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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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At least 1,495 records · Page 83
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices

The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or non-trivial high-dimensional classical optimization. In this work, we introduce a quantum solver of contracted eigenvalue equations, the quantum analogue of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

The Schwarz Alternating Method for the Seamless Coupling of Nonlinear Reduced Order Models and Full Order Models

Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of a set of partial differential equations (PDEs) using conventional techniques (e.g., finite element, finite volume, finite difference methods) where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the “plug-and-play” integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method [41]. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multiscale coupling of finite element FOMs in solid mechanics [35, 36], we present a new extension of the Schwarz framework that enables FOM-ROM and ROM-ROM coupling, following a domain decomposition of the physical geometry on which a PDE is posed. In order to maintain efficiency and achieve computation speed-ups, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting (ECSW) approach [13]. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Machine Learning for Real-time Fusion Plasma Behavior Prediction and Manipulation

This project set out with an ambitious goal: to develop and apply machine learning-based methods of discovering new controllers and operating regimes for achieving better performing plasmas in tokamaks. As summarized in the dozens of papers below, the project was a huge success. It developed several machine learning components and integrated them into a single ML system that was added to the DIII-D PCS. That functionality was demonstrated over a series of experiments on DIII-D. A summary of these results is given here with the details in the published papers below.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Security Analysis of a Class of Secured Spread Spectrum Systems

Abstract—A method of adding physical layer security to a class of spread spectrum systems has been recently proposed. In this paper, we look into the rate at which an eavesdropper may gain information about the system to decipher the data symbols. The Shannon mutual information is used to measure the rate of information that may be gained by an eavesdropper. The k-nearest neighbors (k-NN) method is used to obtain the estimates of relevant entropy values which will be then used to quantify the rate of information recovery as more data are being transmitted. It turns out that such information recovery requires adoption of special methods that avoid any destructive bias in the estimates. Details of these methods are also presented.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Quantum simulation of second-quantized Hamiltonians in compact encoding

We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. Here, we show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the φ 4 -theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2006

Jet Noise Diagnostics Supporting Statistical Noise Prediction Methods

The primary focus of my presentation is the development of the jet noise prediction code JeNo with most examples coming from the experimental work that drove the theoretical development and validation. JeNo is a statistical jet noise prediction code, based upon the Lilley acoustic analogy. Our approach uses time-average 2-D or 3-D mean and turbulent statistics of the flow as input. The output is source distributions and spectral directivity. NASA has been investing in development of statistical jet noise prediction tools because these seem to fit the middle ground that allows enough flexibility and fidelity for jet noise source diagnostics while having reasonable computational requirements. These tools rely on Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solutions as input for computing far-field spectral directivity using an acoustic analogy. There are many ways acoustic analogies can be created, each with a series of assumptions and models, many often taken unknowingly. And the resulting prediction can be easily reverse-engineered by altering the models contained within. However, only an approach which is mathematically sound, with assumptions validated and modeled quantities checked against direct measurement will give consistently correct answers. Many quantities are modeled in acoustic analogies precisely because they have been impossible to measure or calculate, making this requirement a difficult task. The NASA team has spent considerable effort identifying all the assumptions and models used to take the Navier-Stokes equations to the point of a statistical calculation via an acoustic analogy very similar to that proposed by Lilley. Assumptions have been identified and experiments have been developed to test these assumptions. In some cases this has resulted in assumptions being changed. Beginning with the CFD used as input to the acoustic analogy, models for turbulence closure used in RANS CFD codes have been explored and compared against measurements of mean and rms velocity statistics over a range of jet speeds and temperatures. Models for flow parameters used in the acoustic analogy, most notably the space-time correlations of velocity, have been compared against direct measurements, and modified to better fit the observed data. These measurements have been extremely challenging for hot, high speed jets, and represent a sizeable investment in instrumentation development. As an intermediate check that the analysis is predicting the physics intended, phased arrays have been employed to measure source distributions for a wide range of jet cases. And finally, careful far-field spectral directivity measurements have been taken for final validation of the prediction code. Examples of each of these experimental efforts will be presented. The main result of these efforts is a noise prediction code, named JeNo, which is in middevelopment. JeNo is able to consistently predict spectral directivity, including aft angle directivity, for subsonic cold jets of most geometries. Current development on JeNo is focused on extending its capability to hot jets, requiring inclusion of a previously neglected second source associated with thermal fluctuations. A secondary result of the intensive experimentation is the archiving of various flow statistics applicable to other acoustic analogies and to development of time-resolved prediction methods. These will be of lasting value as we look ahead at future challenges to the aeroacoustic experimentalist.

Bridges, James E.↗
DOE OSTI2025

Nonlinear simulation of under-resolved flows with shocks

Here, we consider the numerical simulation of advection-dominated flows whose wide range of physical length scales exceed the memory capacity of finite computers. Simulating flows with shocks and turbulence presented challenges for the earliest computers that were quickly overcome by the development of new numerical methodology. Principal among those new ideas were artificial viscosity and finite volume methods, concepts that remain in common use today. We begin by describing the history of those methods, the innovators and their motivations. We then describe the development of finite scale theory, a reformulation of Navier–Stokes theory that exposes the physical principles on which artificial viscosity is based. We discuss the essential properties of the finite scale equations, the observer, unresolved kinetic energy and inviscid energy dissipation. We briefly consider the implementation of the finite scale equations on the computer from the point of view of Gisin’s conjectures about finite information.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Predicting Energetics Materials’ Crystalline Density from Chemical Structure by Machine Learning

To expedite new molecular compound development, a long-sought goal within the chemistry community has been to predict molecules’ bulk properties of interest a priori to synthesis from a chemical structure alone. In this work, we demonstrate that machine learning methods can indeed be used to directly learn the relationship between chemical structures and bulk crystalline properties of molecules, even in the absence of any crystal structure information or quantum mechanical calculations. We focus specifically on a class of organic compounds categorized as energetic materials called high explosives (HE) and predicting their crystalline density. An ongoing challenge within the chemistry machine learning community is deciding how best to featurize molecules as inputs into machine learning models—whether expert handcrafted features or learned molecular representations via graph-based neural network models—yield better results and why. We evaluate both types of representations in combination with a number of machine learning models to predict the crystalline densities of HE-like molecules curated from the Cambridge Structural Database, and we report the performance and pros and cons of our methods. Our message passing neural network (MPNN) based models with learned molecular representations generally perform best, outperforming current state-of-the-art methods at predicting crystalline density and performing well even when testing on a data set not representative of the training data. However, these models are traditionally considered black boxes and less easily interpretable. Here, to address this common challenge, we also provide a comparison analysis between our MPNN-based model and models with fixed feature representations that provides insights as to what features are learned by the MPNN to accurately predict density.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

TROPHY: A Topologically Robust Physics-Informed Tracking Framework for Tropical Cyclones

Tropical cyclones (TCs) are among the most destructive weather systems. Realistically and efficiently detecting and tracking TCs are critical for assessing their impacts and risks. In particular, the eye is a signature feature of a mature TC. Therefore, knowing the eyes’ locations and movements is crucial for both operational weather forecasts and climate risk assessments. Recently, a multilevel robustness framework has been introduced to study the critical points of time-varying vector fields. The framework quantifies the robustness (i.e., structural stability) of critical points across varying neighborhoods. By relating the multilevel robustness with critical point tracking, the framework has demonstrated its potential in cyclone tracking. An advantage is that it identifies cyclonic features using only 2D wind vector fields, which is encouraging as most tracking algorithms require multiple dynamic and thermodynamic variables at different altitudes. A disadvantage is that the framework does not scale well computationally for datasets containing a large number of cyclones. Herein this paper introduces a topologically robust physics-informed tracking framework (TROPHY) for TC tracking. The main idea is to integrate physical knowledge of TC to drastically improve the computational efficiency of multilevel robustness framework for large-scale climate datasets. First, during preprocessing, we propose a physics-informed feature selection strategy to filter 90% of critical points that are short-lived and have low stability, thus preserving good candidates for TC tracking. Second, during in-processing, we impose constraints during the multilevel robustness computation to focus only on physics-informed neighborhoods of TCs. We apply TROPHY to 30 years of 2D wind fields from reanalysis data in ERA5 and generate a number of TC tracks. In comparison with the observed tracks, we demonstrate that TROPHY can capture TC characteristics (e.g., frequency, intensity, duration, latitudes with maximum intensity, and genesis) that are comparable to and sometimes even better than a well-validated TC tracking algorithm that requires multiple dynamic and thermodynamic scalar fields.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Uncertainty-aware photovoltaic generation estimation through fusion of physics with harmonics information using Bayesian neural networks

We develop an aggregate photovoltaic generation estimation methodology that uses diverse inputs and can reason on its current input-dependent predictive uncertainty. Named PVPHEst, for PhotoVoltaic Physics- & Harmonics-driven Estimator, the resulting tool is intelligently weighing and fusing information carried by the output of physics models, harmonics, and line sensors, using Bayesian neural networks and related techniques aimed at solving machine learning problems with intrinsic uncertainty quantification. As each of the three input classes carries heterogeneous information that only sheds light on one facet of the estimation problem but its value can diminish in the face of diverse grid phenomena, PV-PHEst with its estimation and uncertainty reasoning capabilities perform a nontrivial and potentially mission-critical task of value to grid operators.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Achieving Realistic Runoff in the Western United States with a Land Surface Model Forced by Dynamically Downscaled Meteorology

Abstract In this study, we calibrate a regional climate model’s (RCM) underlying land surface model (LSM). In addition to providing a realistic representation of runoff across the hydroclimatically diverse western United States, this is done to take advantage of the RCM’s ability to physically resolve meteorological forcing data in ungauged regions, and to prepare the calibrated hydrologic model for tight coupling, or the ability to represent land surface–atmosphere interactions, with the RCM. Specifically, we use a 9-km resolution meteorological forcing dataset across the western United States, from the fifth generation ECMWF Reanalysis (ERA5) downscaled by the Weather Research Forecasting (WRF) regional climate model, as an offline forcing for Noah-Multiparameterization (Noah-MP). We detail the steps involved in producing an LSM capable of accurately representing runoff, including physical parameterization selection, parameter calibration, and regionalization to ungauged basins. Based on our model evaluation from 1954 to 2021 for 586 basins with daily natural streamflow, the streamflow bias is reduced from 24.2% to 4.4%, and the median daily Nash–Sutcliffe efficiency (NSE) is improved from 0.12 to 0.36. When validating against basins with monthly natural streamflow data, we obtain a similar reduction in bias and a median monthly NSE improvement from 0.18 to 0.56. In this study, we also discover the optimal setup when using a donor-basin method to regionalize parameters to ungauged basins, which can vary by 0.06 NSE for unique designs of this regionalization method. Significance Statement This study provides useful guidance for improving a land surface model to accurately represent runoff across a spatially extensive and hydroclimatically diverse region (the western United States). The land surface model is updated to represent runoff more accurately at gauged basins, and then additionally updated for basins without observational data using a mathematical approach called the donor-basin method. We make use of a regional climate model’s reanalysis-derived meteorological data and its underlying land surface model to achieve realistic runoff. The calibrated land surface model can thus be tightly coupled in subsequent studies in a manner that should more accurately reflect runoff conditions. Findings from this study will serve as a useful reference for the atmospheric (and hydrologic) modeling communities and their ability to represent large-scale hydrology accurately.

Meteorology & Atmospheric Sciences↗
DOE OSTI2024

Evidential Deep Learning for Probabilistic Modelling of Extreme Storm Events

Uncertainty quantification (UQ) methods play an important role in reducing errors in weather forecasting. Conventional approaches in UQ for weather forecasting rely on generating an ensemble of forecasts from physics-based simulations to estimate the uncertainty. However, it is computationally expensive to generate many forecasts to predict real-time extreme weather events. Evidential Deep Learning (EDL) is an uncertainty-aware deep learning approach designed to provide confidence about its predictions using only one forecast. It treats learning as an evidence acquisition process where more evidence is interpreted as increased predictive confidence. We apply EDL to storm forecasting using real-world weather datasets and compare its performance with traditional methods. Our findings indicate that EDL not only reduces computational overhead but also enhances predictive uncertainty. This method opens up novel opportunities in research areas such as climate risk assessment, where quantifying the uncertainty about future climate is crucial.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Leveraging prior mean models for faster Bayesian optimization of particle accelerators

Tuning particle accelerators is a challenging and time-consuming task that can be automated and carried out efficiently using suitable optimization algorithms, such as model-based Bayesian optimization techniques. One of the major advantages of Bayesian algorithms is the ability to incorporate prior information about beam physics and historical behavior into the model used to make control decisions. In this work, we examine incorporating prior accelerator physics information into Bayesian optimization algorithms by utilizing fast executing, neural network models trained on simulated or historical datasets as prior mean functions in Gaussian process models. We show that in ideal cases, this technique substantially increases convergence speed to optimal solutions in high-dimensional tuning parameter spaces. Additionally, we demonstrate that even in non-ideal cases, where prior models of beam dynamics do not exactly match experimental conditions, the use of this technique can still enhance convergence speed. Finally, we demonstrate how these methods can be used to improve optimization in practical applications, such as transferring information gained from beam dynamics simulations to online control of the LCLS injector, and transferring knowledge gained from experimental measurements across different operating modes, such as accelerating different ion species at the ATLAS heavy ion accelerator.

43 PARTICLE ACCELERATORS↗
DOE OSTI2024

Comparative Assessment of U-Net-Based Deep Learning Models for Segmenting Microfractures and Pore Spaces in Digital Rocks

Segmentation of high-resolution X-ray microcomputed tomography (µCT) images is crucial in digital rock physics (DRP), affecting the characterization and analysis of microscale phenomena in the porous media. The complexity of geological structures and nonideal scanning conditions pose significant challenges to conventional image segmentation approaches. Motivated by the recent increasing popularity of deep learning (DL) techniques in image processing, this work undertakes a comparative study of DL models, specifically U-Net and its variants, for segmenting multiple targets with distinguished features in digital rocks, including discrete fracture networks (DFNs), pore spaces, and solid rock. Particularly, DFNs have a smaller volumetric fraction over others, bringing in a substantial challenge of imbalanced segmentation. The primary focus is to evaluate the architecture and feature enhancement strategies of various DL models, including U-Net, attention U-Net, residual U-Net, U-Net++, and residual U-Net++. The models were designed as 2.5D, utilizing a central 2D image and its two adjacent upper and lower 2D images as input to provide a pseudo-3D context. In addition, because the ground truth of segmentation was unknown for real-world digital rocks, we created a benchmark data set following the inverse operations of segmentation. The data synthesis started from the label images (i.e., solid rock, pore spaces, and DFNs), followed by simulating partial volume blurring, adding random background noise, and introducing ring artifacts to mimic real raw X-ray µCT images. The data set, which included various rock types (i.e., sandstone and artificial data), scanning resolution, and magnitudes of noise and artifacts, was divided into training and testing data sets with a 90% and 10% ratio, respectively. Moreover, in addition to the conventional pixel-wise evaluation metrics, the physics-based metric of the lattice-Boltzmann method (LBM) simulated permeability provided more comprehensive assessments. The results demonstrated that the residual connections, nested architectures, and redesigned skip connections contribute to the model performance and give the residual U-Net++ the highest accuracy. The improvements were mainly on the boundaries and small targets, especially the DFNs, which dominate the interconnectivity and therefore affect the permeability greatly. This study also rigorously evaluated the efficiency and generalization of each model, demonstrating that the sophisticated architectures achieved excellent practicability and maintained robust performance on completely unseen data, ensuring their suitability for diverse and challenging DRP applications.

58 GEOSCIENCES↗
NASA NTRS2009

Compatibility of Segments of Thermoelectric Generators

A method of calculating (usually for the purpose of maximizing) the power-conversion efficiency of a segmented thermoelectric generator is based on equations derived from the fundamental equations of thermoelectricity. Because it is directly traceable to first principles, the method provides physical explanations in addition to predictions of phenomena involved in segmentation. In comparison with the finite-element method used heretofore to predict (without being able to explain) the behavior of a segmented thermoelectric generator, this method is much simpler to implement in practice: in particular, the efficiency of a segmented thermoelectric generator can be estimated by evaluating equations using only hand-held calculator with this method. In addition, the method provides for determination of cascading ratios. The concept of cascading is illustrated in the figure and the definition of the cascading ratio is defined in the figure caption. An important aspect of the method is its approach to the issue of compatibility among segments, in combination with introduction of the concept of compatibility within a segment. Prior approaches involved the use of only averaged material properties. Two materials in direct contact could be examined for compatibility with each other, but there was no general framework for analysis of compatibility. The present method establishes such a framework. The mathematical derivation of the method begins with the definition of reduced efficiency of a thermoelectric generator as the ratio between (1) its thermal-to-electric power-conversion efficiency and (2) its Carnot efficiency (the maximum efficiency theoretically attainable, given its hot- and cold-side temperatures). The derivation involves calculation of the reduced efficiency of a model thermoelectric generator for which the hot-side temperature is only infinitesimally greater than the cold-side temperature. The derivation includes consideration of the ratio (u) between the electric current and heat-conduction power and leads to the concept of compatibility factor (s) for a given thermoelectric material, defined as the value of u that maximizes the reduced efficiency of the aforementioned model thermoelectric generator.

Snyder, G. Jeffrey↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Maximizing efficiency of dataset compression for machine learning potentials with information theory

Machine learning interatomic potentials (MLIPs) balance high accuracy and lower costs compared to density functional theory calculations, but their performance often depends on the size and diversity of training datasets. Large datasets improve model accuracy and generalization but are computationally expensive to produce and train on, while smaller datasets risk discarding rare but important atomic environments and compromising MLIP accuracy/reliability. Here, we develop an information-theoretical framework to quantify the efficiency of dataset compression methods and propose an algorithm that maximizes this efficiency. By framing atomistic dataset compression as an instance of the minimum set cover (MSC) problem over atom-centered environments, our method identifies the smallest subset of structures that contains as much information as possible from the original dataset while pruning redundant information. The approach is extensively demonstrated on the GAP-20 and TM23 datasets and validated on 64 varied datasets from the ColabFit repository. Across all cases, MSC consistently retains outliers, preserves dataset diversity, and reproduces the long-tail distributions of forces even at high compression rates, outperforming other subsampling methods. Furthermore, MLIPs trained on MSC-compressed datasets exhibit reduced error for out-of-distribution data even in low-data regimes. We explain these results using an outlier analysis and show that such quantitative conclusions could not be achieved with conventional dimensionality reduction methods. The algorithm is implemented in the open-source QUESTS package and can be used for several tasks in atomistic modeling, from data subsampling, outlier detection, and training improved MLIPs at a lower cost.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Adaptive Mesh Refinement for Parallel in Time Methods

The project applied the multigrid-reduction-in-time (MGRIT) algorithm to an existing sub-cycled adaptive mesh refinement (AMR) code to investigate the performance of flows dominated by inertial physics. Previous work demonstrated good performance from MGRIT+AMR applied to flows dominated by diffusive physics. Consistent with previous experience, inertial physics negatively affected convergence rates and performance. Efforts to circumvent this issue by appealing to the physics of turbulence were investigated. It has been demonstrated that scales can be effectively transferred between multigrid levels for a turbulent flow resulting in a) partial convergence observed and b) nearly identical results to sequential time-stepping. Performance improvements have not yet been demonstrated - attempts at coarsening the grid on coarser MG levels compromises the solution quality and leads to divergence. This report summarize the accomplishments for the time-frame from 10/5/2020 to 12/31/2020 with an informal no-cost-extension to 05/20/2021

97 MATHEMATICS AND COMPUTING↗