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DOE OSTI2025

FedCSpc: A Cross-Silo Federated Learning System With Error-Bounded Lossy Parameter Compression

Cross-Silo federated learning is widely used for scaling deep neural network (DNN) training over data silos from different locations worldwide while guaranteeing data privacy. Communication has been identified as the main bottleneck when training large-scale models due to large-volume model parameters and gradient transmission across public networks with limited bandwidth. Most previous works focus on gradient compression, while limited work tries to compress parameters that can not be ignored and extremely affect communication performance during the training. Here, to bridge this gap, we propose FedCSpc: an efficient cross-silo federated learning system with an XAI-driven adaptive parameter compression strategy for large-scale model training. Our work substantially differs from existing gradient compression techniques due to the distinct data features of gradient and parameter. The key contributions of this paper are fourfold. (1) Our designed FedCSpc proposes to compress the parameter during the training using the state-of-the-art error-bounded lossy compressor – SZ3. (2) We develop an adaptive compression error bound adjustment algorithm to guarantee the model accuracy effectively. (3) We exploit an efficient approach to utilize the idle CPU resources of clients to compress the parameters. (4) We perform a comprehensive evaluation with a wide range of models and benchmarks on a GPU cluster with 65 GPUs. Results show that FedCSpc can achieve the same model accuracy as FedAvg while reducing the data volume of parameters and gradients in communication by up to 7.39× and 288×, respectively. With 32 clients on a 4 Gb size model, FedCSpc significantly outperforms FedAvg in wall-clock time in the emulated WAN environment (at the bandwidth of 1 Gbps or lower without loss of generality).

SZ3↗

DOE OSTI2025

Applying Machine Learning and Bayesian Inference to Identify and Locate Moving Anthropogenic Sources Using Distributed Acoustic Sensing Data

Distributed acoustic sensing (DAS) systems, which use existing telecommunication fibers, offer high‐resolution capabilities ideal for recording anthropogenic sources. However, the complexity of urban environments and the large amount of data recorded by DAS require automated methods to efficiently detect and categorize anthropogenic sources. Here, we evaluate how well three machine learning models (k‐nearest neighbor [k‐NN], convolutional neural networks, and recurrent‐convolutional neural networks) can identify various anthropogenic sources recorded by DAS. Our findings reveal that both k‐NN and neural network methods perform well in high signal‐to‐noise ratio (SNR) settings. However, their accuracy decreases at SNRs <4. We also use Kalman filtering, a form of Bayesian inference, on backprojected locations of these sources to recover locations that generally fall within standard smartphone Global Positioning System errors. By combining machine learning and Kalman filter results, we calculate a multidimensional model of moving anthropogenic sources. These results demonstrate the potential of DAS data in urban seismology for accurately identifying and locating such sources. Depending on the research objectives, these sources can be further studied or filtered out to improve the quality of seismic data for earthquake studies. Such methods provide a valuable tool for urban seismology and seismic hazard analysis.

Luckie, Thomas William [Sandia National Laboratori↗

DOE OSTI2023

Eulerian finite element simulations of the drop weight impact test with a dislocation Density-based continuum model

During a drop weight impact test, the kinetic energy of the falling weight is transferred into the sample resting on the anvil. The plastic deformation in the sample is an important mechanism for the dissipation of the input kinetic energy. We use Eulerian finite element analysis to simulate the deformation and temperature evolution in a 1,3,5-trinitro-1,3,5-triazine (RDX) sample consisting of multiple crystals. In Eulerian finite element simulations, the mesh moves relative to the material. After every change of position between the mesh and the material, the state variables are interpolated to the new mesh position, i.e., advection. In an effort to reduce the advection errors, we use a rate form of a dislocation density-based continuum model by Luscher et al. Here, the simulations predict localization of plastic deformation, and plastic dissipation as a significant source of heat generation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

A Modeling Toolkit for Comparing AC and DC Electrical Distribution Efficiency in Buildings

