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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 1,063 records · Page 59
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Effects of water-based binders on electrochemical performance of manganese dioxide cathode in mild aqueous zinc batteries

In the majority of rechargeable batteries including lithium ion batteries (LIBs), polyvinylidene fluoride (PVdF) binders are the most commonly used binder for both anode and cathode. However, using PVdF binder requires the organic solvent of N-methyl-2-pyrrolidone (NMP) which is expensive, volatile, combustible, toxic, and has poor recyclability. Therefore, switching to aqueous electrode processing routes with non-toxic binders would provide a great leap forward towards the realization of ideally fully sustainable and environmentally friendly electrochemical energy storage devices. Various water-soluble binders (aqueous binders) were characterized and compared to the performance of conventional PVDF. Our study demonstrates that the electrochemical performance of Zn/MnO2 aqueous batteries is significantly improved by using sodium carboxymethyl cellulose (CMC) binder. Additionally, CMC binders offers desirable adhesion, good wettability, homogeneous material distribution, and strong chemical stability at certain pH levels (3.5~5) without any decomposition for long-cycle life.

25 ENERGY STORAGE↗
DOE OSTI2022

NETL’s Techno-Economic Models for Assessing CO2 Pipeline Transport and Geologic Storage

Presentation at Society of Petroleum Engineers (SPE) Workshop: Future Energy Roadmap – Navigating Through the Energy Transition, held in Galveston, Texas, August 22-23, 2022. The presentation provides an overview of the techno-economic models NETL has developed for assessing performance characteristics and cost drivers for CO2 pipeline transport (FECM/NETL CO2 Transport Cost Model or CO2_T_COM), CO2 saline storage (FECM/NETL CO2 Saline Storage Cost Model or CO2_S_COM), and oil production and CO2 storage using CO2 enhanced oil recovery (EOR) (FE/NETL CO2 Prophet Model or CO2_Prophet and FE/NETL Onshore CO2 EOR Cost Model or CO2_E_COM). A high-level description of each model is presented along with useful outputs that can be generated with each model. These tools can be used individually to evaluate the economic opportunity for specific CCUS components, or they can be used in tandem to assess an integrated CCUS value chain.

Morgan, David↗
DOE OSTI2023

Performance assessment of active insulation systems in residential buildings for energy savings and peak demand reduction

Active insulation systems (AISs) in buildings are envelopes that integrate thermal insulation, thermal energy storage, and controls. Although different designs for AISs have been proposed in the literature, a comprehensive analysis of feasible AISs is lacking. This paper discusses the energy performance, peak demand reduction potential, and performance characteristics of an AIS that uses a concrete wall as thermal mass sandwiched between two solid-state thermal switches (STSs). These STSs change their thermal conductivity using an on/off metal switch to create or break a thermal bridge across the STS. This paper first describes the experimental setup, used to determine the ratio of thermal resistance during R-high (low thermal conductivity) and R-low (high thermal conductivity) states of the STSs. This ratio was then used in whole-building energy simulations to evaluate the performance of AIS walls across different climate zones with/without a freeze timer of 60 min. The timer was added to reduce the number of switches of STSs from one state to another, and hence the energy needed for these switches. Analysis of the switching frequency and interval of STSs, thermal conductivity of walls, impact of wall orientation, and heat transfer through the wall from the use of AIS at different climate zones/locations were performed. The simulation results show that the AIS can achieve energy savings ranging from ~980 to 2,290 kWh in a single-family home with a floor area of ~220 m 2 compared with an IECC 2018 baseline. We found the energy savings was higher in dry climate zones which represent 17% of residential buildings in the United States, compared to humid or marine climate.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Data Structure Alchemy

In an increasingly more data-driven world, the project set out to uncover the first principles of data-structure design, chart the immense design space they form, and build automation that can synthesize an optimal structure, or even a whole storage engine, for any given workload, hardware platform, and cost target. Data structures are at the center of every computational system and are directly responsible for its performance. Two core technical thrusts were defined: 1) Mapping design spaces for key data-centric abstractions (filters, hash functions, storage-engine layouts, neural-network topologies, blockchain protocols, image layouts, etc.). 2) Developing search & synthesis algorithms, initially analytical cost models, later neural-guided bi-level optimisers that navigate sextillions of candidate designs in seconds and materialise the best one as ready‐to-run code. This report distills the key insights, accomplishments, and impact.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

