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DOE OSTI2022

Strategies for breaking molecular scaling relationships for the electrochemical CO 2 reduction reaction

The electrocatalytic CO 2 reduction reaction (CO 2 RR) is a promising strategy for converting CO 2 to fuels and value-added chemicals using renewable energy sources. Molecular electrocatalysts show promise for the selective conversion of CO 2 to single products with catalytic activity that can be tuned through synthetic structure modifications. However, for the CO 2 RR by traditional molecular catalysts, beneficial decreases in overpotentials are usually correlated with detrimental decreases in catalytic activity. This correlation is sometimes referred to as a “molecular scaling relationship”. Overcoming this inverse correlation between activity and effective overpotential remains a challenge when designing new, efficient molecular catalyst systems. Here, in this perspective, we discuss some of the concepts that give rise to the molecular scaling relationships in the CO 2 RR by molecular catalysts. We then provide an overview of some reported strategies from the last decade for breaking these scaling relationships. We end by discussing strategies and progress in our own research designing efficient molecular catalysts with redox-active ligands that show high activity at low effective overpotentials for the CO 2 RR.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Impacts of Solar Land Availability on the Evolution of the U.S. Power System

Long-term planning scenarios of the United States electricity system have not identified solar resource availability to be a limiting factor in how much solar capacity might be deployed. However, there are some smaller regions where siting challenges indicate solar deployment could be limited by solar resource availability. In this work, we use geospatial modeling tools to develop a restrictive siting scenario that significantly reduces the amount and location of utility-scale solar resources. We use that restrictive siting regime, along with other previously defined siting scenarios, to examine how solar resource availability impacts the buildout of the electricity system using business-as-usual and decarbonization futures in a national-scale capacity expansion model. The restrictive siting scenario uses observed siting pressures to exclude many types of land on which development is allowed today and results in a greater than 90% reduction in solar resource potential-to approximately 3000 GW. Even with this drastic reduction, utility-scale solar capacity grows by more than 3 times through 2050 in the business-as-usual scenarios and by more than 6 times in the decarbonization scenarios, though less solar is deployed in scenarios with lower available solar resource. We quantify how system costs and generation mixes change as solar resource availability changes.

14 SOLAR ENERGY↗

DOE OSTI2025

Technical, Economic, Energetic, and Environmental Evaluation of Pretreatment Strategies for Scaling Control in Brackish Water Desalination Brine Treatment

Effective pretreatment is essential for achieving long-term stable operation and high water recovery during the desalination of alternative waters. This study developed a process modeling approach for technical, economic, energetic, and environmental assessments of pretreatment technologies to identify the impacts of each technology treating brackish water desalination brine with high scaling propensity. The model simulations evaluated individual pretreatment technologies, including chemical softening (CS), chemical coagulation (CC), electrocoagulation (EC), and ion exchange (IX). In addition, combinations of these pretreatment technologies aiming at the effective reduction of key scaling constituents such as hardness and silica were investigated. The three evaluation parameters in this assessment consist of levelized cost of water (LCOW, $/m 3 ), specific energy consumption and cumulative energy demand (SEC|CED, kWh/m 3 ), and carbon dioxide emissions (CO 2 , kg CO 2-eq /m 3 ). The case study evaluated in this work was the desalination brine from the Kay Bailey Hutchison Desalination Plant (KBHDP) with a total dissolved solids (TDS) concentration of 11,000 mg/L and rich in hardness and silica. The evaluation of individual pretreatment units from the highest to lowest LCOW, SEC|CED, and CO 2 emissions in the KBHDP brine was IX > CS > EC > CC, CS > IX > EC > CC, and CC > CS > EC > IX, respectively. In the case of pretreatment combinations for the KBHDP, the EC + IX treatment combination was shown to be the best in terms of the LCOW and CO 2 emissions. The modeling and evaluation of these pretreatment units provide valuable guidance on the selection of cost-effective, energy-efficient, and environmentally sustainable pretreatment technologies tailored to desalination brine applications for minimal- or zero-liquid discharge.

