data for terrestrial samples. The partition coefficient of Li between olivine and melt in some lunar basalts with low Mg# (Mg# < 0.75 in olivine, or < ~0.5 in melt) is higher than published data for terrestrial * Corresponding author. Email address: youxue@umich.edu 2 samples, which is attributed to the dependence of DLi on Mg# and the lack of literature DLi data with low Mg#. The partition coefficient of V in lunar basalts is measured to be 0.17 to 0.74, significantly higher than that in terrestrial basalts (0.003 to 0.21), which can be explained by the lower oxygen fugacity in lunar basalts. The significantly higher DV can explain why V is less enriched in evolved lunar basalts than terrestrial basalts. The partition coefficient of Cr between olivine and basalt melt in the Moon is 0.11 to 0.62, which is lower than those in terrestrial settings by a factor of approximately 2. This is surprising because previous authors showed that Cr partition coefficient is independent of fO2. A quasi-thermodynamically based model is developed to correlate Cr partition coefficient to olivine and melt composition and fO2. The lower Cr partition coefficient between olivine and basalt in the Moon can lead to more Cr enrichment in the lunar magma ocean, as well as more Cr enrichment in mantle-derived basalts in the Moon. Hence, even though Cr is typically a compatible element in terrestrial basalts, it is moderately incompatible in primitive lunar basalts, with a similar degree of incompatibility as V based on partition coefficients in this work, as also evidenced by the relatively constant V/Cr ratio of 0.039 ± 0.011 in lunar basalts. The confirmation of constant V/Cr ratio is important for constraining concentrations of Cr (slightly volatile and siderophile) and V (slightly siderophile) in the bulk silicate Moon.
On bottleneck partitioning k-ary n-cubes
Graph partitioning is a topic of extensive interest, with applications to parallel processing. In this context graph nodes typically represent computation, and edges represent communication. One seeks to distribute the workload by partitioning the graph so that every processor has approximately the same workload, and the communication cost (measured as a function of edges exposed by the partition) is minimized. Measures of partition quality vary; in this paper we consider a processor's cost to be the sum of its computation and communication costs, and consider the cost of a partition to be the bottleneck, or maximal processor cost induced by the partition. For a general graph the problem of finding an optimal partitioning is intractable. In this paper we restrict our attention to the class of k-art n-cube graphs with uniformly weighted nodes. Given mild restrictions on the node weight and number of processors, we identify partitions yielding the smallest bottleneck. We also demonstrate by example that some restrictions are necessary for the partitions we identify to be optimal. In particular, there exist cases where partitions that evenly partition nodes need not be optimal.
High-order partitioned spectral deferred correction solvers for multiphysics problems
We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1], [2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem modeled as a system of coupled ordinary differential equations (ODEs), coupled through coupling terms that can depend on the state of each subsystem; therefore the method applies to both a semi-discretized system of partial differential equations (PDEs) or problems naturally modeled as coupled systems of ODEs. The sufficient conditions to build arbitrarily high-order partitioned SDC schemes are derived. Based on these conditions, various of partitioned SDC schemes are designed. The stability of the first-order partitioned SDC scheme is analyzed in detail on a coupled, linear model problem. We show that the scheme is conditionally stable, and under conditions on the coupling strength, the scheme can be unconditionally stable. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems with moderate coupling strength. They include a stiff linear system of ODEs, advection-diffusion-reaction systems, and fluid-structure interaction problems with both incompressible and compressible flows, where we verify the design order of the SDC schemes and study various stability properties. We also directly compare the accuracy, stability, and cost of the proposed partitioned SDC solver with the partitioned IMEX method in [1], [2] on this suite of test problems. The results suggest that the high-order partitioned SDC solvers are more robust than the partitioned IMEX solvers for the numerical examples considered in this work, while the IMEX methods require fewer implicit solves.
AutoAtlas: Neural Network for 3D Unsupervised Partitioning and Representation Learning
Here we present a novel neural network architecture called AutoAtlas for fully unsupervised partitioning and representation learning of 3D brain Magnetic Resonance Imaging (MRI) volumes. AutoAtlas consists of two neural network components: one neural network to perform multi-label partitioning based on local texture in the volume, and a second neural network to compress the information contained within each partition. We train both of these components simultaneously by optimizing a loss function that is designed to promote accurate reconstruction of each partition, while encouraging spatially smooth and contiguous partitioning, and discouraging relatively small partitions. We show that the partitions adapt to the subject specific structural variations of brain tissue while consistently appearing at similar spatial locations across subjects. AutoAtlas also produces very low dimensional features that represent local texture of each partition. We demonstrate prediction of metadata associated with each subject using the derived feature representations and compare the results to prediction using features derived from FreeSurfer anatomical parcellation. Since our features are intrinsically linked to distinct partitions, we can then map values of interest, such as partition-specific feature importance scores onto the brain for visualization.
