A modified Monte-Carlo quadrature. II.
Modified Monte Carlo procedure using partitioning method and stratified sampling
MONTE CARLO METHOD↗
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Results for “partitioned methods”
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General theory of the transverse dielectric constant of III-V semiconducting compounds
A general model of the transverse dielectric constant of III-V compounds is developed using a hybrid method which combines the kp method with a nonlocal pseudopotential calculation. In this method the Brillouin zone is partitioned into three regions by expanding the energy bands and matrix elements about the F, X, and L symmetry points. The real and imaginary parts of the dielectric constant are calculated as a sum of the individual contributions of each region. By using this partition method, it is possible to get good insight into the dependence of the dielectric constant on the shape of the band structure.
Kahen, K. B.↗
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.
Nupur [Univ. of Tennessee, Knoxville, TN (United S↗
Formal Functional Test Designs with a Test Representation Language
This article discusses the application of the Category-Partition Method to the test design phase. The method provides a formal framework for reducing the total number of possible test cases to a minimum logical subset for effective testing. An automatic tool and a formal language have been developed to implement the method and produce the specification of test cases
software↗
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.
Kennedy, Christopher A.↗
Understanding the Impact of Unobservable Variables on the Performance of Predictive Models: The Need for Feature Space Partitioning and Fusion
When developing predictive models over a dataset, the model is globally optimized across the entire feature space to learn a decision boundary. However, when unobservable variables—which cannot be measured or estimated—interact with the observable variables, this can negatively impact the optimization applied to the decision boundary since the data samples introduced by unobservable variables may have little to no association with the applied global optimization. This, consequently, penalizes the entire decision boundary and model performance. This paper examines some of the detrimental effects of unobservable variables, particularly their role in creating new modes in the distribution of observable variables and reducing the separability of class distributions. Such challenges result in skewed or warped decision boundaries and decreased accuracy of model predictions, particularly for interpretable models like logistic regression and decision trees. Through two illustrative case examples, we highlight the need to address the challenges imposed by unobservable variables. We propose a strategy to mitigate these challenges by creating local regions within the feature space through partitioning. This enables the optimization of local models within the regions to overcome the impact of unobservability in different feature space localities. Research into a more sophisticated partitioning strategy and where the partition should be relative to the sample of interest is left as future work. Through the analysis of the impact of unobservability and the development of a partitioning method, we demonstrate the clear need for a partitioning strategy that integrates knowledge from multiple local models to estimate risk factors using information fusion. Thus, we establish the foundation and motivation for using partitioning and information fusion to overcome the effects of unobservability in predictive models. Formal fusion methods, such as Dempster-Shafer theory, can better leverage the information from local regions to improve the performance of interpretable predictive models in the presence of unobservable variables.
Time Series Data↗
Spectral deferred correction methods for high-order accuracy in poroelastic problems
In this work, we investigate high-order accuracy in time integration by examining two operator splitting methods for poroelastic problems: the two-pass and the spectral deferred correction (SDC) methods. To enhance the order of accuracy, the two-pass method partitions a coupled operator symmetrically, whereas the SDC method corrects truncation errors by establishing an error equation. These high-order methods are applied to underlying solution strategies, i.e., monolithic, fixed-stress sequential, and undrained sequential methods. We observe that semi-discretized systems from spatial discretization have forms similar to those of index-1 differential algebraic equations (DAEs), causing order reduction against the two-pass method when it is used in conjunction with either the monolithic or sequential method. On the other hand, the SDC in conjunction with the monolithic method exhibits the desired second-order accuracy in poroelastic problems while increasing the order of accuracy for index-1 DAEs. However, the SDC in conjunction with either of the two sequential methods does not achieve the desired order of accuracy, and maintains first order because the flow equation for poroelasticity has an additional approximation associated with the volumetric strain rate term, which does not yield exactly the same forms as those of conventional DAEs. Thus, the monolithic SDC method can achieve higher-order accuracy, but may require higher computational costs because it involves solving matrix systems larger than those for the sequential methods.
02 PETROLEUM↗
Research in Computational Astrobiology
We report on several projects in the field of computational astrobiology, which is devoted to advancing our understanding of the origin, evolution and distribution of life in the Universe using theoretical and computational tools. Research projects included modifying existing computer simulation codes to use efficient, multiple time step algorithms, statistical methods for analysis of astrophysical data via optimal partitioning methods, electronic structure calculations on water-nuclei acid complexes, incorporation of structural information into genomic sequence analysis methods and calculations of shock-induced formation of polycylic aromatic hydrocarbon compounds.
Chaban, Galina↗
The calculation of upper and lower bounds of energy eigenvalues in perturbation theory by means of partitioning techniques.
