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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 199 records · Page 11

Challenges in High-Assurance Runtime Verification

Safety-critical systems are growing more complex and becoming increasingly autonomous. Runtime Verification (RV) has the potential to provide protections when a system cannot be assured by conventional means, but only if the RV itself can be trusted. In this paper, we proffer a number of challenges to realizing high-assurance RV and illustrate how we have addressed them in our research. We argue that high-assurance RV provides a rich target for automated verification tools in hope of fostering closer collaboration among the communities.

Goodloe, Alwyn E.↗

Classic and Quantum Task-Based Intelligent Runtime for QIRs Running on Multiple QPUs

High-performance computing systems are rapidly evolving into heterogeneous platforms that fuse quantum accelerators with traditional classical processing units (CPUs) and graphical processing units (GPUs). This convergence calls for runtimes capable of managing both classical and quantum workloads in a unified manner. We introduce an intelligent, task-based runtime that marries the Intelligent RuntIme System (IRIS) asynchronous scheduler with a quantum programming stack through the Quantum Intermediate Representation Execution Engine (QIR-EE). Our design allows programs written in the quantum intermediate representation (QIR) to be dispatched concurrently to a variety of back-ends, including multiple quantum simulators and nascent quantum processors, enabling genuine hybrid execution on a single node. To illustrate its practicality, we partition a 4-qubit and 20-qubit circuit into three sub-circuits using quantum circuit cutting via the QCut library. Each sub-circuit is simulated independently by the QIR-EE driver within IRIS, after which a classical post-processing step merges the simulation results to recover the outcome of the original full-circuit computation. This case study demonstrates how finer task granularity can enable the parallel execution and lower the simulation burden per quantum task while preserving overall accuracy, highlighting the feasibility of our hybrid approach.

Miniskar, Narasinga Rao [ORNL] (ORCID:000000018259↗

CodeFlow: A Code Generation System for Flash-X Orchestration Runtime

We propose the CodeFlow toolchain for Flash-X that realizes the “recipe-to-source” code transformation for Flash-X simulations and that is necessary to achive performance portability. We design a high-level language to express operations of simulations in so-called recipes, which are given as input to the toolchain. The tools of the CodeFlow pipeline include code transformation with tree-based source code representation techniques and code orchestration and generation based on control flow graphs. The generated source code utilizes a new runtime, developed for Flash-X, that orchestrates dynamic and asynchronous data movement and task execution. The functionality of CodeFlow is demonstrated using a hydrodynamic problem with a strong shock.

97 MATHEMATICS AND COMPUTING↗

BEAM Technology Flight Demonstration

As technologies advance, their growing complexity makes them harder to maintain. Detection methods for isolating and identifying impending problems are needed to balance this complexity. Through comparison of signal pairs from onboard sensors, the Beacon-based Exception Analysis For Multimissions (BEAM) algorithm can identify and help classify deviations in system operation from a data-trained statistical model. The goal of this task is to mature BEAM and validate its performance on a flying test bed. A series of F-18 flight demonstrations with BEAM monitoring engine parameters in real time was used to demonstrate in-the-field readiness. Captured F-18 and simulated F-18 engine data were used in model creation and training. The algorithm was then ported to the embedded system with a data buffering, file writing, and data-time-stamp monitoring shell to reduce the impact of embedded system faults on BEAM'S ability to correctly identify engine faults. Embedded system testing identified hardware related restrictions and contributed to iterative improvements in the code's runtime performance. The system was flown with forced engine flameouts and other pilot induced faults to simulate operation out of the norm. Successful detection of these faults, confirmed through post-flight data analysis, helped BEAM achieve TRL6.

Wang, David↗

Monitoring Distributed Real-Time Systems: A Survey and Future Directions

Runtime monitors have been proposed as a means to increase the reliability of safety-critical systems. In particular, this report addresses runtime monitors for distributed hard real-time systems. This class of systems has had little attention from the monitoring community. The need for monitors is shown by discussing examples of avionic systems failure. We survey related work in the field of runtime monitoring. Several potential monitoring architectures for distributed real-time systems are presented along with a discussion of how they might be used to monitor properties of interest.