Dissecting the Genetic Architecture of Carbon Partitioning in Sorghum Using Multiscale Phenotypes
Carbon partitioning in plants may be viewed as a dynamic process composed of the many interactions between sources and sinks. The accumulation and distribution of fixed carbon is not dictated simply by the sink strength and number but is dependent upon the source, pathways, and interactions of the system. As such, the study of carbon partitioning through perturbations to the system or through focus on individual traits may fail to produce actionable developments or a comprehensive understanding of the mechanisms underlying this complex process. Using the recently published sorghum carbon-partitioning panel, we collected both macroscale phenotypic characteristics such as plant height, above-ground biomass, and dry weight along with microscale compositional traits to deconvolute the carbon-partitioning pathways in this multipurpose crop. Multivariate analyses of traits resulted in the identification of numerous loci associated with several distinct carbon-partitioning traits, which putatively regulate sugar content, manganese homeostasis, and nitrate transportation. Using a multivariate adaptive shrinkage approach, we identified several loci associated with multiple traits suggesting that pleiotropic and/or interactive effects may positively influence multiple carbon-partitioning traits, or these overlaps may represent molecular switches mediating basal carbon allocating or partitioning networks. Conversely, we also identify a carbon tradeoff where reduced lignin content is associated with increased sugar content. The results presented here support previous studies demonstrating the convoluted nature of carbon partitioning in sorghum and emphasize the importance of taking a holistic approach to the study of carbon partitioning by utilizing multiscale phenotypes.
Randomized Cholesky Preconditioning for Graph Partitioning Applications
A graph is a mathematical representation of a network; we say it consists of a set of vertices, which are connected by edges. Graphs have numerous applications in various fields, as they can model all sorts of connections, processes, or relations. For example, graphs can model intricate transit systems or the human nervous system. However, graphs that are large or complicated become difficult to analyze. This is why there is an increased interest in the area of graph partitioning, reducing the size of the graph into multiple partitions. For example, partitions of a graph representing a social network might help identify clusters of friends or colleagues. Graph partitioning is also a widely used approach to load balancing in parallel computing. The partitioning of a graph is extremely useful to decompose the graph into smaller parts and allow for easier analysis. There are different ways to solve graph partitioning problems. For this work, we focus on a spectral partitioning method which forms a partition based upon the eigenvectors of the graph Laplacian (details presented in Acer, et. al.). This method uses the LOBPCG algorithm to compute these eigenvectors. LOBPCG can be accelerated by an operator called a preconditioner. For this internship, we evaluate a randomized Cholesky (rchol) preconditioner for its effectiveness on graph partitioning problems with LOBPCG. We compare it with two standard preconditioners: Jacobi and Incomplete Cholesky (ichol). This research was conducted from August to December 2021 in conjunction with Sandia National Laboratories.
Stencils and problem partitionings: Their influence on the performance of multiple processor systems
Given a discretization stencil, partitioning the problem domain is an important first step for the efficient solution of partial differential equations on multiple processor systems. Partitions are derived that minimize interprocessor communication when the number of processors is known a priori and each domain partition is assigned to a different processor. This partitioning technique uses the stencil structure to select appropriate partition shapes. For square problem domains, it is shown that non-standard partitions (e.g., hexagons) are frequently preferable to the standard square partitions for a variety of commonly used stencils. This investigation is concluded with a formalization of the relationship between partition shape, stencil structure, and architecture, allowing selection of optimal partitions for a variety of parallel systems.
Stencils and problem partitionings - Their influence on the performance of multiple processor systems
Given a discretization stencil, partitioning the problem domain is an important first step for the efficient solution of partial differential equations on multiple processor systems. Partitions are derived that minimize interprocessor communication when the number of processors is known a priori and each domain partition is assigned to a different processor. This partitioning technique uses the stencil structure to select appropriate partition shapes. For square problem domains, it is shown that non-standard partitions (e.g., hexagons) are frequently preferable to the standard square partitions for a variety of commonly used stencils. This investigation is concluded with a formalization of the relationship between partition shape, stencil structure, and architecture, allowing selection of optimal partitions for a variety of parallel systems.