14 SOLAR ENERGY↗

DOE OSTI2023

Optimizing Communication in 2D Grid-Based MPI Applications at Exascale

The new reality of exascale computing faces many challenges in achieving optimal performance on large numbers of nodes. A key challenge is the efficient utilization of the message-passing interface (MPI), a critical component for process communication. This paper explores communication optimization strategies to harness the GPU-accelerated architectures of these supercomputers. We focus on MPI applications where processors form a two-dimensional process grid, a common arrangement in applications involving dense matrix operations. This configuration offers a unique opportunity to implement innovative strategies to improve performance and maintain effective load distribution. We study two applications— Dist-FW (Apsp:all-pair-shortest-path) and HPL-MxP (LU factorization with Mixed precision)—on two accelerated systems: Summit (IBM Power and NVIDIA V100) and Frontier (AMD EPYC and MI250X). These supercomputers are operated by the Oak Ridge Leadership Computing Facility (OLCF) and are currently ranked #1 and #5 on the Top500 list. We show how to scale up both applications to exascale levels and tackle the MPI challenges related to implementation, synchronization, and performance. We also compare the performance of several communication strategies at an unprecedented scale. Accurately predicting application performance becomes crucial for cost reduction as the computational scale grows. To address this, we suggest a hyperbolic model as a better alternative to the traditional one-sided asymptotic model for predicting future application performance at such large scales.

Lu, Hao↗

DOE OSTI2024

Atomic-scale identification of active sites of oxygen reduction nanocatalysts

Heterogeneous nanocatalysts play a crucial role in both the chemical and energy industries. Despite substantial advancements in theoretical, computational and experimental studies, identifying their active sites remains a major challenge. Here we utilize atomic electron tomography to determine the three-dimensional atomic structure of PtNi and Mo-doped PtNi nanocatalysts for the electrochemical oxygen reduction reaction. We then employ the experimental atomic structures as input to first-principles-trained machine learning to identify the active sites of the nanocatalysts. Through the analysis of the structure–activity relationships, we formulate an equation termed the local environment descriptor, which balances the strain and ligand effects to provide physical and chemical insights into active sites in the oxygen reduction reaction. The ability to determine the three-dimensional atomic structure and chemical composition of realistic nanoparticles, combined with machine learning, could transform our fundamental understanding of the active sites of catalysts and guide the rational design of optimal nanocatalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Multi-Scale Modeling and Prototype Development for Electrochemical CO2 Reduction (CRADA Final Report)

In this CRADA project, Lawrence Livermore National Laboratory, Stanford University, SLAC National Laboratory, and TotalEnergies collaboratively executed a multidisciplinary investigation of electrochemical reduction of CO2 to produce sustainable fuels and chemicals. Overall, the project led to an increased understanding of the fundamental processes involved in CO2 electrolysis, from the atomistic scale to the full electrolyzer device scale, ultimately leading to design guidelines for CO2 electrolyzers that will help in their future commercialization. As the model systems, Ag- and Cu-based catalysts were investigated in various forms depending on the electrochemical platform that was utilized to study the activity, selectivity, and durability towards electrochemical CO2 reduction. By employing experimental, theoretical, and computational techniques, the project team experimentally validated multi-physics models, evaluated the experimental levers that lead to increased electrolyzer reaction selectivity and energy efficiency, and used computational optimization to design higher performance electrodes. The learnings of this project were extensively documented in publicly available peer-reviewed journal publications and conference presentations, which serve as a foundation for further work to build from.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Passive cooling designs to improve heat resilience of homes in underserved and vulnerable communities

Disadvantaged communities face a growing threat to staying safe during heat waves, especially during coincident power outages. This study develops a methodology to evaluate the effectiveness of passive cooling measures (those that operate without power) to improve residential building heat resilience. Building performance is simulated for representative homes and on district scales in two disadvantaged communities in Fresno, California. Eleven passive measures are evaluated using four heat resilience metrics with and without grid power. Results show performance of the mitigation measures varies by building characteristics, surrounding environment, and power scenario. The two most effective measures were installing solar-control window films and adding roof insulation. For pre-1978 single-family homes, these two measures can reduce unmet degree-hours (UDH) indoors by 12% and 11% respectively without grid power, or 28% and 37% with grid power. Their respective UDH reductions at district scale typically range 8% — 20% and 4% — 12% without grid power, or 14% — 44% and 8% — 51% with grid power. Top floors have higher overheating risk than lower floors during extreme heat events with coincident power outages. Natural ventilation can help, reducing UDH by 21% — 26%. The methodology and findings from this study can help cities, communities, and utilities develop effective and targeted strategies to promote greater residential heat resilience.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2020