Goodloe, Alwyn E.↗

The Minos Computing Library: Efficient Parallel Programming for Extremely Heterogeneous Systems

Hardware specialization has become the silver bullet to achieve efficient high performance, from Systems-on-Chip systems, where hardware specialization can be ``extreme'', to large-scale HPC systems. As the complexity of the systems increases, so does the complexity of programming such architectures in a portable way. This work introduces the Minos Computing Library (MCL), as system software, programming model, and programming model runtime that facilitate programming extremely heterogeneous systems. MCL supports the execution of several multi-threaded applications within the same compute node, performs asynchronous execution of application tasks, efficiently balances computation across hardware resources, and provides performance portability. We show that code developed on a personal desktop automatically scales up to fully utilize powerful workstations with 8 GPUs and down to power-efficient embedded systems. MCL provides up to 17.5x speedup over OpenCL on NVIDIA DGX-1 systems and up to 1.88x speedup on single-GPU systems. In multi-application workloads, MCL dynamically resource allocation provides up to 2.43x performance improvement over manual, static allocation of computing resources.

Gioiosa, Roberto↗

Copilot 3

Ultra-critical systems require high-level assurance, which cannot always be guaranteed in compile time. The use of runtime verification (RV) enables monitoring these systems in runtime, to detect property violations early and limit their potential consequences. The introduction of monitors in ultra-critical systems poses a challenge, as failures and delays in the RV subsystem could affect other subsystems and threaten the mission as a whole. This paper presents Copilot 3, a runtime verification framework for real-time embedded systems. Copilot monitors are written in a compositional, stream-based language with support for a variety of Temporal Logics (TL), which results in robust, high-level specifications that are easier to understand than their traditional counterparts. The framework translates monitor specifications into C code with static memory requirements, which can be compiled to run on embedded hardware. This paper presents version 3 of the Copilot language, demonstrates its suitability with a number of examples, and discusses its use in larger applications. Additionally, it describes the framework?s architecture, its implementation as a Domain Specific Language (DSL) embedded in Haskell, and the progress of the project over the years.

Ivan Perez↗

LibPowerMon: A vendor-neutral system monitoring library

As power becomes an important resource to provision for modern HPC systems it becomes crucial to obtain deeper insights into application's power and thermal characteristics and to correlate application context with important system-level measurements. Existing profiling tools to monitor power and thermal measurements either operate at a granularity that is not fine enough to correlate with application-level events that describe application context or are not equipped to sample application-level events transparently. We introduce LibPowerMon, a lightweight, vendor-neutral user-level profiling framework to simultaneously sample user-specified application events and system-level metrics at up to 1 kHz sampling intervals. At the application level, LibPowerMon provides a source-level phase markup interface to capture application context. At runtime level, LibPowerMon records MPI and OpenMP events and can be extended to include other runtimes. At the system level, LibPowerMon samples power, thermal and relevant metrics, and provides an extensible interface to system's control capabilities. In this release, LibPowerMon supports monitoring and control functionalities on the Intel, IBM Power9 and Nvidia Volta architectures.

MARATHE, ANIRUDDHAP↗

HIPLZ: Enabling performance portability for exascale systems

While heterogeneous computing has emerged as a dominant trend in current and future High-Performance Computing (HPC) systems, it is also widely recognized that this shift has led to increased software complexity due to a proliferation of programming systems for different heterogeneous processors. One such example is the Heterogeneous-Compute Interface for Portability from AMD (HIP ), which is composed of a C Runtime API and C++ Kernel Language. Many HPC applications will likely use HIP on future exascale systems (e.g., Frontier and El Capitan), but HIP currently only targets AMD and NVIDIA processors. This limitation creates challenges for users who would also like to run their applications on exascale systems based on other architectures (e.g., Aurora, which is based on Intel hardware) that are currently not targeted by HIP . In this paper, we introduce the design and implementation of HIPLZ , a compiler and runtime system that uses the Intel Level Zero API to support HIP on Intel GPU architectures. We discuss the design of HIPLZ , derived from HIPCL (an implementation of HIP on top of OpenCL ), and portability issues that occur from using the Level Zero runtime as a backend. We evaluate our implementation by running several performance benchmarks and mini-apps written in HIP on Intel architectures using HIPLZ . Our results show that this approach provides competitive performance relative to Intel's OpenCL implementations on Intel Gen9 and UHD Graphics 770 GPUs, while providing good coverage of features needed by HPC applications. Overall, this approach is a promising demonstration of enabling performance portability for exascale systems.

97 MATHEMATICS AND COMPUTING↗

Lightweight runtime checking of C programs with RTC

The C Programming Language is known for being an efficient language that can be compiled on almost any architecture and operating system. However the absence of dynamic safety checks and a relatively weak type system allows programmer oversights that are hard to spot. In this paper, we present RTC, a runtime monitoring tool that instruments unsafe code and monitors the program execution. RTC is built on top of the ROSE compiler infrastructure. RTC finds memory bugs and arithmetic overflows and underflows, and run-time type violations. Most of the instrumentations are directly added to the source file and only require a minimal runtime system. As a result, the instrumented code remains portable. In tests against known error detection benchmarks, RTC found 98% of all memory related bugs and had zero false positives. Finally, in performance tests conducted with well known algorithms, such as binary search and MD5, we determined that our tool has an average run-time overhead rate of 9.7× and memory overhead rate of 3.5×.

97 MATHEMATICS AND COMPUTING↗

Optimizing checkpoint data placement with guaranteed burst buffer endurance in large-scale hierarchical storage systems

Non-volatile devices, such as SSDs, will be an integral part of the deepening storage hierarchy on large-scale HPC systems. These devices can be on the compute nodes as part of a distributed burst buffer service or they can be external. Wherever they are located in the hierarchy, one critical design issue is the SSD endurance under the write-heavy workloads, such as the checkpoint I/O for scientific applications. For these environments, it is widely assumed that checkpoint operations can occur once every 60 min and for each checkpoint step as much as half of the system memory can be written out. Unfortunately, for large-scale HPC applications, the burst buffer SSDs can be worn out much more quickly given the extensive amount of data written at every checkpoint step. One possible solution is to control the amount of data written by reducing the checkpoint frequency. However, a direct effect caused by reduced checkpoint frequency is the increased vulnerability window of system failures and therefore potentially wasted computation time, especially for large-scale compute jobs.

Computer Science↗

Importance Sampling Model-Based Diffusion for Trajectory Optimization

Trajectory optimization for robotic systems remains a challenging problem. This is especially true for robotic systems featuring nonlinear dynamics and many degrees of freedom. Data-based or model-free diffusion has recently been popularized in the fields of artificial intelligence and trajectory optimization. Model-Based Diffusion provides a data-free method of trajectory optimization, trained at runtime on a system dynamics model, suitable for high-dimensional models. This paper examines how importance sampling can enhance the performance of Model-Based Diffusion for trajectory optimization. Here, we quantify the benefits of importance sampling across three long horizon planning tasks. These results show as much as a 13x improvement in sample efficiency depending on environment and optimization parameters.

Golembeski, Seth [Georgia Institute of Technology,↗

Achieving balanced execution through runtime detection of performance variation

Systems, apparatuses, and methods for achieving balanced execution in a multi-node cluster through runtime detection of performance variation are described. During a training phase, performance counters and an amount of time spent waiting for synchronization is monitored for a plurality of tasks for each node of the multi-node cluster. These values are utilized to generate a model which correlates the values of the performance counters to the amount of time spent waiting for synchronization. Once the model is built, the values of the performance counters are monitored for a period of time at the start of each task, and these values are input into the model. The model generates a prediction of whether a given node is on the critical path. If the given node is predicted to be on the critical path, the power allocation of the given node is increased.

97 MATHEMATICS AND COMPUTING↗

Launch Complex 34, SWMU Cc054 2021 DNAPL Source Zone Operations, Maintenance, and Monitoring, and Hot Spot 6 Air Sparge System Annual Performance Monitoring Report Cape Canaveral Space Force Station, Florida