Experimental Constraints on the Partitioning Behavior of F, Cl, and OH Between Apatite and Basaltic Melt
The mineral apatite is present in a wide range of planetary materials. The presence of volatiles (F, Cl, and OH) within its crystal structure (X-site) have motivated numerous studies to investigate the partitioning behavior of F, Cl, and OH between apatite and silicate melt with the end goal of using apatite to constrain the volatile contents of planetary magmas and mantle sources. A number of recent experimental studies have investigated the apatite-melt partitioning behavior of F, Cl, and OH in magmatic systems. Apatite-melt partitioning of volatiles are best described as exchange equilibria similar to Fe-Mg partitioning between olivine and silicate melt. However, the partitioning behavior is likely to change as a function of temperature, pressure, oxygen fugacity, apatite composition, and melt composition. In the present study, we have conducted experiments to assess the partitioning behavior of F, Cl, and OH between apatite and silicate melt over a pressure range of 0-6 gigapascals, a temperature range of 950-1500 degrees Centigrade, and a wide range of apatite ternary compositions. All of the experiments were conducted between iron-wustite oxidation potentials IW minus 1 and IW plus 2 in a basaltic melt composition. The experimental run products were analyzed by a combination of electron probe microanalysis and secondary ion mass spectrometry (NanoSIMS). Temperature, apatite crystal chemistry, and pressure all play important roles in the partitioning behavior of F, Cl, and OH between apatite and silicate melt. In portions of apatite ternary space that undergo ideal mixing of F, Cl, and OH, exchange coefficients remain constant at constant temperature and pressure. However, exchange coefficients vary at constant temperature (T) and pressure (P) in portions of apatite compositional space where F, Cl, and OH do not mix ideally in apatite. The variation in exchange coefficients exhibited by apatite that does not undergo ideal mixing far exceeds the variations induced by changes in temperature (T) or pressure (P) . In regions where apatite undergoes ideal mixing of F, Cl, and OH, temperature has a stronger effect than pressure on the partitioning behavior, but both are important. Furthermore, fluorine becomes less compatible in apatite with increasing pressure and temperature. We are still in the process of analyzing our experimental run products, but we plan to quantify the effects of P and T on apatite-melt partitioning of F, Cl, and OH.
Partitioning of Large-Scale Power Electronics-Based Power Systems for Small-Signal Stability Analysis
The nodal admittance matrix (NAM)-based approach is suitable for analyzing the small-signal stability of large-scale power electronics-based power systems (PEPSs) as it preserves the system structure by utilizing the admittance matrix. Previously, NAM-based area partition has been proposed, which divides the system into various subareas and interconnections for easier analysis of the low-dimension matrix compared to the entire system-based high-dimension matrix. However, no partition algorithm has been presented for the NAM-based area partition method. This paper focuses on implementing the spectral partitioning algorithm for partitioning large-scale PEPSs into a low-dimension matrix to reduce the computation complexity of the analysis. These spectral components facilitate data transformation into a new space, enabling the application of traditional clustering methods like k-means. To evaluate the performance of the partitioning method, the subareas and interconnections obtained from the spectral clustering algorithm are incorporated into the NAM-based area partition method for a large system with 140 buses. The computational times of the original method, where the NAM-based criterion is directly applied to the entire system, are compared with those of the NAM-based partition method in MATLAB. PSCAD simulations of the whole system and the obtained subareas are conducted to validate the effectiveness of the proposed algorithm.
Spectral Clustering-Based Partitioning of Large-Scale Power Electronics-Based Power Systems for Small-Signal Stability Analysis
The nodal admittance matrix (NAM)-based approach is well-suited for small-signal stability analysis of large-scale power electronics-based power systems (PEPSs), as it preserves the system structure through its admittance matrix. Previous studies have explored partitioning such systems into subareas and interconnections to reduce computational burden; however, they lacked a formal algorithmic procedure for determining feasible partitions. While several grid partitioning methods, such as those based on graph theory or machine learning, exist in the literature, they cannot be directly applied to NAM-based analysis due to differing objectives and constraints. Here, this paper addresses this gap by presenting a systematic, step-by-step procedure for applying a spectral partitioning algorithm that yields a division of the system into subareas suitable for NAM-based analysis. The computational complexity of the proposed method is also derived to demonstrate its efficiency and justify the practicality of the resulting subarea decomposition. The performance of the partitioning method is evaluated by applying the spectral clustering-derived subareas and interconnections to the NAM-based partitioning approach on a 140-bus system. Computational times for the full-system and partitioned NAM analyses are compared using MATLAB. Additionally, PSCAD simulations of the complete system and partitioned subareas are carried out to verify the effectiveness of the proposed method.
Partitioned exponential methods for coupled multiphysics systems
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. Here, the traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.