Influence of Silane Coupling Agents on Filler Network Structure and Stress-Induced Particle Rearrangement in Elastomer Nanocomposites

Filled rubber materials are key in many technologies having broad impact on the economy and sustainability, the most obvious being tire technology. Adding filler dramatically improves the strength of rubber by reinforcement and tailoring the type of filler and the chemistry of the interface between filler and rubber matrix is important for optimizing performance metrics such as fuel efficiency. In a highly loaded, silica-filled, crosslinked model rubber closely mimicking commercial materials, both the filler network structure and the dynamics of the silica filler particles change when the silica surface is modified with silane coupling agents. Reduction in size scales characteristic of the structure are quantified using ultra-small-angle X-ray scattering (USAXS) measurements and the particle dynamics probed with X-ray photon correlation spectroscopy (XPCS). While the structure averaged over the scattering volume changes little with aging after step strain, the dynamics slow appreciably in a manner that varies with the treatment of the silica filler. The evolution of filler particle dynamics depends on the chemical functionality at the silica surface and observing these differences suggests a way of thinking about the origins of hysteresis in nanoparticle-reinforced rubbers. These microscopic filler dynamics are correlated with the macroscopic stress relaxation of the filled materials. The combination of static and dynamic X-ray scattering techniques with rheological measurements is a powerful approach for elucidating the microscopic mechanisms of rubber reinforcement.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Magnetic and Optical Properties of Au–Co Solid Solution and Phase-Separated Thin Films and Nanoparticles

The chemical composition and morphology of Au x Co 1–x thin films and nanoparticles are controlled via a combination of cosputtering, pulsed laser-induced dewetting (PLiD), and annealing, leading to tunable magnetic and optical properties. Regardless of chemical composition, the as-deposited thin films and as-PLiD nanoparticles are found to possess a face-centered cubic (FCC) Au x Co 1–x solid-solution crystal structure. Annealing results in large phase-separated grains of Au and Co in both the thin films and nanostructures for all chemical compositions. The magnetic and optical properties are characterized via vibrating sample magnetometry (VSM), ellipsometry, optical transmission spectroscopy, and electron energy loss spectroscopy (EELS). Despite the exceptionally high magnetic anisotropy inherent to Co, the presence of sufficient Au (72 atom %) in the Au x Co 1–x solid solution results in superparamagnetic thin films. Among the as-PLiD nanoparticle samples, an increased Co composition leads to a departure from traditional ferromagnetism in favor of wasp-waisted hysteresis caused by magnetic vortices. Phase separation resulting from annealing leads to ferromagnetism for all compositions in both the thin films and nanoparticles. The optical properties of Au x Co 1–x nanostructures are also largely influenced by the chemical morphology, where the Au x Co 1–x intermixed solid solution has significantly damped plasmonic performance relative to pure Au and comparable to pure Co. Phase separation greatly enhances the quality factor, optical absorption, and electron energy loss spectroscopy (EELS) signatures. The enhancement of the localized surface plasmon resonances (LSPRs) scales with the reduction in Co composition, despite EELS evidence that excitation of the Co portions of a nanoparticle can provide a similar, and in some instances enhanced, LSPR resonance compared to Au. This behavior, however, is seemingly limited to the LSPR dipole mode, while higher-order modes are greatly damped by a Co aloof position. As a result, this observed magneto-plasmonic functionality and tunability could be applicable in biomedicine, namely, cancer therapeutics.

annealing (metallurgy)↗

DOE OSTI2024

Combining High-Throughput Experiments and Active Learning to Characterize Deep Eutectic Solvents