This Annual Performance Monitoring Report (PMR) for the Dense Non-Aqueous Phase Liquid (DNAPL) Source Zone (DSZ) and Hot Spot 6 (HS 6) Air Sparge (AS) System presents the results of Year 12 operation of the hydraulic containment (HC) Interim Measure (IM), the results of performance monitoring direct-push technology (DPT) sampling and monitoring well sampling conducted in the DSZ, and the results of operations and performance sampling of the HS 6 AS IM at Launch Complex 34 (LC34), located at Cape Canaveral Space Force Station (CCSFS), Florida. Site-wide biennial LTM sampling was not conducted during this reporting period and is scheduled to be conducted in December 2022. The timeframe for activities documented in this PMR extends from April 1, 2021 to March 31, 2022. LC34 has been designated Solid Waste Management Unit CC054 under the Kennedy Space Center (KSC) Resource Conservation and Recovery Act Corrective Action Program. The objective of the HC IM at LC34 is to contain the DSZ and deep dissolved-phase trichloroethene (TCE) high concentration plume via operation of a hydraulic containment system (HCS). The pre-IM design 300 micrograms per liter (μg/L) TCE groundwater contour was used to establish the deep zone capture area for deep recovery wells, and the shallow zone capture area was defined by the DSZ. The system began operating in 2010, and in 2015, the system was expanded to provide HC for areas within the 300 μg/L TCE groundwater isocontours of HS 3 and 4. In 2018 and 2019, an investigation was conducted to re-characterize the DSZ, which included investigating TCE mass in Layer 7. This data was subsequently used to optimize the pumping rates of the HCS to more adequately capture residual contaminant mass. The operational period for Year 12 of the HCS was from April 1, 2021 to March 31, 2022. Operational runtime for the system was 94 percent during Year 12, with downtime events attributed to planned maintenance, system repairs, and power outages. As of March 31, 2022, a total of 285,712,801 gallons of groundwater containing 84,933 pounds of VOCs have been removed by the HCS. Influent concentrations of TCE have decreased since startup from approximately 280,000 µg/L (January 2010) to 12,000 µg/L (March 2022). During the reporting period, all effluent concentrations from the HCS (aqueous and vapor) were below regulatory reporting limits, indicating the system continues to operate as intended. Performance monitoring was conducted in December 2021 within the DSZ to evaluate TCE contamination. Groundwater samples were collected via DPT at nine locations, consistent with previous events in 2017, 2018, 2019, and 2020. Full vertical profile sampling was completed at each DPT from 8 to 98 ft bls, at 5 foot intervals. The DPT performance monitoring results are summarized in this PMR. The results revealed TCE remains at concentrations greater than 11,000 µg/L in the DSZ (1-percent solubility, indicative of DNAPL) at eight of the nine DPT locations at depths ranging from 8 to 98 ft bls. An overall increasing trend of TCE concentrations was observed in DPT samples during this reporting period, which may be due to several recovery wells that were turned off during the AS Pilot Study in the DSZ that operated from July 2021 to February 2022 (documented separately from this report). The maximum TCE concentration in 2021 was 15,400,000 µg/L in the 58 ft bls depth interval at DPT597 (previous maximum result in 2020 was 1,690,000 at 48 ft bls at DPT596). During the 2021 DPT event, the overall majority of TCE contamination was identified in the 58 ft bls interval (below Layer 4), where in the previous year the majority of mass was observed in Layer 4. This trend appears to indicate continued mass discharge from Layer 4 (fine-grained unit). In addition to DPT sampling, monitoring well samples were collected from deep wells in the DSZ area (Layers 7 and 8) to verify vertical delineation. All monitoring well results were non-detect or below cleanup levels, with exception of one well (IW0162, screened 105 to 115 ft bls, which is below the existing recovery well capture zone) where TCE was identified above cleanup target levels. The HS 6 AS system remained operational during the reporting period covered under this report. The HS 6 AS IM was initiated in 2018 with 160 AS wells, and expanded in 2019 with an additional 140 AS wells. Quarterly performance monitoring was reduced to semi-annual prior to this operational period. The results of the HS 6 system operation and semi-annual performance monitoring are summarized in this report. Semi-annual monitoring results collected in April and October 2021 show concentrations of contaminants of concern (cis-1,2-dichloroethene, trans-1,2- dichloroethene, and vinyl chloride) are generally decreasing and not impacting the surface water drainage canal, indicating the HS 6 IM is meeting objectives. Overall, the tasks associated with Year 12 operation of the HC IM and operation of the HS 6 AS IM were performed in accordance with the recommendations of the 2020 LC34 (Year 11) Operations, Maintenance, and Monitoring Report for DNAPL Source Zone, Site Wide LongTerm Monitoring, and Hot Spot 6 Air Sparging System PMR (NASA, 2021c). Evaluation of results from the HC IM and HS 6 IM show that these systems are operating as designed and meeting performance objectives.

