Publications

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The purpose of this project was to investigate the use of Bayesian methods for the estimation of the reliability of complex systems. The goals were to find methods for dealing with continuous data, rather than simple pass/fail data; to avoid assumptions of specific probability distributions, especially Gaussian, or normal, distributions; to compute not only an estimate of the reliability of the system, but also a measure of the confidence in that estimate; to develop procedures to address time-dependent or aging aspects in such systems, and to use these models and results to derive optimal testing strategies. The system is assumed to be a system of systems, i.e., a system with discrete components that are themselves systems. Furthermore, the system is 'engineered' in the sense that each node is designed to do something and that we have a mathematical description of that process. In the time-dependent case, the assumption is that we have a general, nonlinear, time-dependent function describing the process. The major results of the project are described in this report. In summary, we developed a sophisticated mathematical framework based on modern probability theory and Bayesian analysis. This framework encompasses all aspects of epistemic uncertainty and easily incorporates steady-state and time-dependent systems. Based on Markov chain, Monte Carlo methods, we devised a computational strategy for general probability density estimation in the steady-state case. This enabled us to compute a distribution of the reliability from which many questions, including confidence, could be addressed. We then extended this to the time domain and implemented procedures to estimate the reliability over time, including the use of the method to predict the reliability at a future time. Finally, we used certain aspects of Bayesian decision analysis to create a novel method for determining an optimal testing strategy, e.g., we can estimate the 'best' location to take the next test to minimize the risk of making a wrong decision about the fitness of a system. We conclude this report by proposing additional fruitful areas of research.

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This document describes how to obtain, install, use, and enjoy a better life with OVIS version 2.0. The OVIS project targets scalable, real-time analysis of very large data sets. We characterize the behaviors of elements and aggregations of elements (e.g., across space and time) in data sets in order to detect anomalous behaviors. We are particularly interested in determining anomalous behaviors that can be used as advance indicators of significant events of which notification can be made or upon which action can be taken or invoked. The OVIS open source tool (BSD license) is available for download at ovis.ca.sandia.gov. While we intend for it to support a variety of application domains, the OVIS tool was initially developed for, and continues to be primarily tuned for, the investigation of High Performance Compute (HPC) cluster system health. In this application it is intended to be both a system administrator tool for monitoring and a system engineer tool for exploring the system state in depth. OVIS 2.0 provides a variety of statistical tools for examining the behavior of elements in a cluster (e.g., nodes, racks) and associated resources (e.g., storage appliances and network switches). It calculates and reports model values and outliers relative to those models. Additionally, it provides an interactive 3D physical view in which the cluster elements can be colored by raw element values (e.g., temperatures, memory errors) or by the comparison of those values to a given model. The analysis tools and the visual display allow the user to easily determine abnormal or outlier behaviors. The OVIS project envisions the OVIS tool, when applied to compute cluster monitoring, to be used in conjunction with the scheduler or resource manager in order to enable intelligent resource utilization. For example, nodes that are deemed less healthy, that is, nodes that exhibit outlier behavior in some variable, or set of variables, that has shown to be correlated with future failure, can be discovered and assigned to shorter duration or less important jobs. Further, applications with fault-tolerant capabilities can invoke those mechanisms on demand, based upon notification of a node exhibiting impending failure conditions, rather than performing such mechanisms (e.g. checkpointing) at regular intervals unnecessarily.

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This is a progress report on polynomial system solving for statistical modeling. This is a progress report on polynomial system solving for statistical modeling. This quarter we have developed our first model of shock response data and an algorithm for identifying the chamber cone containing a polynomial system in n variables with n+k terms within polynomial time - a significant improvement over previous algorithms, all having exponential worst-case complexity. We have implemented and verified the chamber cone algorithm for n+3 and are working to extend the implementation to handle arbitrary k. Later sections of this report explain chamber cones in more detail; the next section provides an overview of the project and how the current progress fits into it.

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In this report, we present the novel functionality of parallel tetrahedral mesh refinement which we have implemented in MOAB. This report details work done to implement parallel, edge-based, tetrahedral refinement into MOAB. The theoretical basis for this work is contained in [PT04, PT05, TP06] while information on design, performance, and operation specific to MOAB are contained herein. As MOAB is intended mainly for use in pre-processing and simulation (as opposed to the post-processing bent of previous papers), the primary use case is different: rather than refining elements with non-linear basis functions, the goal is to increase the number of degrees of freedom in some region in order to more accurately represent the solution to some system of equations that cannot be solved analytically. Also, MOAB has a unique mesh representation which impacts the algorithm. This introduction contains a brief review of streaming edge-based tetrahedral refinement. The remainder of the report is broken into three sections: design and implementation, performance, and conclusions. Appendix A contains instructions for end users (simulation authors) on how to employ the refiner.