The Nuclear Fuel Cycle: Safe Management of Spent Nuclear Fuel

The aim for storage of spent nuclear fuel (SNF) either in wet or in dry storage systems is to ensure general safety objective s are met throughout a desired storage period. Staff at the Savannah River National Laboratory (SRNL), in collaborations with partners at other national laboratories, industry research organizations, and the University of South Carolina (UofSC), have performed materials aging testing and analyses, and have established nuclear materials aging management programs to support extended periods of safe storage of research reactor (RR) SNF and of commercial power reactor (PR) SNF pending ultimate disposal. Several example challenges include susceptibility of aluminum SNF from research reactors to corrosion in poor quality water (wet storage). In dry storage, aluminum SNF can release hydrogen via radiolysis of the hydrated oxides on the aluminum cladding. Austenitic stainless steel canisters used for dry storage are susceptible to chloride-induced stress corrosion cracking (outside-in attack) that threaten the confinement boundary provided by the canister. This paper further describes these challenges, among others, and the formulated solutions to support extended safe storage of SNF.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2024

Strategically Modified Ligand Incorporating Mixed Phosphonate and Carboxylate Groups to Enhance Performance in All-Iron Redox Flow Batteries

Iron redox flow batteries (Fe-RFBs) hold significant promise for achieving cost-effectiveness and utilizing abundant materials for stationary energy storage applications. Here, a design of a novel Fe complex utilizing a nitrogenous phosphonate/carboxylate mixed ligand, N,N-Bis(phosphonomethyl)glycine (BMPG), is presented to achieve high performance Fe anolyte. Compared to its all-phosphonate form, nitrilotri(methylphosphonic acid) (NTMPA), the new complex Fe(BPMG) 2 demonstrates a negatively shifted redox potential, resulting in ≈0.07 V (≈10%) increase in battery output voltage. Full battery testing paired with ferrocyanide catholyte demonstrates stable cycling (capacity degradation <0.0001%/cycle) over 730 consecutive charge/discharge cycles with Coulombic Efficiency of 100% at a current density of 20 mA cm -2 under near neutral pH (≈8). Of particular interest, density functional theory (DFT) studies and operando Raman measurements provide strong evidence supporting a molecular structure in BPMG, which reveals the mixed phosphonate/carboxylate groups in BPMG maintain the octahedral coordination of the Fe ion center with phosphonates exclusively, while leaving the carboxylate unbound for both Fe(II) and Fe(III) complexes. This structural similarity between BPMG-based Fe(II) and Fe(III) complexes effectively mitigates the slow redox reaction kinetics observed in Fe(NTMPA) 2 anolyte, where significant ligand reorientation occurs between Fe(II) and Fe(III) complexes.

25 ENERGY STORAGE↗
DOE OSTI2020

EFRC Accomplishments 2014-2020: NorthEast Center for Chemical Energy Storage (NECCES)

The mission of NECCES is to develop an understanding of how key electrode reactions occur in cathode materials for Li based batteries, and how they can be controlled to improve electrochemical performance, from the atomistic level to the macroscopic level through the life-time of the operating battery. This mission will allow the determination of the ultimate limits of intercalation reactions for chemical energy storage. The approach and specific research goals of NECCES are to: 1. Attain reversible multi-electron transfer in a cathode material using lithium, using the model compound ε -VOPO 4 ; 2. Close the gap between the theoretical and practical energy density for intercalation compounds, using the model compound LiNi 0.85 Co 0 .10Al 0.05 O 2 ; 3. Understand the transition from cation redox to anion redox electrochemistry; 4. Understand performance limiting transport in positive electrode structures from the local through the meso to the macroscale.

25 ENERGY STORAGE↗
DOE OSTI2024

Enhancing Building Energy Efficiency through Advanced Sizing and Dispatch Methods for Energy Storage

Energy storage and electrification of buildings hold great potential for future decarbonization. However, there are several technical and economic barriers that prevent large-scale adoption and integration of energy storage in buildings. These barriers include integration with building control systems, high capital costs, and the necessity to identify and quantify value streams for different stakeholders. To overcome these obstacles, it is crucial to develop advanced sizing and dispatch methods to assist planning and operational decision-making for integrating energy storage in buildings. This work develops simple and flexible optimal sizing and dispatch framework for thermal energy storage (TES) and battery energy storage (BES) systems in large-scale office building. The optimal sizes of TES, BES, as well as other building assets are determined in a joint manner instead of sequentially to avoid sub-optimal solutions. The interaction between the sizing at the planning stage and hourly or sub-hourly dispatch at the operating stage is explicitly modeled. The solution is determined considering both capital costs in optimal sizing and operational benefits in optimal dispatch. Comprehensive assessments are performed using simulation studies to quantify potential energy, economic, and emission benefits by different utility tariffs and climate locations, to improve our understanding of the techno-economic performance of different TES and BES systems, and to identify barriers for adopting energy storage for buildings. Finally, the proposed framework will provide guidance to a broad range of stakeholders to properly design energy storage in buildings and maximizes potential benefits, thereby advancing affordable building energy storage deployment and helping us accelerating the transition towards a cleaner and more equitable energy economy.

Yu, Mingyung↗
DOE OSTI2024

Interfacial Engineering Using Covalent Organic Frameworks in Polymer Composites for High‐Temperature Electrostatic Energy Storage

Abstract The use of inorganic nanofillers has been an effective method to improve high‐temperature capacitive performance of dielectric polymers, though there are unmet challenges such as undesirable organic–inorganic compatibility, and low efficiencies and energy densities. Herein, a surface functionalization strategy using covalent organic frameworks (COFs) is employed to address such challenges in realizing high‐performing polymer composites. Specifically, core–shell structured nanoparticles, where ZrO 2 nanoparticles act as the core and a COF material forms the shell, are constructed and composited with the polyetherimide (PEI) matrix. The design leverages the high electron affinity ( E A ) of the outer COF shell to create energy traps, thereby capturing free charges and limiting electrical conduction. Concurrently, the low E A and wide bandgap of the ZrO 2 core introduce energy barriers to impede charge injection and migration. This orchestrated “energy level cascade” results in a marked reduction of leakage current and energy loss. The resulting polymer composite showcases an impressive discharged energy density of 6.21 J cm −3 at an efficiency above 90%, with a maximum discharged energy density reaching 7.43 J cm −3 at 150 °C. These performance metrics position the PEI/ZrO 2 @COF polymer composite to surpass or be on par with state‐of‐the‐art high‐temperature PEI composites and other advanced polymer dielectrics.

Xie, Zongliang↗
DOE OSTI2021

Aerosol Deposition Inside a Spent Fuel Transportation and Storage Canister (Interim Report)

This report documents the progress in preparing the ANSYS/Fluent ® and MELCOR models to perform characterization studies of aerosols dispersal and deposition within a select spent fuel cask system. Steady state thermal-hydraulic cask response is modeled with both codes, at present, while the MELCOR source code is being modified to allow imposed thermal-hydraulic conditions with aerosol physics calculations. This will allow the MELCOR model to assume the thermal-hydraulic calculation from the ANSYS/Fluent ® , while only computing the aerosol physics. Detailed results are presented on the thermal-hydraulic analysis of the MAGNASTOR ® cask for the current ANSYS/Fluent ® model, with convergent conditions observed over two fidelities. While the MELCOR computation computes the steady state conditions, they differ sufficiently from that computed with ANSYS/Fluent ® . The MELCOR analysis include a set of UO 2 sources to investigate system response for interim reporting. The airborne concentrations and evolving distributions are presented. Model development is anticipated to continue as additional components impacting flow conditions or aerosol deposition within the MAGNASTOR ® cask and Westinghouse fuel assemblies are identified.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

Reinforcement-based Program Induction in a Neural Virtual Machine

We present a neural virtual machine that can be trained to perform algorithmic tasks. Rather than combining a neural controller with non-neural memory storage as has been done in the past, this architecture is purely neural and emulates tape-based memory via fast associative weights (onestep learning). Here we formally define the architecture, and then extend the system to learn programs using recurrent policy gradient reinforcement learning based on examples of program inputs labeled with corresponding output targets, which are compared against actual output to generate a sparse reward signal. We describe the policy gradient training procedure used, and report its empirical performance on a number of smallscale list processing tasks, such as finding the maximum list element, filtering out certain elements, and reversing the order of the elements. These results show that program induction via reinforcement learning is possible using sparse rewards and solely neural computations.

Katz, Garrett E.↗
DOE OSTI2022

Evaluation of bio-inspired flow fields in a mediated Li-S flow battery for grid energy storage

Lithium-sulfur is a redox flow battery with high energy density for applications in safe, reliable, and lasting scaling of energy. However, lithium-based batteries often encounter platting as a problem thanks to poor Li-ions deposition after cycling. Aiming to reduce this impact, a uniform and continuous flow of ions is needed. On this work, novel bio-inspired flow fields in the electrochemical cell were tested to improve ions flowability and lithium platting control, ultimately enhancing battery performance and life. To secure Li-S efficient, low-cost, and secure energy storage capabilities, we chose a configuration with decamethylferrocene and cobaltocene acting as redox mediators, Li metal as anode and sulfur kept in a separate catholyte reservoir. Flow test and battery results insinuated a beneficial influence of bio-inspired designs in flowing electrolyte uniformly with less pressure and pump power in comparison to other conventional designs used in the industry, with an encouraging ability to approach a cheap, safe, and reliable Li-S grid energy storage.

25 ENERGY STORAGE↗
DOE OSTI2024

Studying Wind Loading on CSP Collectors to Improve Performance and Reliability

Electricity generation through Concentrating Solar-thermal Power (CSP) adds the advantage of thermal energy storage and heat production. CSP collector costs constitute nearly one-third of total plant costs. Managing wind loading, especially dynamic wind loading caused by the turbulent wind flow, is a significant design challenge. Traditional collector designs rely on wind tunnel experiments and numerical simulations, which do not fully capture dynamic effects. Here, we provide an overview of NREL's activities on wind loading on CSP collectors, aiming to improve their reliability and cost-efficiency. We present insights from field measurements at operational parabolic trough and power-tower CSP plants. Over two years, we measured turbulent wind fields and resulting structural loads in a parabolic trough plant, revealing wind, turbulence, and collector field interactions. Similar measurements are ongoing at the Crescent Dunes heliostat field. These campaigns show how atmospheric turbulence, wind direction, and collector orientation affect dynamic wind loading. Upwind collector structures can generate turbulent structures, causing fluctuating loads at downstream collectors, impacting fatigue lifetime and optical efficiency. Preliminary results indicate that interior collectors experience higher turning moments than those at the edges, suggesting higher drive wear. At Crescent Dunes, we use instrumented heliostats at the field's edge and interior to study the translation of turbulent wind to dynamic structural loads. The campaign is still ongoing, and we will present first findings. Another focus of our work is tying the experimental wind loading findings to optical performance of the collectors, using simulation tools. For example, we derive torsional errors of the parabolic troughs - the angular offset from the sun position along a collector row. This error is impacted by wind-induced mirror deformations. Future work will focus on continuing the measurement efforts and combining them with computational fluid dynamic simulations. By providing detailed field measurements and validated models, our efforts aim to enhance understanding of wind loading on CSP collectors, improving their structural integrity and optical performance.

CSP↗
DOE OSTI2022

Phase Stability and Kinetics of Topotactic Dual Ca 2+ –Na + Ion Electrochemistry in NaSICON NaV 2 (PO 4 ) 3

Recent reports of reversible calcium plating and stripping have rekindled interest in the development of Ca-ion batteries (CIBs) as next-generation energy storage devices. This technology has the potential to overcome the limitations of conventional Li-ion batteries, but CIBs are plagued by a paucity of suitable cathode materials. To date, NaSICON-structured NaV 2 (PO 4 ) 3 has been demonstrated as a successful cathode candidate, exhibiting reversible (de)intercalation of 0.6 mol Ca 2+ along with stable cycling performance. However, a complex multiphase mixture forms on discharge so the Ca-ion charge storage mechanism in the NaSICON framework is poorly understood. Here in this work, we report on an investigation of the structure and/or Na + /Ca 2+ environment(s) of a variety of chemically prepared NaSICON Ca x Na y V 2 (PO 4 ) 3 phases which were characterized using synchrotron XRD, SEM-EDS, 23 Na NMR, and TEM. Highly calciated CaV 2 (PO 4 ) 3 , Ca 1.5 V 2 (PO 4 ) 3 , and CaNaV 2 (PO 4 ) 3 phases can be prepared at high temperature, but -unlike Ca 0.6 NaV 2 (PO 4 ) 3 -these materials are electrochemically inactive. To better understand the fundamental factors impacting successful Ca 2+ electrochemistry in this system, DFT was employed to examine the Ca x Na y V 2 (PO 4 ) 3 phase diagram and Ca 2+ diffusion mechanism. Theoretical insights show that phase separation into Na-rich and Ca-rich phases is a reason for the capacity limitation and demonstrate that Na + ions in the host materials assist the migration of neighboring Ca 2+ ions, enabling reversible electrochemistry in Ca x Na y V 2 (PO 4 ) 3 . This investigation of fundamental principles affecting reversible Ca 2+ (de)intercalation in Ca x Na y V 2 (PO 4 ) 3 allows for the development of design principles to enable the discovery of a variety of successful cathodes for CIBs.

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

Assessing the Optical Performance Impact of Tracking Error in an Operational Concentrated Solar Power Plant Using Monte Carlo Ray-Tracing Simulation

Concentrating Solar Power (CSP) provides firm and dispatchable electricity due to its thermal storage and hybridization capabilities, which supports the decarbonization of our energy supply. Of the various CSP technologies, parabolic trough collectors are the most mature, with over 500 MW operating worldwide. The optical performance of parabolic trough systems is sensitive to tracking error, which is defined as the angular offset of a collector away from the sun position in the transversal plane. Tracking error commonly occurs due to non-continuous adjustment of the trough angle to point toward the sun, but other factors such as gravity, heating, and wind loading can also contribute to tracking error. Researchers have explored the impact of tracking error on optical performance both numerically and experimentally, but existing studies do not include measurements from operational utility-scale power plants. Tilt angle measurements of parabolic troughs at operational utility-scale power plants illustrate spatial variations in optical performance and include various sources of tracking error such as gravity, heating, and wind loading. To fully characterize wind driven loads on parabolic troughs, we are conducting a long-term field measurement campaign at the Nevada Solar One CSP plant located in Boulder City, NV, which has a nominal capacity of 72 MW and 0.5 hours of full-load storage. We record load measurements on four outer trough rows, collecting support structure bending moments, drive torque moments, dynamic accelerations of the spaceframe, mirror displacement, and tilt angles. Using the tilt measurements acquired at 20 Hz frequency, we calculate the deviation between the nominal sun position and the tracker angle. Using a Monte-Carlo ray-tracing simulation software, we assess the impact of the tracker angle deviation on optical performance throughout the diurnal cycle at various spatial locations within the CSP plant. In our view, this first-of-a-kind study will provide important guidance for future trough designs that reduce the impact of various sources of tracking error on performance.

concentrating solar power↗
DOE OSTI2025

Composite biopolymer electrolytes for high-performance reversible zinc-air batteries

Zinc-air batteries (ZABs) are regarded as one of the most promising candidates for next-generation energy storage systems due to their inherent safety, high energy density, and low cost. In this study, we optimized the composition of gel polymer electrolytes (GPEs), which play a crucial role in enhancing ZAB reversibility due to their water retention capabilities. The GPEs are synthesized using a combination of polyacrylic acid (PAA), polyacrylamide (PAM), and biopolymer kappa-carrageenan (KC), with and without additional additives. Among these, the composite GPE containing KC, PAA, and PAM demonstrated superior ionic conductivity (6.68 mS/cm at 30 °C) and enhanced water retention compared to the PAA–PAM-based GPE. Furthermore, this composite electrolyte operates within a wider electrochemical window of −1.0 V–2.2 V. A bifunctional catalyst, nickel–iron layered double hydroxide (NiFe-LDH), is electrochemically synthesized on a modified activated carbon cloth (CC). The catalyst loading is optimized based on oxygen evolution reaction (OER) performance, showing an overpotential of 555 mV. The oxygen reduction reaction (ORR) occurred at a half-wave potential of 0.620 V, which is lower than that of Pt/C (0.852 V), indicating better catalytic activity. The KC based cell exhibit better cycling stability than PAA–PAM-based GPE. The electrochemical performance of ZAB cells is evaluated under various atmospheres: ambient air and humid conditions and temperatures. Notably, in a humid environment, the cells achieved extended cycling stability, operating for over 500 h—significantly longer than under open-air conditions. Overall, this study highlights the importance of tailored GPE design and efficient bifunctional catalysts to enhance the performance and longevity of zinc-air batteries for long-duration energy storage applications.

Catalyst↗
DOE OSTI2021

FY2020 performance assessment annual review for the E-area low-level waste facility

The Savannah River Site (SRS) E-Area Low-Level Waste Facility (ELLWF) consists of six types of disposal units described in the Performance Assessment (PA) (WSRC, 2008): Low Activity Waste Vault (LAWV), Intermediate Level Vault (ILV), Trenches [Slit Trenches (STs), Engineered Trenches (ETs), and Component-in-Grout (CIG) Trenches], and Naval Reactor Component Disposal Areas (NRCDAs). The ELLWF is a part of the Solid Waste Management Facility (SWMF). SWMF is managed and operated by the SRS Management and Operations prime contractor, Savannah River Nuclear Solutions (SRNS). Within SRNS, the Solid Waste Management (SWM) organization is responsible for operating the SWMF, and the Savannah River National Laboratory (SRNL) is the technical agency responsible for preparing and maintaining the PA. SWMF operations have been performed at SRS since 1952. The mission of the SWMF is to provide storage, processing, disposal, and shipment of radioactive, hazardous, and mixed waste. The SWMF is committed to treat, store, and dispose of these waste products in a manner that protects the environment and the health and safety of the facility worker, the co-located worker, and the offsite general public. Wastes handled in the SWMF include low level waste, transuranic waste, hazardous waste, Toxic Substances Control Act waste, and mixed waste (containing both hazardous and radioactive constituents). The FY2020 PA Annual Review for the ELLWF affirms that the disposal facility continued to operate within the bounds of the current PA and Composite Analysis (CA) baseline and the subsequent SA’s and satisfied all the requirements, conditions, and limitations identified in the 2008 DAS (DOE 2008a), RWMB SRNL_STI-2020-00588 Revision 0 vi (McGill, 2020), and ELLWF Low-Level Waste Acceptance Criteria (SRS-1S, 2014). This annual review affirms that the supporting studies performed in FY2020 do not alter the conclusions of the ELLWF PA (WSRC, 2008) and that there is a reasonable expectation that the ELLWF will meet the performance objectives delineated in DOE Manual 435.1-1 (DOE 2011)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2020

Multigrid deflation for Lattice QCD

Computing the trace of the inverse of large matrices is typically addressed through statistical methods. Deflating out the lowest eigenvectors or singular vectors of the matrix reduces the variance of the trace estimator. This work summarizes our efforts to reduce the computational cost of computing the deflation space while achieving the desired variance reduction for Lattice QCD applications. Previous efforts computed the lower part of the singular spectrum of the Dirac operator by using an eigensolver preconditioned with a multigrid linear system solver. Despite the improvement in performance in those applications, as the problem size grows the runtime and storage demands of this approach will eventually dominate the stochastic estimation part of the computation. In this work, we propose to compute the deflation space in one of the following two ways. First, by using an inexact eigensolver on the Hermitian, but maximally indefinite, operator. Second, by exploiting the fact that the multigrid prolongator for this operator is rich in components toward the lower part of the singular spectrum. We show experimentally that the inexact eigensolver can approximate the lower part of the spectrum even for ill-conditioned operators. Also, the deflation based on the multigrid prolongator is more efficient to compute and apply, and, despite its limited ability to approximate the fine level spectrum, it obtains similar variance reduction on the trace estimator as deflating with approximate eigenvectors from the fine level operator.

97 MATHEMATICS AND COMPUTING↗