trichloroethene↗

Launch Complex 34, SWMU CC054 2023 DNAPL Source Zone Operations, Maintenance, and Monitoring, Site-Wide Long-Term Monitoring, and Hot Spot 6 Air Sparge System Annual Performance Monitoring and Phase Two Expansion Construction Completion Report Cape Canaveral Space Force Station, Florida

This Annual Performance Monitoring Report (PMR) for the Dense Non-Aqueous Phase Liquid (DNAPL) Source Zone (DSZ), Site-Wide Long-Term Monitoring (LTM), and Hot Spot 6 (HS 6) Air Sparge (AS) System presents the results of Year 14 operations and performance monitoring of the hydraulic containment (HC) Interim Measure (IM), details associated with construction and implementation of the HS 6 AS system expansion (Phase Two), and the results of operations and performance sampling of the HS 6 AS IM at Launch Complex 34 (LC34), located at Cape Canaveral Space Force Station (CCSFS), Florida. The timeframe for activities documented in this PMR extends from April 1, 2023 to March 31, 2024. LC34 has been designated Solid Waste Management Unit CC054 under the Kennedy Space Center (KSC) Resource Conservation and Recovery Act Corrective Action Program. The objective of the HC IM at LC34 is to contain the shallow and deep DSZ and surrounding dissolved-phase trichloroethene (TCE) high concentration plume via operation of a hydraulic containment system (HCS). The pre-IM design 300 micrograms per liter (μg/L) TCE groundwater contour was used to establish the deep zone capture area for deep recovery wells, and the shallow zone capture area was defined by the DSZ. The system began operating in 2010, and in 2015, the system was expanded to provide HC for areas within the 300 μg/L TCE groundwater isocontours of HS 3 and 4. In 2018 and 2019, an investigation was conducted to recharacterize the DSZ, which included investigating TCE mass in Layer 7. This data was subsequently used to optimize the pumping rates of the HCS and install additional recovery wells in Layer 7 to more adequately capture residual contaminant mass. The operational period for Year 14 of the HCS was from April 1, 2023 to March 31, 2024. Operational runtime for the system was 94 percent during Year 14, with downtime events attributed to planned maintenance, system repairs, and power outages. As of March 31, 2024, a total of 344,849,634 cumulative gallons of groundwater containing 94,656 pounds of chlorinated volatile organic compounds (CVOCs) have been removed by the HCS. During the reporting period covered under this report, the HCS recovered 31,176,393 gallons and approximately 6,319 pounds of CVOC mass. Total combined influent concentrations of TCE have decreased since startup from approximately 280,000 µg/L (January 2010) to 25,000 µg/L (March 2024). During the reporting period, all effluent concentrations from the HCS (aqueous and vapor) were below regulatory reporting limits, indicating the system continues to operate as intended. Performance monitoring was conducted in January 2024 within the DSZ to evaluate TCE contamination. Groundwater samples were collected via DPT at nine locations, consistent with previous events between 2017 and 2022. Full vertical profile sampling was completed at each DPT from 8 to 98 feet below land surface (bls), at 5-foot intervals. The DPT performance monitoring results are summarized in this PMR. The results revealed TCE remains at concentrations greater than 11,000 µg/L in the DSZ (1-percent solubility, indicative of DNAPL) at eight of the nine DPT locations and at depths ranging from 28 to 98 feet bls. An overall decreasing trend of TCE concentrations was observed in DPT samples during this reporting period, which is a reduction from the previous event (December 2022) and the peak event in December 2021, where TCE percentages appeared to increase in all depth zones because several recovery wells were turned off during the AS pilot study in the DSZ. The maximum TCE concentration in January 2024 was 1,600,000 µg/L in the 53 feet bls depth interval at DPT594 (previous maximum result in 2022 was 1,800,000 µg/L in the 48 feet bls depth interval at DPT599). This maximum concentration in the 53 feet bls depth interval is in the deep capture zone. During the January 2024 DPT event, the largest portion of TCE mass was observed in the 48 feet bls interval above/within Layer 4. This trend remains consistent with previous years and appears to indicate continued mass discharge from Layer 4 (fine-grained unit). In addition to DPT sampling, annual groundwater samples were collected from 11 deep monitoring wells in the DSZ area (Layers 7 and 8) in December 2023 to verify vertical and horizontal delineation. Three of the wells were also sampled biweekly to evaluate operations of recovery well RW21D (screened 86 to 106 feet bls), which was installed in January 2023. Of the Layer 7/8 monitoring sampled only annually, results were non-detect or less than groundwater cleanup target levels