The high tunability of deep eutectic solvents (DESs) stems from the ease of changing their precursors and relative compositions. However, measuring the physicochemical properties across large composition and temperature ranges, necessary to properly design target-specific DESs, is tedious and error-prone and represents a bottleneck in the advancement and scalability of DES-based applications. As such, active learning (AL) methodologies based on Gaussian processes (GPs) were developed in this work to minimize the experimental effort necessary to characterize DESs. Owing to its importance for large-scale applications, the reduction of DES viscosity through the addition of a low-molecular-weight solvent was explored as a case study. A high-throughput experimental screening was initially performed on nine different ternary DESs. Then, GPs were successfully trained to predict DES viscosity from its composition and temperature, showcasing the ability of these stochastic, nonparametric models to accurately describe the physicochemical properties of complex mixtures. Finally, the ability of GPs to provide estimates of their own uncertainty was leveraged through an AL framework to minimize the number of data points necessary to obtain accurate viscosity modes. This led to a significant reduction in data requirements, with many systems requiring only five independent viscosity data points to be properly described.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Emerging signals of declining forest resilience under climate change

Forest ecosystems depend on their capacity to withstand and recover from natural and anthropogenic perturbations (that is, their resilience). Experimental evidence of sudden increases in tree mortality is raising concerns about variation in forest resilience, yet little is known about how it is evolving in response to climate change. Here we integrate satellite-based vegetation indices with machine learning to show how forest resilience, quantified in terms of critical slowing down indicators, has changed during the period 2000–2020. We show that tropical, arid and temperate forests are experiencing a significant decline in resilience, probably related to increased water limitations and climate variability. By contrast, boreal forests show divergent local patterns with an average increasing trend in resilience, probably benefiting from warming and CO 2 fertilization, which may outweigh the adverse effects of climate change. These patterns emerge consistently in both managed and intact forests, corroborating the existence of common large-scale climate drivers. Reductions in resilience are statistically linked to abrupt declines in forest primary productivity, occurring in response to slow drifting towards a critical resilience threshold. Approximately 23% of intact undisturbed forests, corresponding to 3.32 Pg C of gross primary productivity, have already reached a critical threshold and are experiencing a further degradation in resilience. Together, these signals reveal a widespread decline in the capacity of forests to withstand perturbation that should be accounted for in the design of land-based mitigation and adaptation plans

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Human-caused long-term changes in global aridity

Widespread aridification of the land surface causes substantial environmental challenges and is generally well documented. However, the mechanisms underlying increased aridity remain relatively underexplored. Here, we investigated the anthropogenic and natural factors affecting long-term global aridity changes using multisource observation-based aridity index, factorial simulations from the Coupled Model Intercomparison Project phase 6 (CMIP6), and rigorous detection and attribution (D&A) methods. Our study found that anthropogenic forcings, mainly rising greenhouse gas emissions (GHGE) and aerosols, caused the increased aridification of the globe and each hemisphere with high statistical confidence for 1965–2014; the GHGE contributed to drying trends, whereas the aerosol emissions led to wetting tendencies; moreover, the bias-corrected CMIP6 future aridity index based on the scaling factors from optimal D&A demonstrated greater aridification than the original simulations. These findings highlight the dominant role of human effects on increasing aridification at broad spatial scales, implying future reductions in aridity will rely primarily on the GHGE mitigation.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo

We review recent advances in the capabilities of the open source ab initio Quantum Monte Carlo (QMC) package QMCPACK and the workflow tool Nexus used for greater efficiency and reproducibility. The auxiliary field QMC (AFQMC) implementation has been greatly expanded to include k-point symmetries, tensor-hypercontraction, and accelerated graphical processing unit (GPU) support. These scaling and memory reductions greatly increase the number of orbitals that can practically be included in AFQMC calculations, increasing the accuracy. Advances in real space methods include techniques for accurate computation of bandgaps and for systematically improving the nodal surface of ground state wavefunctions. Results of these calculations can be used to validate application of more approximate electronic structure methods, including GW and density functional based techniques. To provide an improved foundation for these calculations, we utilize a new set of correlation-consistent effective core potentials (pseudopotentials) that are more accurate than previous sets; these can also be applied in quantum-chemical and other many-body applications, not only QMC. These advances increase the efficiency, accuracy, and range of properties that can be studied in both molecules and materials with QMC and QMCPACK.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Robustness of inflation to kinetic inhomogeneities

We investigate the effects of large inhomogeneities in both the inflaton field and its momentum. We find that in general, large kinetic perturbations reduce the number of e-folds of inflation. In particular, we observe that inflationary models with sub-Planckian characteristic scales are not robust even to kinetic energy densities that are sub-dominant to the potential energy density, unless the initial field configuration is sufficiently far from the minimum. This strengthens the results of our previous work. In inflationary models with super-Planckian characteristic scales, despite a reduction in the number of e-folds, inflation is robust even when the potential energy density is initially sub-dominant. For the cases we study, the robustness of inflation strongly depends on whether the inflaton field is driven into the reheating phase by the inhomogeneous scalar dynamics.

gravity↗

DOE OSTI2020

Electromagnetic Shielding Design for 200 kW Stationary Wireless Charging of Light-Duty EV

Wireless power transfer (WPT) is a developing technology with the advantage of convenience and flexible charging. SAE recommended practice J2954 defines typical size and geometry with aluminum or ferrite plate shielding to limit leakage electromagnetic (EM) fields for WPT with power levels lower than 22 kVA from the input side. However, as the WPT power goes up to 100- or 200-kW level, EM safety surrounding the WPT becomes a critical concern. To address this oncoming safety challenge, a novel ferrite shielding design is proposed in this paper. Different misalignment scenarios in accordance with definitions in SAE J2954 are also taken into consideration to ensure EM safety under various operation scenarios. Simulation results, which are preliminarily verified by magnetic field measurements at 1.1 m from the center of the vehicle side coil under 100 kW operation, indicate that the magnetic field leakage can be maintained below the limits defined in SAE J2954 for 200 kW operation. A 3.3 kW scale-down test was also carried out and compared to parallel scale-down simulations. 26.8% field emission reduction is observed from the scale-down test, which supports the effectiveness of the proposed shielding design.

Zhang, Bo↗

DOE OSTI2023

IACMI Project 4.7: Pultruded Textile Carbon Fiber for Spar Caps (Final Report)

The primary objective of this project was to demonstrate the potential to significantly reduce the cost of wind turbine blades with carbon fiber reinforced polymer (CFRP) structure. Applicability of textile carbon fibers (TCF) were evaluated for use in pultruded spar cap (SC) elements as a path to cost reduction for utility scale wind turbine blades. In earlier work for the Department of Energy (DOE) Wind Energy Technologies Office (WETO), a collaboration of Sandia National Laboratory (SNL), Oak Ridge National Laboratory (ORNL), and Montana State University has demonstrated potential for pultruded TCF to compete with infused fiberglass and commercially available carbon fiber pultruded sections for spar cap construction. In the design cases evaluated, the TCF sections fared well when compared on cost per unit composite stiffness and cost per unit composite compressive strength for those designs [1]. Both stiffness and compressive strength tend to be key factors in the design of blade composite Spar Cap which carry the bulk of the blade structural loads in bending. Spar Cap design tends to distribute largely symmetric tensile and compressive stresses to opposite sides of the spar structure, but since carbon fiber composite compressive strength is typically 20-50% lower than tensile strength, the compressive loading reaches failure levels well before the tensile loading. Stiffness is critical in containing the large tip deflection in high wind loading situations. However, materials and process development were very limited in the earlier study and the work in this project was expanded to make the comparative information more representative of what will be required in order to make further inroads towards implementation. Similar to that study, this project team confirmed that the primary materials of interest for pultruded spar cap elements should be thermoset (TS) resins reinforced by carbon fibers, utilizing as high a percentage of TCF as practical to benchmark cost and performance against commercial carbon fibers. To make the closest comparison possible and eliminate specific test article size, resin selection, and equipment/operational nuance effects, the team planned to pultrude sections with 100% commercially available carbon fiber (Panex 35 carbon fiber from Zoltek) as well as samples utilizing high fractions of TCF. The resin system chosen was based on formulations recommended by large wind industry supplier Hexion and consisted of Hexion resin RSL-4597, curing agent CCA-138, and internal mold release additive 117, along with common kaolin filler ASP400P from BASF. As commonly deployed in spar cap configurations, the team had a mold built to pultrude a rectangular spar cap element of 100mm width and 3mm thickness. The extremely limited number of samples produced for the earlier study were produced with a “generic” epoxy utilized for a variety of applications by the pultruder contracted to produce test articles for demonstration purposes. More importantly, those samples were produced at a fiber fraction only slightly over 50%. Based on feedback from our industrial advisory team for that project and strongly recommended by this project team, the consensus is that it is highly desirable to obtain fiber fractions of 65-68% for significant penetration in wind blade spar cap. Although this requirement has yet to be exhaustively confirmed in readily available information, this was established as a project goal and informally decided we needed to exceed 60% fiber fraction to gain serious industry consideration. Previous TCF pultrusion trials have been challenged by the lack of robust TCF packages, resulting in non-uniform tension across and between tows, as well as excess labor and waste for removal of interleaved paper. The non-uniform tension and associated intermingling of tows in textile acrylic fiber tows and associated difficulties created from broken filaments in carbon fiber conversion inhibit the ordered packing necessary to enhance fiber fraction elevation. (These “cross-overs” are not considered undesirable for textile applications and there is some sense that they might be advantageous for those applications). In addition to work that is ongoing at the acrylic fiber manufacturers to improve their formats, The Institute of Advanced Composites Manufacturing Innovation (IACMI) Project 6.12 (report PA16-0349-6.12-01) [2] has developed and demonstrated a more robust packaging and creeling approach that at least partially addresses these issues, thus improving control of the TCF feed into the pultrusion unit. It was hoped that these and other improvements currently being implemented would allow us to achieve fiber fractions at least approaching these fiber fraction targets. During this project, sections utilizing 100% commercially available carbon fiber reinforcement were produced as a baseline, as well as sections reinforced with about 94% TCF and the balance being commercially available fiber for comparison. The most important finding was that similar to results reported in the earlier WETO-funded project and results from tests of TCF reported at IACMI meetings, this work demonstrated that sections pultruded with TCF in an epoxy resin frequently utilized in actual spar cap production had stiffness and compressive strengths largely comparable to similar sections pultruded with a commercially available carbon fiber also frequently utilized in the wind industry. Although the amount of that data is limited, some of the tensile strength results were actually closer than would have been expected based on fiber strength results provided by the TCF and commercial fiber producers. The actual test data are reported and discussed in detail in Section 5. The pultruded sections dominated by TCF reinforcement were approximately 8-10% lower in fiber fraction than for the sections produced using commercial fiber alone, making direct comparison difficult. The COVID-19 project has provided significant insight into the current state-of-the-art with various TCF product forms. The data obtained in this project will guide the planned improvements at the precursor level, especially in attaining uniform tensioning and payout to facilitate enhanced fiber fractions and overall processability of the TCF composites. The project team is providing guidance to stakeholders concerning the attributes, needs, and potential demand for TCF in wind blade spar caps. Results achieved in this project are consistent with findings in the related work cited [1] and support this guidance and the high potential for this product type. TCF precursor-producing partners continue to express interest in enhancing their product forms and the team looks forward to working with these improved materials as they become available.

42 ENGINEERING↗

DOE OSTI2020

Electromagnetic Shielding Design for 200 kW Stationary Wireless Charging of Light-Duty EV

25 ENERGY STORAGE↗

DOE OSTI2021

Scalable Yet Rigorous Floating-Point Error Analysis

Automated techniques for rigorous floating-point round-off error analysis are a prerequisite to placing important activities in HPC such as precision allocation, verification and code optimization on a formal footing. Yet existing techniques cannot provide tight bounds for expressions beyond a few dozen operators; barely enough for HPC. In this work, we offer an approach embedded in a new tool called SATIRE that scales error analysis by four orders of magnitude compared to today’s best-of-class tools. We explain how three key ideas underlying SATIRE help it attain such scale: path strength reduction, bound optimization and abstraction. SATIRE provides tight bounds and rigorous guarantees on significantly larger expressions with well over a hundred thousand operators, covering important examples including FFT, matrix multiplication and PDE stencils.

Das, Arnab↗