We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification, and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly \scrO (N), and the number of iterations grows slowly with N.
Data fields sampled on irregularly spaced points arise in many science and engineering applications. For regular grids, Convolutional Neural Networks (CNNs) gain benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds using Generalized Moving Least Squares (GMLS). GMLS is a nonparametric meshfree technique for estimating linear bounded functionals from scattered data, and has emerged as an effective technique for solving partial differential equations (PDEs). By parameterizing the GMLS estimator, we obtain learning methods for linear and non-linear operators with unstructured stencils. The requisite calculations are local, embarrassingly parallelizable, and supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform operator regression, develop predictive dynamical models, and obtain feature extractors for engineering quantities of interest. The results show the promise of these architectures as foundations for data-driven model development in scientific machine learning applications.
American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FEDSM
Laros, James H.; Visintainer, Robert; Furlan, John; Pagalthivarthi, Krishnan V.; Garman, Mohamed; Cutright, Aaron; Wang, Yan
Wear prediction is important in designing reliable machinery for slurry industry. It usually relies on multi-phase computational fluid dynamics, which is accurate but computationally expensive. Each run of the simulations can take hours or days even on a high-performance computing platform. The high computational cost prohibits a large number of simulations in the process of design optimization. In contrast to physics-based simulations, data-driven approaches such as machine learning are capable of providing accurate wear predictions at a small fraction of computational costs, if the models are trained properly. In this paper, a recently developed WearGP framework [1] is extended to predict the global wear quantities of interest by constructing Gaussian process surrogates. The effects of different operating conditions are investigated. The advantages of the WearGP framework are demonstrated by its high accuracy and low computational cost in predicting wear rates.
Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyper-differential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multi-physics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.
Porosity in additively manufactured metals can reduce material strength and is generally undesirable. Although studies have shown relationships between process parameters and porosity, monitoring strategies for defect detection and pore formation are still needed. In this paper, instantaneous anomalous conditions are detected in-situ via pyrometry during laser powder bed fusion additive manufacturing and correlated with voids observed using post-build micro-computed tomography. Large two-color pyrometry data sets were used to estimate instantaneous temperatures, melt pool orientations and aspect ratios. Machine learning algorithms were then applied to processed pyrometry data to detect outlier images and conditions. It is shown that melt pool outliers are good predictors of voids observed post-build. With this approach, real time process monitoring can be incorporated into systems to detect defect and void formation. Alternatively, using the methodology presented here, pyrometry data can be post processed for porosity assessment.
Concern about memory errors has been widespread in high-performance computing (HPC) for decades. These concerns have led to significant research on detecting and correcting memory errors to improve performance and provide strong guarantees about the correctness of the memory contents of scientific simulations. However, power concerns and changes in memory architectures threaten the viability of current approaches to protecting memory (e.g., Chipkill). Returning to less protective error-correcting codes (ECC), e.g., single-error correction, double-error detection (SECDED), may increase the frequency of memory errors, including silent data corruption (SDC). SDC has the potential to silently cause applications to produce incorrect results and mislead domain scientists. We propose an approach for exploiting unnecessary bits in pointer values to support encoding the pointer with a Reed-Solomon code. Encoding the pointer allows us to provides strong capabilities for correcting and detecting corruption of pointer values. In this paper, we provide a detailed description of how we can exploit unnecessary pointer bits to store Reed-Solomon parity symbols. We evaluate the performance impacts of this approach and examine the effectiveness of the approach against corruption. Our results demonstrate that encoding and decoding is fast (less than 45 per event) and that the protection it provides is robust (the rate of miscorrection is less than 5% even for significant corruption). The data and analysis presented in this paper demonstrates the power of our approach. It is fast, tunable, requires no additional per-pointer storage resources, and provides robust protection against pointer corruption.
This article describes a parallel implementation of a two-level overlapping Schwarz preconditioner with the GDSW (Generalized Dryja–Smith–Widlund) coarse space described in previous work [12, 10, 15] into the Trilinos framework; cf. [16]. The software is a significant improvement of a previous implementation [12]; see Sec. 4 for results on the improved performance.
We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal, and potential energy budgets, ensuring an energy-consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit-explicit time stepping algorithm.
Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
Wind energy is stochastic in nature; the prediction of aerodynamic quantities and loads relevant to wind energy applications involves modeling the interaction of a range of physics over many scales for many different cases. These predictions require a range of model fidelity, as predictive models that include the interaction of atmospheric and wind turbine wake physics can take weeks to solve on institutional high performance computing systems. In order to quantify the uncertainty in predictions of wind energy quantities with multiple models, researchers at Sandia National Laboratories have applied Multilevel-Multifidelity methods. A demonstration study was completed using simulations of a NREL 5MW rotor in an atmospheric boundary layer with wake interaction. The flow was simulated with two models of disparate fidelity; an actuator line wind plant large-eddy scale model, Nalu, using several mesh resolutions in combination with a lower fidelity model, OpenFAST. Uncertainties in the flow conditions and actuator forces were propagated through the model using Monte Carlo sampling to estimate the velocity defect in the wake and forces on the rotor. Coarse-mesh simulations were leveraged along with the lower-fidelity flow model to reduce the variance of the estimator, and the resulting Multilevel-Multifidelity strategy demonstrated a substantial improvement in estimator efficiency compared to the standard Monte Carlo method.
We propose a very large family of benchmarks for probing the performance of quantum computers. We call them volumetric benchmarks (VBs) because they generalize IBM's benchmark for measuring quantum volume [1]. The quantum volume benchmark defines a family of square circuits whose depth d and width w are the same. A volumetric benchmark defines a family of rectangular quantum circuits, for which d and w are uncoupled to allow the study of time/space performance trade-offs. Each VB defines a mapping from circuit shapes - (w, d) pairs - to test suites C(w, d). A test suite is an ensemble of test circuits that share a common structure. The test suite C for a given circuit shape may be a single circuit C, a specific list of circuits {C1... CN} that must all be run, or a large set of possible circuits equipped with a distribution Pr(C). The circuits in a given VB share a structure, which is limited only by designers' creativity. We list some known benchmarks, and other circuit families, that fit into the VB framework: several families of random circuits, periodic circuits, and algorithm-inspired circuits. The last ingredient defining a benchmark is a success criterion that defines when a processor is judged to have “passed” a given test circuit. We discuss several options. Benchmark data can be analyzed in many ways to extract many properties, but we propose a simple, universal graphical summary of results that illustrates the Pareto frontier of the d vs w trade-off for the processor being benchmarked.
The data from the multi-modal transportation test conducted in 2017 demonstrated that the inputs from the shock events during all transport modes (truck, rail, and ship) were amplified from the cask to the spent commercial nuclear fuel surrogate assemblies. These data do not support common assumption that the cask content experiences the same accelerations as the cask itself. This was one of the motivations for conducting 30 cm drop tests. The goal of the 30 cm drop test is to measure accelerations and strains on the surrogate spent nuclear fuel assembly and to determine whether the fuel rods can maintain their integrity inside a transportation cask when dropped from a height of 30 cm. The 30 cm drop is the remaining NRC normal conditions of transportation regulatory requirement (10 CFR 71.71) for which there are no data on the actual surrogate fuel. Because the full-scale cask and impact limiters were not available (and their cost was prohibitive), it was proposed to achieve this goal by conducting three separate tests. This report describes the first two tests — the 30 cm drop test of the 1/3 scale cask (conducted in December 2018) and the 30 cm drop of the full-scale dummy assembly (conducted in June 2019). The dummy assembly represents the mass of a real spent nuclear fuel assembly. The third test (to be conducted in the spring of 2020) will be the 30 cm drop of the full-scale surrogate assembly. The surrogate assembly represents a real full-scale assembly in physical, material, and mechanical characteristics, as well as in mass.
Trusting simulation output is crucial for Sandia’s mission objectives. Here, we rely on these simulations to perform our high-consequence mission tasks given national treaty obligations. Other science and modeling applications, while they may have high-consequence results, still require the strongest levels of trust to enable using the result as the foundation for both practical applications and future research. To this end, the computing community has developed workflow and provenance systems to aid in both automating simulation and modeling execution as well as determining exactly how was some output was created so that conclusions can be drawn from the data. Current approaches for workflows and provenance systems are all at the user level and have little to no system level support making them fragile, difficult to use, and incomplete solutions. The introduction of container technology is a first step towards encapsulating and tracking artifacts used in creating data and resulting insights, but their current implementation is focused solely on making it easy to deploy an application in an isolated “sandbox” and maintaining a strictly read-only mode to avoid any potential changes to the application. All storage activities are still using the system-level shared storage. This project explores extending the container concept to include storage as a new container type we call data pallets. Data Pallets are potentially writeable, auto generated by the system based on IO activities, and usable as a way to link the contained data back to the application and input deck used to create it.
Knowledge graph embedding (KGE) learns latent vector representations of named entities (i.e., vertices) and relations (i.e., edge labels) of knowledge graphs. Herein, we address two problems in KGE. First, relations may belong to one or multiple categories, such as functional, symmetric, transitive, reflexive, and so forth; thus, relation categories are not exclusive. Some relation categories cause non-trivial challenges for KGE. Second, we found that zero gradients happen frequently in many translation based embedding methods such as TransE and its variations. To solve these problems, we propose i) converting a knowledge graph into a bipartite graph, although we do not physically convert the graph but rather use an equivalent trick; ii) using multiple vector representations for a relation; and iii) using a new hinge loss based on energy ratio(rather than energy gap) that does not cause zero gradients. We show that our method significantly improves the quality of embedding.