Recently, there has been considerable research interest in the potential for DC distribution systems in buildings instead of the traditional AC distribution systems. Due to the need for performing power conversions between DC and AC electricity, DC distribution may provide electrical efficiency advantages in some systems. To support comparative evaluations of AC-only, DC-only, and hybrid AC/DC distribution systems in buildings, a new modeling toolkit called the Building Electrical Efficiency Analysis Model (BEEAM) was developed and is described in this paper. To account for harmonics in currents or voltages arising from nonlinear devices, the toolkit implements harmonic power flow, along with nonlinear device behavioral descriptions derived from empirical measurements. This paper describes the framework, network equations, device representations, and an implementation of the toolkit in an open source software package, including a component library and graphical interface for creating circuits. Simulations of electrical behavior and device and system efficiencies using the toolkit are compared with experimental measurements of a small office environment in a variety of operating and load configurations. A detailed analysis of uncertainty estimation is also provided. Key findings were that a comparison of predicted versus measured efficiencies and power losses in the validation testbed using the initial toolkit implementation predicted device- and system-level efficiencies with reasonably good accuracy under both balanced and unbalanced AC scenarios. An uncertainty analysis also revealed that the maximum estimated error for system efficiency across all scenarios was 3%, and measured and modeled system efficiency agreed within the experimental uncertainty in approximately half of the scenarios. Based on the correspondence between simulation and measurement, the toolkit is proposed by the authors as a potentially useful tool for comparing efficiency in AC, DC, and hybrid AC/DC distribution systems in buildings.

DC distribution↗

DOE OSTI2020

Effective Opacity of the Intergalactic Medium from Galaxy Spectra Analysis

We measure the effective opacity (τ{sub eff}) of the intergalactic medium from the composite spectra of 281 Lyman-break galaxies in the redshift range 2 ≲ z ≲ 3. Our spectra are taken from the COSMOS Lyα Mapping And Tomographic Observations survey derived from the Low Resolution Imaging Spectrometer on the W.M. Keck I telescope. We generate composite spectra in two redshift intervals and fit them with spectral energy distribution (SED) models composed of simple stellar populations. Extrapolating these SED models into the Lyα forest, we measure the effective Lyα opacity (τ{sub eff}) in the 2.02 ≤ z ≤ 2.44 range. At z = 2.22, we estimate τ{sub eff} =0.159±0.001 from a power-law fit to the data. These measurements are consistent with estimates from quasar analyses at z < 2.5 indicating that the systematic errors associated with normalizing quasar continua are not substantial. We provide a Gaussian processes model of our results and previous τ{sub eff} measurements that describes the steep redshift evolution in τ{sub eff} from z = 1.5–4.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Online adaptive basis refinement and compression for reduced-order models via vector-space sieving

In many applications, projection-based reduced-order models (ROMs) have demonstrated the ability to provide rapid approximate solutions to high-fidelity full-order models (FOMs). However, there is no a priori assurance that these approximate solutions are accurate; their accuracy depends on the ability of the low-dimensional trial basis to represent the FOM solution. As a result, ROMs can generate inaccurate approximate solutions, e.g., when the FOM solution at the online prediction point is not well represented by training data used to construct the trial basis. To address this fundamental deficiency of standard model-reduction approaches, this work proposes a novel online-adaptive mechanism for efficiently enriching the trial basis in a manner that ensures convergence of the ROM to the FOM, yet does not incur any FOM solves. The mechanism is based on the previously proposed adaptive $h$-refinement method for ROMs, but improves upon this work in two crucial ways. First, the proposed method enables basis refinement with respect to any orthogonal basis (not just the Kronecker basis), thereby generalizing the refinement mechanism and enabling it to be tailored to the physics characterizing the problem at hand. Second, the proposed method provides a fast online algorithm for periodically compressing the enriched basis via an efficient proper orthogonal decomposition (POD) method, which does not incur any operations that scale with the FOM dimension. We report these two features allow the proposed method to serve as (1) a failsafe mechanism for ROMs, as the method enables the ROM to satisfy any prescribed error tolerance online (even in the case of inadequate training), and (2) an efficient online basis-adaptation mechanism, as the combination of basis enrichment and compression enables the basis to adapt online while controlling its dimension.