Quantum mechanical perturbation theory calculation of upper and lower bounds of energy eigenvalues using partitioning methods
PERTURBATION THEORY↗
A Multistage Stochastic Transmission Expansion Algorithm for Wide-Area Planning under Uncertainty
The overall objective for this project was to develop and demonstrate a set of methods for solving the transmission investment problem for a large network considering many possible scenarios of future conditions and multiple decision points when investments can be made. Project sub-objectives achieved this goal through a succession of extending the methods to apply to problems with increasing complexity or additional features, including the number of decision points, whether generation and transmission are co-optimized, and whether AC or DC power flow is used. A transmission model was developed for the Western Electric Coordinating Council (WECC) region, the high-voltage transmission system that serves the western third of the continental U.S. Using a dataset provided by WECC and by researchers from John Hopkins University, we have validated and demonstrated the model and used it to compare the new method for solving multi-stage stochastic transmission planning to several state-of-the-art techniques. The project has resulted in several key outcomes and achievements: The covariance-based method for choosing a small set of hours to represent short-term variability has superior performance in terms of accuracy to existing methods, including K-means clustering and Importance Sampling; The combined partitioning method for long-term uncertainty with the nested clustering approach for choosing representative hours for each long-term group has superior accuracy for equivalent computational effort compared with existing methods; Using the partitioning/clustering method combined with Sample Average Approximation provides both statistical bounds on the quality of the solution and at the same time, a complete investment plan for all contingencies in the full uncertainty set; no existing methods can provide both at the same time; The method is demonstrated to work well for choosing both transmission and generation investments; A variant on the method allows for both scenario selection and simultaneous correction for the error from the DC power flow approximation to provide a tractable method for AC power flow-based transmission planning under uncertainty; The method applied to the WECC case study demonstrates the additional value to the system operator and the consumer of identifying flexible investment options in the near-term decisions. In particular, the case study exhibits significant option value in postponing some transmission additions that appear useful but in some long-term system states create new congestion problems.
24 POWER TRANSMISSION AND DISTRIBUTION↗
A unified formulation of splitting-based implicit time integration schemes
Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. Here this work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Algorithms for parallel flow solvers on message passing architectures
The purpose of this project has been to identify and test suitable technologies for implementation of fluid flow solvers -- possibly coupled with structures and heat equation solvers -- on MIMD parallel computers. In the course of this investigation much attention has been paid to efficient domain decomposition strategies for ADI-type algorithms. Multi-partitioning derives its efficiency from the assignment of several blocks of grid points to each processor in the parallel computer. A coarse-grain parallelism is obtained, and a near-perfect load balance results. In uni-partitioning every processor receives responsibility for exactly one block of grid points instead of several. This necessitates fine-grain pipelined program execution in order to obtain a reasonable load balance. Although fine-grain parallelism is less desirable on many systems, especially high-latency networks of workstations, uni-partition methods are still in wide use in production codes for flow problems. Consequently, it remains important to achieve good efficiency with this technique that has essentially been superseded by multi-partitioning for parallel ADI-type algorithms. Another reason for the concentration on improving the performance of pipeline methods is their applicability in other types of flow solver kernels with stronger implied data dependence. Analytical expressions can be derived for the size of the dynamic load imbalance incurred in traditional pipelines. From these it can be determined what is the optimal first-processor retardation that leads to the shortest total completion time for the pipeline process. Theoretical predictions of pipeline performance with and without optimization match experimental observations on the iPSC/860 very well. Analysis of pipeline performance also highlights the effect of uncareful grid partitioning in flow solvers that employ pipeline algorithms. If grid blocks at boundaries are not at least as large in the wall-normal direction as those immediately adjacent to them, then the first processor in the pipeline will receive a computational load that is less than that of subsequent processors, magnifying the pipeline slowdown effect. Extra compensation is needed for grid boundary effects, even if all grid blocks are equally sized.
Vanderwijngaart, Rob F.↗
Structural optimization by generalized, multilevel decomposition
The developments toward a general multilevel optimization capability and results for a three-level structural optimization are described. The method partitions a structure into a number of substructuring levels where each substructure corresponds to a subsystem in the general case of an engineering system. The method is illustrated by a portal framework that decomposes into individual beams. Each beam is a box that can be further decomposed into stiffened plates. Substructuring for this example spans three different levels: (1) the bottom level of finite elements representing the plates; (2) an intermediate level of beams treated as substructures; and (3) the top level for the assembled structure. The three-level case is now considered to be qualitatively complete.