GCTLs in December 2023, with the exception of one well, IW45D2, which had a cis-1,2-dichloroethene (cDCE), detection greater than the GCTL. Of the three wells sampled biweekly during the operational period, the well located closest to Layer 7 recovery well RW21D (IW44D2, screened 105 to 115 feet bls) had concentrations of TCE, cDCE and vinyl chloride (VC) greater than GCTLs throughout the operational period, but displayed a decreasing trend since the peak concentrations in September 2023. The maximum TCE concentration during this operational period was 190,000 µg/L at IW44D2 in September 2023, but reduced to 700 µg/L in March 2024, indicating the HCS is still effectively removing mass from the source area. Expansion of the HCS and addition of new recovery wells is ongoing and will continue to be evaluated as the groundwater recovery scheme is optimized. Details of the expansion and optimization will be provided in a future PMR. The HS 6 AS IM was initiated in 2018 with 160 AS wells and expanded in 2019 with another 140 AS wells. An additional expansion of the HS 6 AS IM was completed during the reporting period covered under this report and details of the construction implementation and startup of the expansion are detailed in Section III of this report. The new expansion, referred to as Phase Two, was implemented between August 17, 2022 and August 28, 2023, and included the installation of 190 air sparge wells to treat an additional 11.2 acres. The original configuration (referred to as Phase One) operated until Phase Two came online, then all but 52 AS wells were turned off so the components could be moved and utilized in the Phase Two area. The 52 AS wells that remain on are in a barrier configuration preventing contaminated groundwater from impacting the treated area. The HS 6 AS system (both Phase One and Two) operated normally during the reporting period covered under this report. Semi-annual performance monitoring of the Phase One configuration was conducted in April and November 2023, consistent with previous years. For the Phase Two configuration, 21 new monitoring wells were installed and sampled quarterly, with a baseline event in July 2023, and quarterly events in November 2023 and February 2024 summarized in this report. Semi-annual monitoring results collected in April and October 2023 show concentrations of contaminants of concern (cDCE, trans-1,2-dichloroethene, and VC) have decreased to less than GCTLs in nearly all wells and not impacting the surface water drainage canal, indicating the HS 6 IM continues to meet objectives. The baseline and quarterly sampling for the Phase Two configuration indicate generally decreasing concentrations in wells within and around the perimeter of the treatment area. At least two more quarters of monitoring will be conducted and once those results are evaluated a reduced the sampling frequency may be considered. Overall, the tasks associated with Year 14 operation of the HC IM and operation of the HS 6 AS IM were performed in accordance with recommendations included in the previous 2022 LC34 (Year 13) PMR. Evaluation of results from the HC IM and HS 6 IM show that these systems are operating as designed and meeting performance objectives.

groundwater remediation↗

CALORIE: A Constraint Language and Optimizing Runtime for Exascale Power Management (Final Report)

This final technical report summarizes the key accomplishments on the CALORIE project, a DOE Early Career award received by PI Henry Hoffmann at the University of Chicago. CALORIE’s main goal was to create principled methodologies, tools, and practices to help scientists and high-performance computing (HPC) operators maximize the performance and insights obtained from scientific computing applications in the face of exascale power constraints. The project accomplished these goals by completing the following objectives: • Designing a language for describing application goals (including constraints and objectives) and system capabilities. The application goals will include things like which simulations or simulations plus in situ analysis will be run together, what the power constraints are, and what requirements there are for in situ analysis (for example, a desired frame rate for visualization). The system capabilities include all components that can be adjusted to tradeoff power and performance. • Designing a runtime system that takes specified goals and capabilities and dynamically determines what capabilities to use to meet the goals. This runtime adapts to changes in goals, application behavior, or available capabilities to automatically maintain the goals despite unexpected disturbances. • Developing a foundational understanding of how power constraints affect the problem of scheduling applications in large-scale systems. This report provides an overview of the accomplishments related to each of these key objectives.

97 MATHEMATICS AND COMPUTING↗