42 ENGINEERING↗

DOE OSTI2025

Rapid SACR Observations of Convection at Bankhead National Forest (RAPID) Field Campaign Report

Improving our representation of convective cell processes requires better quantification of convective clouds throughout their entire life cycle. This includes gaining a clearer understanding of the controls on key convective cloud properties, such as updraft intensity, particle size distributions, rainfall rates, and hydrometeor species. Our inability to improve convective cloud process modeling stems, in part, from a limited understanding of convective cell properties, particularly given how rapidly these storms evolve. This lack of detailed observations in the most intense and organized convective storms is especially significant, as large errors remain in representing these clouds, which are critical for severe weather prediction and Earth system model performance. Cloud and precipitation radars are essential tools for studying cloud microphysics and dynamics, particularly in deeper convective clouds. The recent U.S. Department of Energy Atmospheric Radiation Measurement (ARM) User Facility’s third Mobile Facility (AMF3) Bankhead National Forest (BNF) deployment provides a unique opportunity to investigate important land–atmosphere interactions, as well as the environmental controls on deep convective cloud processes, in a location favorable for frequent convection. We operate the X/Ka-band Scanning ARM Cloud Radar (X/Ka SACR) using a scan strategy optimized to capture these rapidly evolving clouds and their properties, thereby improving studies of deep convective cloud processes. This effort is strengthened by a complementary and coordinated partnership with ongoing university and multi-agency radar activities collocated in north Alabama—a unique opportunity to examine clouds and precipitation from a lifetime-centric perspective.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Sub-percentage measure of distances to redshift of 0.1 by a new cosmic ruler

ABSTRACT Distance-redshift diagrams probe expansion history of the Universe. We show that the stellar mass-binding energy (massE) relation of galaxies proposed in our previous study offers a new distance ruler at cosmic scales. By using elliptical galaxies in the main galaxy sample of the Sloan Digital Sky Survey Data Release 7, we construct a distance-redshift diagram over the redshift range from 0.05 to 0.2 with the massE ruler. The best-fit dark energy density is 0.675 ± 0.079 for flat Λ-cold dark matter (ΛCDM) model, consistent with those by other probes. At the median redshift of 0.11, the median distance is estimated to have a fractional error of 0.34 per cent, much lower than those by supernova (SN) Ia and baryonic acoustic oscillation (BAO) and even exceeding their future capability at this redshift. The above low-$\mathit{ z}$ measurement is useful for probing dark energy that dominates at the late Universe. For a flat dark energy equation of state model (flat wCDM), the massE alone constrains w to an error that is only a factor of 2.2, 1.7, and 1.3 times larger than those by BAO, SN Ia, and cosmic microwave background (CMB), respectively.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Optimizing Geant4 Hadronic Models

Geant4, the leading detector simulation toolkit used in high energy physics, employs a set of physics models to simulate interactions of particles with matter across a wide range of energies. These models, especially the hadronic ones, rely largely on directly measured cross-sections and inclusive characteristics, and use physically motivated parameters. However, they generally aim to cover a broad range of possible simulation tasks and may not always be optimized for a particular process or a given material. The Geant4 collaboration recently made many parameters of the models accessible via a configuration interface. This opens a possibility to fit simulated distributions to the thin target experimental datasets and extract optimal values of the model parameters and the associated uncertainties. Such efforts are currently undertaken by the Geant4 collaboration with the goal of offering alternative sets of model parameters, also known as "tunes", for certain applications. The effort should subsequently lead to more accurate estimates of the systematic errors in physics measurements given the detector simulation role in performing the physics measurements. Results of the study are presented to illustrate how Geant4 model parameters can be optimized through applying fitting techniques, to improve the agreement between the Geant4 and the experimental data.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2023

Surface temperature comparison of the Arctic winter MOSAiC observations, ERA5 reanalysis, and MODIS satellite retrieval

Atmospheric model systems, such as those used for weather forecast and reanalysis production, often have significant and systematic errors in their representation of the Arctic surface energy budget and its components. The newly available observation data of the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) expedition (2019/2020) enable a range of model analyses and validation in order to advance our understanding of potential model deficiencies. In the present study, we analyze deficiencies in the surface radiative energy budget over Arctic sea ice in the ERA5 global atmospheric reanalysis by comparing against the winter MOSAiC campaign data, as well as, a pan-Arctic level-2 MODIS ice surface temperature remote sensing product. We find that ERA5 can simulate the timing of radiatively clear periods, though it is not able to distinguish the two observed radiative Arctic winter states, radiatively clear and opaquely cloudy, in the distribution of the net surface radiative budget. The ERA5 surface temperature over Arctic sea ice has a conditional error with a positive bias in radiatively clear conditions and a negative bias in opaquely cloudy conditions. The mean surface temperature error is 4°C for radiatively clear situations at MOSAiC and up to 15°C in some parts of the Arctic. The spatial variability of the surface temperature, given by 4 observation sites at MOSAiC, is not captured by ERA5 due to its spatial resolution but represented in the level-2 satellite product. The sensitivity analysis of possible error sources, using satellite products of snow depth and sea ice thickness, shows that the positive surface temperature errors during radiatively clear events are, to a large extent, caused by insufficient sea ice thickness and snow depth representation in the reanalysis system. A positive bias characterizes regions with ice thickness greater than 1.5 m, while the negative bias for thinner ice is partly compensated by the effect of snow.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Comparison of Machine Learning Approaches for Prediction of the Equivalent Alkane Carbon Number for Microemulsions Based on Molecular Properties

The chemical properties of oils are vital in the design of microemulsion systems. The hydrophilic–lipophilic difference equation used to predict microemulsions’ phase behavior expresses the oils’ physiochemical properties as the equivalent alkane carbon number (EACN). The experimental determination of EACN requires knowledge of the temperature dependence of the microemulsion system and the effects of different surfactant concentrations. Thus, the experimental determination is time-intensive and tedious, requiring days to months for proper separations. Furthermore, the experiments require high purity of chemicals because microemulsions are sensitive to impurities. Our work focuses on the quick and reliable predictions of the EACN with machine learning (ML) models. Due to the immaturity of ML chemical predictions, we compare three graph neural networks (GNNs) and a gradient-boosted tree algorithm, known as XGBoost. The GNNs use the molecular structures represented as simplified molecular-input line-entry system (SMILES) codes for the initial input, which allows us to assess whether geometry optimization is necessary for reliable results. The XGBoost model also begins with the SMILES representations of the molecules but uses molecular descriptors instead of geometry optimizations. As a result, the best model tested (crystal graph convolutional neural network with Merck molecular force field-94) has an error of 1.15 EACN units of the true EACN for unknown data with the errors skewed toward zero and an R² score of 0.9

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

CALPHAD-based ICME design of single-step aging to enhance mechanical strength of WAAM Haynes 282

To match the strength of wire-arc additive manufactured Haynes 282 to its wrought counterpart via a single-step aging heat treatment, the CALPHAD (Calculation of Phase Diagrams) method is integrated with physics-based process-structure-property models and experimental validation. The integrated computational materials engineering (ICME) framework simulates the effects of aging on γ′ and M 23 C 6 precipitation and the resulting yield strength. To improve simulation reliability, the interfacial energies between γ/γ′ and γ/M 23 C 6 carbides were estimated by comparison with precipitation kinetic modeling and measured precipitate sizes. γ′ and M23C6 were found to precipitate simultaneously between 640 and 860 °C, producing microstructures similar to those produced by two-step aging. The optimal γ′ size for peak yield stress was calculated to be 20–23 nm. WAAM Haynes 282 aged at 780 °C for 50 h exceeded the mechanical performance of its wrought counterpart subjected to two-step aging, though desired properties can also be achieved at 800 °C for 16 h or less. The error in yield strength is less than 20 MPa, demonstrating good agreement between the modeling framework and experiments. Creep studies showed that WAAM Haynes 282 exceeded the calculated rupture time, reaching 481 h. This proposed methodology can accelerate the design of aging heat treatments for any γ′-strengthened nickel-base alloy, minimizing the resources required for trial-and-error experiments.

CALPHAD↗

DOE OSTI2024

Modeling athermal phonons in novel materials using the G4CMP simulation toolkit

Understanding phonon and charge propagation in superconducting devices plays an important role in both performing low-threshold dark matter searches and limiting correlated errors in superconducting qubits. The Geant4 Condensed Matter Physics (G4CMP) package, originally developed for the Cryogenic Dark Matter Search (CDMS) experiment, models charge and phonon transport within silicon and germanium detectors and has been validated by experimental measurements of phonon caustics, mean charge-carrier drift velocities, and heat pulse propagation times. Here, in this work, we present a concise framework for expanding the capabilities for phonon transport to a number of other novel substrate materials of interest to the dark matter and quantum computing communities, including sapphire (Al 2 O 3 ), gallium arsenide (GaAs), lithium fluoride (LiF), calcium tungstate (CaWO 4 ), and calcium fluoride (CaF 2 ). We demonstrate the use of this framework in generating phonon transport properties of these materials and compare these properties with experimentally-determined values where available.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2025

Characterizing leaf-scale fluorescence with spectral invariants

Sun-induced chlorophyll fluorescence (SIF) is increasingly recognized as a non-destructive probe for tracking terrestrial photosynthesis. Emerging developments in spectral invariants theory provide an innovative and efficient approach for representing SIF radiative transfer processes at the canopy scale. However, modeling leaf-scale fluorescence based on the spectral invariants properties (SIP) remains underexplored. In this study, the spectral invariants theory is employed for the first time to model the leaf-scale total, backward and forward fluorescence (leaf-SIP SIF). The leaf-SIP SIF model separates the leaf-scale radiative transfer process into two distinct components: the wavelength-dependent one associated with leaf biochemical properties, and the wavelength-independent component linked to leaf structural characteristics. The leaf structure-related effects are characterized by two spectrally invariant parameters: the photon recollision probability (p) and the scattering asymmetry parameter (q), which are parameterized using the directly measurable leaf dry matter. Evaluation against field measurements shows that the proposed leaf-SIP SIF model has a good performance, with coefficient of determination (R 2 ) of 0.89, 0.89, 0.90 and root mean squared errors (RMSE) of 1.28, 0.69, 0.74 Wm -2 µm -1 sr -1 , respectively for the total, backward, and forward fluorescence (660–800 nm). The leaf-SIP SIF model with a more concise formulation demonstrates comparable performance with the widely used Fluspect model. Further, the leaf-SIP SIF model provides a simple and efficient approach for simulating leaf-scale fluorescence, with the potential to be integrated into a unified SIP-based model framework for simulating the radiative transfer processes across the soil-leaf-canopy-atmosphere continuum.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Drop Interactions with the Conical Shock Structure Generated by a Mach 4.5 Projectile

This work presents measurements of liquid drop deformation and breakup time behind approximately conical shock waves and evaluates the predictive capabilities of low-order models and correlations developed using planar shock experiments. A conical shock was approximated by firing a bullet at Mach 4.5 past a vertical column of water drops with a mean initial diameter of $192$ $\mu$m. The time-resolved drop position and maximum transverse dimension were characterized using backlit stereo images taken at 500 kHz. The gas density and velocity fields experienced by the drops were estimated using a Reynolds-averaged Navier–Stokes simulation of the bullet. Classical correlations predict drop breakup times and deformation in error by a factor of 3 or more. The Taylor analogy breakup (TAB) model predicts deformed drop diameters that agree within the confidence bounds of the ensemble-averaged experimental values using a dimensionless constant $C$ 2 = $2$ compared to the accepted value $C$ 2 = $2/3$. In conclusion, results demonstrate existing correlations are inadequate for predicting the drop response to the three-dimensional relaxation of the flowfield downstream of a conical-like shock and suggest the TAB model results represent a path toward improved predictions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Computer-aided design of stability enhanced nicotinamide cofactor biomimetics for cell-free biocatalysis

Cell-free biocatalysis (CFB) is an efficient and environmentally friendly method to synthesize molecules such as pharmaceuticals, biochemicals, and biofuels through the in vitro use of enzyme cascades. These enzymes often require redox cofactors to drive chemical reactions. Natural redox cofactors (NAD(P)H) are expensive to isolate, motivating synthetic nicotinamide cofactor biomimetics (NCBs) as a cost-effective solution. A select handful of NCBs have been identified as potential NAD(P)H alternatives with comparable or improved redox capabilities, however, they display a tendency to degrade in common buffers. In this study, a library of 132 NCB candidates is systematically generated, over 85% of which have not been characterized in the literature, to expand the diversity of currently explored NCBs. The decomposition mechanism of NCBs in phosphate is evaluated using density functional theory (DFT), revealing protonation at the nicotinamide C5 position as a reporter of cofactor stability. Based on this result, we trained a linear regression model on DFT calculated descriptors to predict NCB stability in phosphate buffer, achieving mean absolute error (MAE) and root mean squared error (RMSE) values within computational accuracy. Analysis of key atomic descriptors and qualitative trends in our dataset informed the design of novel NCB candidates we propose with optimized stability. This work enables researchers to predict the relative stability of NCBs before synthesis, thereby streamlining the process to make CFB more affordable and viable at industry scales.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Evaluation of GlassNet for physics-informed machine learning of glass stability and glass-forming ability

Glassy materials form the basis of many modern applications, including nuclear waste immobilization, touch-screen displays, and optical fibers, and also hold great potential for future medical and environmental applications. However, their structural complexity and large composition space make design and optimization challenging for certain applications. Of particular importance for glass processing and design is an estimate of a given composition's glass-forming ability (GFA). However, there remain many open questions regarding the underlying physical mechanisms of glass formation, especially in oxide glasses. It is apparent that a proxy for GFA would be highly useful in glass processing and design, but identifying such a surrogate property has proven itself to be difficult. While glass stability (GS) parameters have historically been used as a GFA surrogate, recent research has demonstrated that most of these parameters are not accurate predictors of the GFA of oxide glasses. Here, in this work, we explore the application of an open-source pre-trained neural network model, GlassNet, that can predict the characteristic temperatures necessary to compute GS with reasonable performance and assess the feasibility of using these physics-informed machine learning (PIML)-predicted GS parameters to estimate GFA. In doing so, we track the uncertainties at each step of the computation—from the original ML prediction errors to the compounding of errors during GS estimation, and finally to the final estimation of GFA. While GlassNet exhibits reasonable accuracy on all individual properties, we observe a large compounding of error in the combination of these individual predictions for the PIML prediction of GS, finding that random forest models offer similar accuracy to GlassNet. We also break down the performance of GlassNet on different glass families and find that the error in GS prediction is correlated with the error in crystallization peak temperature prediction. Lastly, we utilize this finding to assess the relationship between top-performing GS parameters and GFA for two ternary glass systems: sodium borosilicate and sodium iron phosphate glasses. We conclude that to obtain true ML predictive capability of GFA, significantly more data needs to be collected.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. Here, the level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.

54 ENVIRONMENTAL SCIENCES↗