Sobieszczanski-Sobieski, J.↗
RANS-MP: A Portable Parallel Navier-Stokes Solver
RANS-MP, a new implementation of a single-grid Navier-Stokes solver using the diagonalized Beam-Warming approximate-factorization scheme, is presented. This first release of the completely rewritten solver employs the following optimizations: (1) Bi-directional multi-partition method for the ADI solver part; this improves granularity and load balance; (2) Improved cache usage through elimination of non-unit-stride array access (possible in part due to multi-partitioning); (3) Preprocessing of communicating boundary conditions to streamline logic during time stepping; (4) Truly parallel, high-performance I/O using the newly-developed MPI-IO library; (5) Elimination of large amounts of redundant operations through efficient use of workspace. Results of some realistic wing computations on the IBM SP2 computer will be presented. We will demonstrate that excellent absolute performance and scalability are obtained with RANS-MP, even for relatively small grid sizes. Besides high performance, an outstanding feature of RANS-MP is its true portability, due to the use of the portable message passing and I/O libraries MPI and MPI-IO.
VanderWijngaart, Rob F.↗
A Domain-Decomposed A-ϕ Formulation Based on Lagrange Multipliers for Low-Frequency Problems
A domain-decomposed A-ϕ formulation based on Lagrange multipliers is proposed to simulate low-frequency elec- tromagnetic problems. This method partitions the computational domain into smaller subdomains, allowing each subdomain to be independently formulated using Lagrange multipliers as Dirichlet boundary conditions, while ensuring continuity of the fields across the interfaces. A mixed finite element method, utilizing both vector and scalar basis functions, is employed to discretize the formulation, resulting in a global system to be solved. The proposed method is validated using TEAM Problem 7 at 50 Hz, demonstrating its effectiveness in handling complex geometries and addressing the low-frequency breakdown issues commonly encountered in traditional finite element methods.
Hossain, Amzad↗
Fatigue and fracture overview
The accomplishments achieved under the isotropic creep-fatigue crack initiation life prediction program are summarized. A sizeable creep-fatigue crack initiation data base was generated on the nickel-base superalloy, B-1900. Companion constitutive modeling programs have also generated extensive data bases on the same heat of material. The crack initiation results have formed the basis of a new approach to creep-fatigue life prediction. The term Cyclic Damage Accumulation (CDA) was coined for the method, which was evaluated under isothermal, uniaxial conditions. Stringent laboratory verification experiments were used to test the accuracy of the method. Considering the quite limited material property data needed to evaluate the constants in the approach, the prediction accuracy is acceptable. At the expense of the larger data base required, Lewis developed total strain- strainrange partitioning method (TS-SRP) is capable of a higher degree of accuracy.
Halford, Gary R.↗
An alternative representation of the ice canopy for calculating microwave brightness temperatures over a thunderstorm
Passive microwave brightness temperatures (TBs) at 92 and 183 GHz from an aircraft thunderstorm overflight are compared with values calculated from radar-derived hydrometeor profiles and a modified proximity sounding. Two methods for modeling particles in the ice canopy are contrasted. The first is a 'traditional' approach employing Marshall-Palmer ice spheres. The second, or 'alternative', method partitions 20 percent of the ice water content into a Marshall-Palmer component for graupel and hail, and 80 percent into a modified gamma spherical particle size distribution function representing ice crystals. Results from the alternative approach are superior to those from the traditional method in the anvil and mature convective core. In the decaying convective region, the traditional approach yields better agreement with observed magnitudes. Neither method, however, matches the geometry of the observed TB depression associated with the decaying convective core. This is likely due to the presence of graupel, which is not detected as a special signature in radar reflectivity, but does diminish TBs through scattering. Brightness temperatures at the relatively high microwave frequencies considered are shown to be very sensitive to the ice-particle size distribution.
Muller, Bradley M.↗
On the distribution of pitch angles in external galactic spirals NGC 1232 and NGC 5457
A numerical method, originally developed to analyze the morphology of global and local structure in prototype galaxies, is modified for analyzing observed disk-shape galaxies. Two digitized spiral galaxies NGC 1232 and NGC 5457 with varying degrees of contrast between arm and interarm regions are analyzed. A synergism of partitioning methods and a geometric mean least-squares regression algorithm serves to isolate local arm segments, spurs, feathers, and secondary features and to measure their pitch angles and lengths. The global arms are actually highly disjointed, with arm segments frequently revealing pitch angles between 30 and 50 deg, certainly greater than those of the parent arms. Prominent spurs tend to exhibit a much greater pitch angle. The automated mathematical algorithm is shown to have negligible numerical biasing and could be applied to any number of spiral galaxies manifesting flocculent structure, either prototype or observed, and could possibly be used as a tool for classification of multiple-armed-type galaxies.
Russell, William S.↗