This report summarizes the existing statistical engines in VTK/Titan and presents the parallel versions thereof which have already been implemented. The ease of use of these parallel engines is illustrated by the means of C++ code snippets. Furthermore, this report justifies the design of these engines with parallel scalability in mind; then, this theoretical property is verified with test runs that demonstrate optimal parallel speed-up with up to 200 processors.

We present a formula for the pairwise update of arbitrary-order centered statistical moments. This formula is of particular interest to compute such moments in parallel for large-scale, distributed data sets. As a corollary, we indicate a specialization of this formula for incremental updates, of particular interest to streaming implementations. Finally, we provide pairwise and incremental update formulas for the covariance. Centered statistical moments are one of the most widely used tools in descriptive statistics. It is therefore essential for statistical analysis packages that robust and efficient algorithms be devised and implemented. However, robustness and speed of execution, in this context as well as in others, tend to be orthogonal. For instance, it is well known1 that algorithms for calculating centered statistical moments that utilize sum of powers for the sake of execution speed (one-pass algorithms) lead to unacceptable numerical instability.

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This report specifies the way in which Gauss points shall be named and ordered when storing them in an EXODUS II file so that they may be properly interpreted by visualization tools. This naming convention covers hexahedra and tetrahedra. Future revisions of this document will cover quadrilaterals, triangles, and shell elements.

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Current work on the Integrated Stockpile Evaluation (ISE) project is evidence of Sandia's commitment to maintaining the integrity of the nuclear weapons stockpile. In this report, we undertake a key element in that process: development of an analytical framework for determining the reliability of the stockpile in a realistic environment of time-variance, inherent uncertainty, and sparse available information. This framework is probabilistic in nature and is founded on a novel combination of classical and computational Bayesian analysis, Bayesian networks, and polynomial chaos expansions. We note that, while the focus of the effort is stockpile-related, it is applicable to any reasonably-structured hierarchical system, including systems with feedback.

The purpose of this report is to define a standard interface for storing and retrieving novel, non-traditional partial differential equation (PDE) discretizations. Although it focuses specifically on finite elements where state is associated with edges and faces of volumetric elements rather than nodes and the elements themselves (as implemented in ALEGRA), the proposed interface should be general enough to accommodate most discretizations, including hp-adaptive finite elements and even mimetic techniques that define fields over arbitrary polyhedra. This report reviews the representation of edge and face elements as implemented by ALEGRA. It then specifies a convention for storing these elements in EXODUS files by extending the EXODUS API to include edge and face blocks in addition to element blocks. Finally, it presents several techniques for rendering edge and face elements using VTK and ParaView, including the use of VTK's generic dataset interface for interpolating values interior to edges and faces.

Traditional cluster monitoring approaches consider nodes in singleton, using manufacturer-specified extreme limits as thresholds for failure ''prediction''. We have developed a tool, OVIS, for monitoring and analysis of large computational platforms which, instead, uses a statistical approach to characterize single device behaviors from those of a large number of statistically similar devices. Baseline capabilities of OVIS include the visual display of deterministic information about state variables (e.g., temperature, CPU utilization, fan speed) and their aggregate statistics. Visual consideration of the cluster as a comparative ensemble, rather than as singleton nodes, is an easy and useful method for tuning cluster configuration and determining effects of real-time changes.

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Verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of metrics. A metric is a real number assigned to one of these shapes depending on its particular vertex coordinates. These metrics are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. The geometric qualities of these regions is usually strongly tied to the accuracy these solvers are able to obtain in their approximations. The subroutines are written in C++ and have a simple C interface. Each metric may be evaluated individually or in combination. When multiple metrics are evaluated at once, they share common calculations to lower the cost of the evaluation.

