RPI unstructured meshing developments for the FASTMath SciDAC 5 applied math institute
Final report for the Rensselaer Polytechnic Institute (RPI) contributions to the FASTMath SciDAC 5 applied math institute
97 MATHEMATICS AND COMPUTING↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Highlights for DOE ASCR Applied Math Office [Slides]
An efficient Picard-based solver is proposed for a novel energy conserving particle integrator preserving all first-order guiding center drifts and correct gyroradius for large time steps in arbitrary (non-uniform) magnetic fields. This research enables the efficient deployment of the novel asymptotic preserving (AP) particle orbit integrator into modern energy-conserving, implicit particle-in-cell codes, delivering a truly multiscale simulation capability.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
AEOLUS: Advances in Experimental Design, Optimal Control, and Learning for Uncertain Complex Systems
Sustained advances in the mathematics of modeling and simulation have resulted in the capability today for routine simulation of a number of large scale complex DOE-relevant systems. As remarkable as this capability for solving the so-called forward problem is, it is typically only the first step-an inner loop within an outer loop that explores the simulation model's parameter space and decision space to characterize uncertainty in the model's predictions, learn unknown model parameters from data, design the most informative experiments, determine optimal control strategies, and create optimal designs. Broadly, what unifies all of these outer loop problems is that they are, in one form or another, optimization problems over parameter/control/design space that are constrained by complex uncertain models. To fully realize the power of scientific simulation as a basis for scientific discovery, technological innovation, and rational decision-making, it is imperative to move beyond simulation to tackle the outer loop of optimization for learning from data, experimental design, and control with complex uncertain models. When the models under consideration are large-scale and complex, and when the optimization variable and uncertain parameter spaces are high (or infinite) dimensional, this constitutes a grand challenge of the highest order, and is intractable with conventional methods. To overcome these challenges, the AEOLUS Center was established to develop a unified mathematical, computational, and statistical framework for (1) Learning predictive models from complex data via Bayesian inference and optimization, and (2) Optimizing experiments, processes, and designs using the resulting uncertain models. These problems are intractable with conventional methods, for several reasons: (1) The simulation problems that govern the inner loops of the optimization problems are expensive to execute (due to severe nonlinearity, heterogeneity, multiphysics/multiscale coupling); (2) The optimization variable and uncertain parameter spaces are high dimensional, often stemming from discretizations of infinite dimensional fields such as initial conditions, sources, or material properties. We argue that the key to overcoming these challenges is to develop new mathematical, computational, and statistical methods that exploit the structure of the Bayesian inference and optimization problems mediated by their underlying complex uncertain models. This structure includes the regularity, sparsity, geometry, low intrinsic dimensionality, and multifidelity nature of the maps from uncertain parameter/optimization variable spaces to the specific objectives targeted: Bayesian inference, optimal experimental design, and optimal control design. Black box methods developed as generic tools are incapable of exploiting this structure. To be successful, we must create, integrate, and cross-fertilize ideas across multiple areas of applied math--including approximation theory, Bayesian inference, data science, experimental design, information theory, machine learning, model reduction, optimal control theory, parallel algorithms, PDE-constrained optimization, randomized algorithms, stochastic optimization, and uncertainty quantification--all while exploiting the structure of the problems at hand. With this goal in mind, we have marshaled a team of leading authorities in these areas. While the methods we develop will be broadly applicable across a wide spectrum of DOE problems in which experiments inform models and the systems those models describe must be optimized under uncertainty, we have chosen a specific area, advanced manufacturing and materials, to drive our work. AMM is characterized by complex models across multiple scales, and is a rich source of challenging problems in inference, experimental design, and optimal control, requiring multifaceted and integrated advances in applied mathematics. As such, AMM serves as an excellent vehicle to motivate and demonstrate the advances in applied mathematics developed by our center.
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Regional-scale fault-to-structure earthquake simulations with the EQSIM framework: Workflow maturation and computational performance on GPU-accelerated exascale platforms
Continuous advancements in scientific and engineering understanding of earthquake phenomena, combined with the associated development of representative physics-based models, is providing a foundation for high-performance, fault-to-structure earthquake simulations. However, regional-scale applications of high-performance models have been challenged by the computational requirements at the resolutions required for engineering risk assessments. The EarthQuake SIMulation (EQSIM) framework, a software application development under the US Department of Energy (DOE) Exascale Computing Project, is focused on overcoming the existing computational barriers and enabling routine regional-scale simulations at resolutions relevant to a breadth of engineered systems. This multidisciplinary software development—drawing upon expertise in geophysics, engineering, applied math and computer science—is preparing the advanced computational workflow necessary to fully exploit the DOE’s exaflop computer platforms coming online in the 2023 to 2024 timeframe. Achievement of the computational performance required for high-resolution regional models containing upward of hundreds of billions to trillions of model grid points requires numerical efficiency in every phase of a regional simulation. This includes run time start-up and regional model generation, effective distribution of the computational workload across thousands of computer nodes, efficient coupling of regional geophysics and local engineering models, and application-tailored highly efficient transfer, storage, and interrogation of very large volumes of simulation data. This article summarizes the most recent advancements and refinements incorporated in the workflow design for the EQSIM integrated fault-to-structure framework, which are based on extensive numerical testing across multiple graphics processing unit (GPU)-accelerated platforms, and demonstrates the computational performance achieved on the world’s first exaflop computer platform through representative regional-scale earthquake simulations for the San Francisco Bay Area in California, USA.
58 GEOSCIENCES↗
A posteriori superlinear convergence bounds for block conjugate gradient
In this paper, we extend to the block case the a posteriori bound showing superlinear convergence of the conjugate gradient method developed by van der Vorst and Vuik in [J. Comput. Applied Math., 48 (1993), pp. 327–341]. That is, we obtain similar bounds but now for the block conjugate gradient method. We also present a series of computational experiments, illustrating the validity of the bound developed here as well as the bound by Simoncini and Szyld from [SIAM Review, 47 (2005), pp. 247–272] using angles between subspaces. Using these bounds, we make some observations on the onset of superlinearity and how this onset depends on the eigenvalue distribution and the block size.
97 MATHEMATICS AND COMPUTING↗
Youth Technology Academy
One of the most important aspects of the YTA Program is that students, working with Tri-C instructors, NASA engineers, and technicians, begin to build relationships with professionals with whom they might not normally have contact. These professionals serve as role models for students and help them to apply math, science and problem-solving skills in real time to construct a competition-ready robot, allowing them to experience the satisfaction and challenges of an engineering or technical career by working diligently to solve problems that may never have been thought possible to solve. Transcending school boundaries, YTA is available to all students who are interested and qualified but who may otherwise never get an opportunity to participate because their school does not offer a robotics program. YTA fills an educational void by offering an exciting engineering and technology experience to a greater number of students than have been able to participate in the past. Our students have been working hard and see the long-term rewards for their efforts. The YTA Team 1270 was a finalist at the 2005 FIRST Robotics Florida Regional and took home second- place honors. Mark Poljak, nominated by the students of YTA Team 1270, received the Woodie Flowers Award at the FIRST 2005 Buckeye Regional competition. This award celebrates effective communication in the art and science of engineering and design. The YTA Team was also invited, as one of fifty-four teams, to participate in this year s FIRST VEX Robotics tournament representing Cuyahoga Community College and NASA GRC (Team 26), held at the FIRST Robotics National Competition in Atlanta, GA. Because of our efforts, community colleges from around the country have taken notice and have asked us to become part of the executive board of the RoboEducators, whose mission is to bring technology training to high school students in preparation for the future workforce.
Source record↗
Flight test design for CH-47 parameter identification
The VTOL Approach and Landing Technology (VALT) program is a significant experimental research program aimed at establishing a data base for rotorcraft operation in a terminal area environment. Work was undertaken to determine helicopter math models suitable for analyzing maneuvers along a VTOL trajectory and to apply these math models to determine the flight test procedures of greatest effectiveness in establishing helicopter dynamic characteristics in this mode of operation. As the principal result of this investigation, a flight test specification is presented for the CH-47 VALT aircraft operating along the specified VTOL trajectory of the VALT program.
Hall, W. E., Jr.↗
Solving Coupled Cluster Equations by the Newton Krylov Method
We describe using the Newton Krylov method to solve the coupled cluster equation. The method uses a Krylov iterative method to compute the Newton correction to the approximate coupled cluster amplitude. The multiplication of the Jacobian with a vector, which is required in each step of a Krylov iterative method such as the Generalized Minimum Residual (GMRES) method, is carried out through a finite difference approximation, and requires an additional residual evaluation. The overall cost of the method is determined by the sum of the inner Krylov and outer Newton iterations. We discuss the termination criterion used for the inner iteration and show how to apply pre-conditioners to accelerate convergence. We will also examine the use of regularization technique to improve the stability of convergence and compare the method with the widely used direct inversion of iterative subspace (DIIS) methods through numerical examples.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A Comparison of Two Balance Calibration Model Building Methods
Simulated strain-gage balance calibration data is used to compare the accuracy of two balance calibration model building methods for different noise environments and calibration experiment designs. The first building method obtains a math model for the analysis of balance calibration data after applying a candidate math model search algorithm to the calibration data set. The second building method uses stepwise regression analysis in order to construct a model for the analysis. Four balance calibration data sets were simulated in order to compare the accuracy of the two math model building methods. The simulated data sets were prepared using the traditional One Factor At a Time (OFAT) technique and the Modern Design of Experiments (MDOE) approach. Random and systematic errors were introduced in the simulated calibration data sets in order to study their influence on the math model building methods. Residuals of the fitted calibration responses and other statistical metrics were compared in order to evaluate the calibration models developed with different combinations of noise environment, experiment design, and model building method. Overall, predicted math models and residuals of both math model building methods show very good agreement. Significant differences in model quality were attributable to noise environment, experiment design, and their interaction. Generally, the addition of systematic error significantly degraded the quality of calibration models developed from OFAT data by either method, but MDOE experiment designs were more robust with respect to the introduction of a systematic component of the unexplained variance.
DeLoach, Richard↗
FTTN: Feature-Targeted Testing for Numerical Properties of NVIDIA & AMD Matrix Accelerators
While NVIDIA has been the dominant provider of GPUs for HPC and ML, now AMD has several offerings of GPUs. This encourages programmers to try out AMD GPUs for new codes and also port existing codes over. Unfortunately, without understanding the floating-point differences between these GPU types, software development or porting can introduce bugs—and currently such an understanding is lacking. The magnitude of this open question becomes clear if one imagines the the number of floating-point precision choices (FP16, FP32, etc.), floating-point formats (standard floats, brain-float, etc.), and execution units available (elementary units, matrix/tensor cores, etc.) Questions such as rounding modes and subnormal support are also important. Most of these answers are unknown today or are hard to access. We provide the first testing-guided approach that answers a significant number of these questions. We also devise tests to reveal internal information (e.g., extra bits kept) to make sure that our findings are reliable. Many of our tests employ systematically generated random-programs, others apply fast-math flags and some involve fused multiplyadd. Especially for tensor/matrix cores, the tests have nontrivial logic that we present Our testing approach is reusable for the plethora of GPUs yet to be introduced. Our findings include up to 7 ulps of difference between NVIDIA and AMD for sin and cos at FP32 precision and 3 ulp at FP64. In our study of matrix cores (NVIDIA) and tensor cores (AMD), we have extensively characterized rounding modes (truncation versus round-to-nearest), the number of extra internal bits kept (whether 3 bits are kept or not), subnormal support for inputs and outputs across four different floating-point formats and across NVIDIA A100 and AMD MI250X GPUs. We believe that this wealth of data becoming available for the first time may help avoid significant porting bugs when migrating code across these platforms.
Li, Xinyi↗
Regression Model Optimization for the Analysis of Experimental Data
A candidate math model search algorithm was developed at Ames Research Center that determines a recommended math model for the multivariate regression analysis of experimental data. The search algorithm is applicable to classical regression analysis problems as well as wind tunnel strain gage balance calibration analysis applications. The algorithm compares the predictive capability of different regression models using the standard deviation of the PRESS residuals of the responses as a search metric. This search metric is minimized during the search. Singular value decomposition is used during the search to reject math models that lead to a singular solution of the regression analysis problem. Two threshold dependent constraints are also applied. The first constraint rejects math models with insignificant terms. The second constraint rejects math models with near-linear dependencies between terms. The math term hierarchy rule may also be applied as an optional constraint during or after the candidate math model search. The final term selection of the recommended math model depends on the regressor and response values of the data set, the user s function class combination choice, the user s constraint selections, and the result of the search metric minimization. A frequently used regression analysis example from the literature is used to illustrate the application of the search algorithm to experimental data.
Ulbrich, N.↗
A mathematical model of a tilt-wing aircraft for piloted simulation
A mathematical model of a tilt-wing aircraft that was used in a piloted, six-degree-of-freedom flight simulation application is described. Two types of control systems developed for the math model are discussed: a conventional, programmed-flap wing-tilt control system and a geared-flap wing-tilt control system. The primary objective was to develop the capability to study tilt-wing aircraft. Experienced Tilt-wing pilots subjectively evaluated the model using programmed-flap control to assess the quality of the simulation. The math model was then applied to study geared-flap control to investigate the possibility of eliminating the need for auxilary pitch-control devices (such as the horizontal tail rotor or tail jet used in earlier tilt-wing designs). This investigation was performed in the moving-base simulation environment, and the vehicle responses with programmed-flap and geared-flap control were compared. The results of the evaluation of the math model are discussed.
Totah, Joseph J.↗
Parachute dynamics and stability analysis
The nonlinear differential equations of motion for a general parachute-riser-payload system are developed. The resulting math model is then applied for analyzing the descent dynamics and stability characteristics of both the drogue stabilization phase and the main descent phase of the space shuttle solid rocket booster (SRB) recovery system. The formulation of the problem is characterized by a minimum number of simplifying assumptions and full application of state-of-the-art parachute technology. The parachute suspension lines and the parachute risers can be modeled as elastic elements, and the whole system may be subjected to specified wind and gust profiles in order to assess their effects on the stability of the recovery system.
Ibrahim, S. K.↗
Effects of simulator variations on the fidelity of a UH-60 Black Hawk simulation
The validity of the vertical motion simulator used to simulate the UH-60A Black Hawk helicopter is evaluated. Aircraft simulation of bob-up, hover turn, and dash/quick stop manuevers are compared to actual flight data. The effects of variations in the simulator's visual and motion characteristics on the simulation data are investigated. Data obtained with the nonintrusive parameter identification procedure and optimal control measures are applied to the helicopter math model used for real-time aircraft simulation. Analysis of the model reveals that the simulated aircraft is underdamped, and the changes in the visual reference have the greatest effect on the ability of the pilot to perform the maneuvers.
Cleveland, W. B.↗
Thermal scale modeling of a manned spacecraft
A manned spacecraft thermal scale modeling program is described. The program consisted of the design, construction, instrumentation, testing, data correlation, and associated analysis of a transient thermal scale model (TSM) of the Subsystem Test Bed (STB). The STB was chosen as a representative manned spacecraft for which detailed thermal vacuum test data were available. The STB is a cylindrical vehicle (15 ft diameter by 8.3 ft high) with four docking hatches and six windows. The pressure shell is insulated with multilayer insulation and protected by meteoroid shields. Four truss assemblies were used to support the STB during testing. It was concluded that thermal scale modeling can be used as an effective thermal design/verification tool for manned spacecraft. Thermal analysis can be used in conjunction with scale model testing to provide a verified math model that can be applied to the prototype manned spacecraft.
Shannon, R. L.↗
Analysis of Sting Balance Calibration Data Using Optimized Regression Models
Calibration data of a wind tunnel sting balance was processed using a candidate math model search algorithm that recommends an optimized regression model for the data analysis. During the calibration the normal force and the moment at the balance moment center were selected as independent calibration variables. The sting balance itself had two moment gages. Therefore, after analyzing the connection between calibration loads and gage outputs, it was decided to choose the difference and the sum of the gage outputs as the two responses that best describe the behavior of the balance. The math model search algorithm was applied to these two responses. An optimized regression model was obtained for each response. Classical strain gage balance load transformations and the equations of the deflection of a cantilever beam under load are used to show that the search algorithm s two optimized regression models are supported by a theoretical analysis of the relationship between the applied calibration loads and the measured gage outputs. The analysis of the sting balance calibration data set is a rare example of a situation when terms of a regression model of a balance can directly be derived from first principles of physics. In addition, it is interesting to note that the search algorithm recommended the correct regression model term combinations using only a set of statistical quality metrics that were applied to the experimental data during the algorithm s term selection process.
Ulbrich, N.↗
Thermodynamic energy balance equations for Space Shuttle Orbiter gas compartment during ascent and re-entry
Thermodynamic energy balance equations are derived and applied to midsection Orbiter-payload atmospheric thermal math models (TMMs) to predict Orbiter component, element, compartment, internal insolation and structure temperatures in support of NASA/JSC mission planning, postflight thermal analysis and payload thermal integration planning. The equations are extended and applied to the forward section, midsection, and aft section of the TMMs for five Orbiter mission phases: prelaunch on pad with purge, lift-off to ascent, re-entry to touchdown, post landing without purge, and post-landing with purge. Predicted results from the 390 node/DFI atmospheric TMM are in good agreement with STS-1 flight measurement data.
Ting, P. C.↗
On the Strong Convergence of Forward-Backward Splitting in Reconstructing Jointly Sparse Signals
We consider the problem of reconstructing an infinite set of sparse, finite-dimensional vectors, that share a common sparsity pattern, from incomplete measurements. This is in contrast to the work (Daubechies et al., Pure Appl. Math. 57(11), 1413–1457, 2004), where the single vector signal can be infinite-dimensional, and (Fornasier and Rauhut, SIAM J. Numer. Anal. 46(2), 577613, 2008), which extends the aforementioned work to the joint sparse recovery of finite number of infinite-dimensional vectors. In our case, to take account of the joint sparsity and promote the coupling of nonvanishing components, we employ a convex relaxation approach with mixed norm penalty ℓ 2,1 . This paper discusses the computation of the solutions of linear inverse problems with such relaxation by a forward-backward splitting algorithm. However, since the solution matrix possesses infinitely many columns, the arguments of Daubechies et al. (Pure Appl. Math. 57(11), 1413–1457, 2004) no longer apply. As such, we establish new strong convergence results for the algorithm, in particular when the set of jointly sparse vectors is infinite.
97 MATHEMATICS AND COMPUTING↗