This report contains an algorithm for decomposing higher-order finite elementsinto regions appropriate for isosurfacing and proves the conditions under which thealgorithm will terminate. Finite elements are used to create piecewise polynomialapproximants to the solution of partial differential equations for which no analyticalsolution exists. These polynomials represent fields such as pressure, stress, and mo-mentim. In the past, these polynomials have been linear in each parametric coordinate.Each polynomial coefficient must be uniquely determined by a simulation, and thesecoefficients are called degrees of freedom. When there are not enough degrees of free-dom, simulations will typically fail to produce a valid approximation to the solution.Recent work has shown that increasing the number of degrees of freedom by increas-ing the order of the polynomial approximation (instead of increasing the number offinite elements, each of which has its own set of coefficients) can allow some typesof simulations to produce a valid approximation with many fewer degrees of freedomthan increasing the number of finite elements alone. However, once the simulation hasdetermined the values of all the coefficients in a higher-order approximant, tools donot exist for visual inspection of the solution.This report focuses on a technique for the visual inspection of higher-order finiteelement simulation results based on decomposing each finite element into simplicialregions where existing visualization algorithms such as isosurfacing will work. Therequirements of the isosurfacing algorithm are enumerated and related to the placeswhere the partial derivatives of the polynomial become zero. The original isosurfacingalgorithm is then applied to each of these regions in turn.3 AcknowledgementThe authors would like to thank David Day and Louis Romero for their insight into poly-nomial system solvers and the LDRD Senior Council for the opportunity to pursue thisresearch. The authors were supported by the United States Department of Energy, Officeof Defense Programs by the Labratory Directed Research and Development Senior Coun-cil, project 90499. Sandia is a multiprogram laboratory operated by Sandia Corporation,a Lockheed-Martin Company, for the United States Department of Energy under contractDE-AC04-94-AL85000.4

This report contains an algorithm for decomposing higher-order finite elements into regions appropriate for isosurfacing and proves the conditions under which the algorithm will terminate. Finite elements are used to create piecewise polynomial approximants to the solution of partial differential equations for which no analytical solution exists. These polynomials represent fields such as pressure, stress, and momentum. In the past, these polynomials have been linear in each parametric coordinate. Each polynomial coefficient must be uniquely determined by a simulation, and these coefficients are called degrees of freedom. When there are not enough degrees of freedom, simulations will typically fail to produce a valid approximation to the solution. Recent work has shown that increasing the number of degrees of freedom by increasing the order of the polynomial approximation (instead of increasing the number of finite elements, each of which has its own set of coefficients) can allow some types of simulations to produce a valid approximation with many fewer degrees of freedom than increasing the number of finite elements alone. However, once the simulation has determined the values of all the coefficients in a higher-order approximant, tools do not exist for visual inspection of the solution. This report focuses on a technique for the visual inspection of higher-order finite element simulation results based on decomposing each finite element into simplicial regions where existing visualization algorithms such as isosurfacing will work. The requirements of the isosurfacing algorithm are enumerated and related to the places where the partial derivatives of the polynomial become zero. The original isosurfacing algorithm is then applied to each of these regions in turn.

Effective monitoring of large computational clusters demands the analysis of a vast amount of raw data from a large number of machines. The fundamental interactions of the system are not, however, well-defined, making it difficult to draw meaningful conclusions from this data, even if one were able to efficiently handle and process it. In this paper we show that computational clusters, because they are comprised of a large number of identical machines, behave in a statistically meaningful fashion. We therefore can employ normal statistical methods to derive information about individual systems and their environment and to detect problems sooner than with traditional mechanisms. We discuss design details necessary to use these methods on a large system in a timely and low-impact fashion.

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In SAND report 2004-1617, we outline a method for edge-based tetrahedral subdivision that does not rely on saving state or communication to produce compatible tetrahedralizations. This report analyzes the performance of the technique by characterizing (a) mesh quality, (b) execution time, and (c) traits of the algorithm that could affect quality or execution time differently for different meshes. It also details the method used to debug the several hundred subdivision templates that the algorithm relies upon. Mesh quality is on par with other similar refinement schemes and throughput on modern hardware can exceed 600,000 output tetrahedra per second. But if you want to understand the traits of the algorithm, you have to read the report!

This is a basic documentation explaining how to use the SANDmath macros with a LATEX 2{var_epsilon} document pertaining to the SANDreport class.

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Finite element meshes are used to approximate the solution to some differential equation when no exact solution exists. A finite element mesh consists of many small (but finite, not infinitesimal or differential) regions of space that partition the problem domain, {Omega}. Each region, or element, or cell has an associated polynomial map, {Phi}, that converts the coordinates of any point, x = ( x y z ), in the element into another value, f(x), that is an approximate solution to the differential equation, as in Figure 1(a). This representation works quite well for axis-aligned regions of space, but when there are curved boundaries on the problem domain, {Omega}, it becomes algorithmically much more difficult to define {Phi} in terms of x. Rather, we define an archetypal element in a new coordinate space, r = ( r s t ), which has a simple, axis-aligned boundary (see Figure 1(b)) and place two maps onto our archetypal element: