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At least 1,171 records · Page 65

Simulation driven adaptive sampling for neutron-diffraction based strain mapping of additively manufactured parts

Neutron diffraction based strain mapping is a useful technique for measuring residual strains in additively manufactured (AM) metal parts. The measurement is traditionally done by scanning the sample in a point-wise raster pattern to extract the strain at each position. Since the overall scan can span several hours, adaptive sampling approaches using Bayesian optimization based on Gaussian process (BO-GP) regression have been introduced—demonstrating that even with a fraction of the typically made measurements the dominant strain patterns in the sample can be reconstructed. However, the parameters of the BO-GP algorithm have to be carefully chosen for best performance, and the movement time between arbitrary points can offset the time savings from a reduced number of measurement locations. In this paper, we propose algorithms to refine the BO-GP based methods by using simulations of strain patterns in AM parts based on the materials and the process used to print them. We demonstrate that the simulated strain patterns can be used to help choose better parameters for the BO-GP based framework—leading to low reconstruction error for the final strain pattern. Furthermore, we show that the strain mapping experiment can be initialized with a sampling pattern learnt from the simulation data and ordered to reduce movement time, dramatically enabling reduction in the overall time required to run the baseline BO-GP method.

Gaussian process regression↗

A Constraint on Primordial B-modes from the First Flight of the Spider Balloon-borne Telescope

We present the first linear polarization measurements from the 2015 long-duration balloon flight of Spider, which is an experiment that is designed to map the polarization of the cosmic microwave background (CMB) on degree angular scales. The results from these measurements include maps and angular power spectra from observations of 4.8% of the sky at 95 and 150 GHz, along with the results of internal consistency tests on these data. While the polarized CMB anisotropy from primordial density perturbations is the dominant signal in this region of sky, Galactic dust emission is also detected with high significance. Galactic synchrotron emission is found to be negligible in the Spider bands. We employ two independent foreground-removal techniques to explore the sensitivity of the cosmological result to the assumptions made by each. The primary method uses a dust template derived from Planck data to subtract the Galactic dust signal. A second approach, which constitutes a joint analysis of Spider and Planck data in the harmonic domain, assumes a modified-blackbody model for the spectral energy distribution of the dust with no constraint on its spatial morphology. Using a likelihood that jointly samples the template amplitude and r parameter space, we derive 95% upper limits on the primordial tensor-to-scalar ratio from Feldman–Cousins and Bayesian constructions, finding r < 0.11 and r < 0.19, respectively. Roughly half the uncertainty in r derives from noise associated with the template subtraction. New data at 280 GHz from Spider's second flight will complement the Planck polarization maps, providing powerful measurements of the polarized Galactic dust emission.

79 ASTRONOMY AND ASTROPHYSICS↗

Probabilistic estimation of depth-resolved profiles of soil thermal diffusivity from temperature time series

Abstract. Improving the quantification of soil thermal and physical properties is key to achieving a better understanding and prediction of soil hydro-biogeochemical processes and their responses to changes in atmospheric forcing. Obtaining such information at numerous locations and/or over time with conventional soil sampling is challenging. The increasing availability of low-cost, vertically resolved temperature sensor arrays offers promise for improving the estimation of soil thermal properties from temperature time series, and the possible indirect estimation of physical properties. Still, the reliability and limitations of such an approach need to be assessed. In the present study, we develop a parameter estimation approach based on a combination of thermal modeling, sliding time windows, Bayesian inference, and Markov chain Monte Carlo simulation to estimate thermal diffusivity and its uncertainty over time, at numerous locations and at an unprecedented vertical spatial resolution (i.e., down to 5 to 10 cm vertical resolution) from soil temperature time series. We provide the necessary framework to assess under which environmental conditions (soil temperature gradient, fluctuations, and trend), temperature sensor characteristics (bias and level of noise), and deployment geometries (sensor number and position) soil thermal diffusivity can be reliably inferred. We validate the method with synthetic experiments and field studies. The synthetic experiments show that in the presence of median diurnal fluctuations ≥ 1.5 ∘C at 5 cm below the ground surface, temperature gradients > 2 ∘C m−1, and a sliding time window of at least 4 d the proposed method provides reliable depth-resolved thermal diffusivity estimates with percentage errors ≤ 10 % and posterior relative standard deviations ≤ 5 % up to 1 m depth. Reliable thermal diffusivity under such environmental conditions also requires temperature sensors to be spaced precisely (with accuracy to a few millimeters), with a level of noise ≤ 0.02 ∘C, and with a bias defined by a standard deviation ≤ 0.01 ∘C. Finally, the application of the developed approach to field data indicates significant repeatability in results and similarity with independent measurements, as well as promise in using a sliding time window to estimate temporal changes in soil thermal diffusivity, as needed to potentially capture changes in bulk density or water content.

54 ENVIRONMENTAL SCIENCES↗

Prospects for Surrogate Neutron Capture Measurements with Radioactive Ion Beams and GODDESS

Neutron capture reactions are responsible for the synthesis of almost all of the elements heavier than iron through the slow s-process, that proceeds close to the line of stability, and the rapid r-process, with very neutron-rich waiting points. Uncertainties in (n,γ) rates in neutron rich nuclei, especially near closed neu- tron shells, can have significant impact [1] on the predictions of final abundances for different astrophysical scenarios for the r process. Understanding (n,γ) rates on neutron-rich fission fragments is also important for nuclear forensics and stockpile stewardship science. Ratkiewicz et al. [2 and references therein] has recently demonstrated that the (d,pγ) reaction is a valid surrogate for (n,γ), where the formation of the compound nucleus from the breakup of the deuteron has been calculated in a reaction model and the subsequent measured gamma-decay probabilities are reproduced with standard level density and strength functions in a Bayesian approach. In parallel to the surrogate validation efforts, we have demonstrated that the (d,pγ) reaction can be measured in inverse kinematics with Gammasphere ORRUBA: Dual Detectors for Experimental Structure Studies (GODDESS) [3] where the Gammasphere array of Compton-suppressed HPGe detectors is coupled to the Oak Ridge Rutgers University Barrel Array of position-sensitive silicon strip detectors. During the commissioning campaign we measured the (d,pγ) reaction with 134 Xe and 95 Mo beams, the latter to demonstrate the surrogate method in inverse kinematics. The present talk will present preliminary results from this campaign including γ-decay probabilities and prospects for surrogate (n,γ) measurements with 143 Ba fission-fragment beams.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Bayesian Optimization of Catalysis with In-Context Learning

Large language models (LLMs) can perform accurate classification with zero or few examples through in-context learning (ICL), allowing the model to observe query-relevant examples at inference time and eliminating the need for additional weight updates to generalize beyond its original training data. We extend this capability to regression with uncertainty estimation using frozen LLMs (e.g., GPT-4o, Gemini), enabling Bayesian optimization (BO) in natural language without explicit model training or feature engineering. We apply this to materials discovery by representing materials as synthesis and testing procedures for use in natural language prompts. This Bayesian, design-first approach prioritizes optimization toward target material properties before detailed characterization, in contrast to conventional experimental workflows that often emphasize characterization of suboptimal materials. On benchmarks like aqueous solubility and oxidative coupling of methane (OCM), BO-ICL matches or outperforms Gaussian processes. In live experiments on the reverse water–gas shift (RWGS) reaction, BO-ICL identifies multimetallic catalysts that approach equilibrium CO yield within 6 and 10 iterations from a pool of 3,700 and 360,000 candidates, respectively. Our method redefines materials representation and accelerates discovery, with broad applications across catalysis, materials science, and AI.

Calibration↗

Autonomous sputter synthesis of thin film nitrides with composition controlled by Bayesian optimization of optical plasma emission

Autonomous experimentation has emerged as an efficient approach to accelerate the pace of material discovery. Although instruments for autonomous synthesis have become popular in molecular and polymer science, solution processing of hybrid materials, and nanoparticles, examples of autonomous tools for physical vapor deposition are scarce yet important for the semiconductor industry. Here, we report the design and implementation of an autonomous workflow for sputter deposition of thin films with controlled composition, leveraging a highly automated sputtering reactor custom-controlled by Python, optical emission spectroscopy (OES), and a Bayesian optimization algorithm. We modeled film composition, measured by x-ray fluorescence, as a linear function of plasma emission lines monitored during co-sputtering from elemental Zn and Ti targets in an N 2 and Ar atmosphere. A Bayesian control algorithm, informed by OES, navigates the space of sputtering power to fabricate films with user-defined compositions by minimizing the absolute error between desired and measured optical emission signals. We validated our approach by autonomously fabricating Zn x Ti 1-x N y films that deviate from the targeted cation composition by a relative ±3.5%, even for 15 nm thin films, demonstrating that the proposed approach can reliably synthesize thin films with a specific composition and minimal human interference. Moreover, the proposed method can be extended to more difficult synthesis experiments where plasma intensity lines depend non-linearly on pressure, or the elemental sticking coefficients strongly depend on the substrate temperature.

36 MATERIALS SCIENCE↗

The Dark Energy Survey Year 3 high-redshift sample: selection, characterization, and analysis of galaxy clustering

ABSTRACT The fiducial cosmological analyses of imaging surveys like DES typically probe the Universe at redshifts z < 1. We present the selection and characterization of high-redshift galaxy samples using DES Year 3 data, and the analysis of their galaxy clustering measurements. In particular, we use galaxies that are fainter than those used in the previous DES Year 3 analyses and a Bayesian redshift scheme to define three tomographic bins with mean redshifts around z ∼ 0.9, 1.2, and 1.5, which extend the redshift coverage of the fiducial DES Year 3 analysis. These samples contain a total of about 9 million galaxies, and their galaxy density is more than 2 times higher than those in the DES Year 3 fiducial case. We characterize the redshift uncertainties of the samples, including the usage of various spectroscopic and high-quality redshift samples, and we develop a machine-learning method to correct for correlations between galaxy density and survey observing conditions. The analysis of galaxy clustering measurements, with a total signal to noise S/N ∼ 70 after scale cuts, yields robust cosmological constraints on a combination of the fraction of matter in the Universe Ωm and the Hubble parameter h, $\Omega _m h = 0.195^{+0.023}_{-0.018}$, and 2–3 per cent measurements of the amplitude of the galaxy clustering signals, probing galaxy bias and the amplitude of matter fluctuations, bσ8. A companion paper (in preparation) will present the cross-correlations of these high-z samples with cosmic microwave background lensing from Planck and South Pole Telescope, and the cosmological analysis of those measurements in combination with the galaxy clustering presented in this work.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Evaluating 3 decades of precipitation in the Upper Colorado River basin from a high-resolution regional climate model

Abstract. Convection-permitting regional climate models (RCMs) have recently become tractable for applications at multi-decadal timescales. These types of models have tremendous utility for water resource studies, but better characterization of precipitation biases is needed, particularly for water-resource-critical mountain regions, where precipitation is highly variable in space, observations are sparse, and the societal water need is great. This study examines 34 years (1987–2020) of RCM precipitation from the Weather Research and Forecasting model (WRF; v3.8.1), using the Climate Forecast System Reanalysis (CFS; CFSv2) initial and lateral boundary conditions and a 1 km × 1 km innermost grid spacing. The RCM is centered over the Upper Colorado River basin, with a focus on the high-elevation, 750 km2 East River watershed (ERW), where a variety of high-impact scientific activities are currently ongoing. Precipitation is compared against point observations (Natural Resources Conservation Service Snow Telemetry or SNOTEL), gridded climate datasets (Newman, Livneh, and PRISM), and Bayesian reconstructions of watershed mean precipitation conditioned on streamflow and high-resolution snow remote-sensing products. We find that the cool-season precipitation percent error between WRF and 23 SNOTEL gauges has a low overall bias (x^ = 0.25 %, s = 13.63 %) and that WRF has a higher percent error during the warm season (x^ = 10.37 %, s = 12.79 %). Warm-season bias manifests as a high number of low-precipitation days, though the low-resolution or SNOTEL gauges limit some of the conclusions that can be drawn. Regional comparisons between WRF precipitation accumulation and three different gridded datasets show differences on the order of ± 20 %, particularly at the highest elevations and in keeping with findings from other studies. We find that WRF agrees slightly better with the Bayesian reconstruction of precipitation in the ERW compared to the gridded precipitation datasets, particularly when changing SNOTEL densities are taken into account. The conclusions are that the RCM reasonably captures orographic precipitation in this region and demonstrates that leveraging additional hydrologic information (streamflow and snow remote-sensing data) improves the ability to characterize biases in RCM precipitation fields. Error characteristics reported in this study are essential for leveraging the RCM model outputs for studies of past and future climates and water resource applications. The methods developed in this study can be applied to other watersheds and model configurations. Hourly 1 km × 1 km precipitation and other meteorological outputs from this dataset are publicly available and suitable for a wide variety of applications.

Geology↗

Multilevel Logistic Regression with Random Slope for Community Annoyance Survey Data

This paper documents recent dose-response modeling work at NASA in anticipation of follow-on work by a contactor. Specifically, this paper compares the results of a Bayesian MLR model with a fixed slope to one with a random slope using WSPR and QSF18 data. Previously reported dose-response modeling efforts of WSPR and QSF18 data have used a MLR model with a fixed slope term. A random slope may more accurately depict the dose-response relationship of individuals in the efforts to produce a population summary dose-response curve. Results of a fixed versus random slope model with WSPR and QSF18 data indicate minimal difference between the modeling methods. The simpler fixed slope model is preferable for these data, but these results do not preclude consideration of a random slope term in modeling efforts of future X-59 community test data.

dose-response↗

Setting the Bar for the Replacement of the Probability of Collision Metric

To date, satellite conjunction assessment (CA) risk analysis has largely embraced the probability of collision (Pc) as the omnibus metric to evaluate collision likelihood, and its use in such assessments has mostly been straightforward: at the point at which a conjunction mitigation decision is required, the calculated Pc is compared to a threshold; and if the calculated Pc exceeds that threshold, then a mitigation action is warranted. With only minor variation, this approach is employed by major CA risk assessment centers (e.g., NASA, EUSST, CNES, JAXA) and is advanced as the preferred method in the published CA best practices handbooks. Despite this near unanimity of operational practice, there is a major strain of secondary literature critical of the Pc and willing to propose alternatives. Alfano (2005) pointed out the ability of the Pc to underrepresent the risk in certain situations and counselled a maximum Pc construct. Carpenter (2017, 2019) reiterated this criticism and proposed using instead a confidence interval on the miss distance. Balch et al. (2019) identified what they argued was a defect in the entire Bayesian Pc construct and believed that the use of a more conservative methodology based on covariance ellipsoid overlap was necessary. Delande (2022) introduced the framework of collision “plausibility” to the risk assessment process and sketched out how this might be used operationally. Elkantassi (2022) published a full development of the miss distance confidence interval approach and applied it to several worked examples. While these different approaches to collision risk assessment do differ in their details, they all converge on two central points: first, the Pc’s failure to give an adequate expression of the risk in dilution region situations is a fatal flaw; and second, a conjunction should be presumed risky and in need of mitigation until the evidence of the situation can establish otherwise. These criticisms, if correct, would counsel a number of modifications to current CA operational practice; as such, they force a re-examination of fundamental aspects of the CA problem, including the following: 1. Is the CA risk assessment a probability problem, a statistics problem, or something else? 2. If it is a statistics problem, does it lend itself naturally to a hypothesis test construction? 3. If it can be construed as a hypothesis test, what form should the null hypothesis take, to wit: what constraints exist on the choice of the null hypothesis, what selections are in best alignment with all of the attendant parameters of the problem, and what is implied philosophically by different choices? 4. What are the implications of using the different proposed risk assessment parameters for CA? This question should be answered both in determining how frequently the dilution region situation cited by the critics of the Pc actually appears in an operationally significant manner and the missed detection and false alarm rates of all of the proposed risk assessment metrics, compared both to the Pc and to each other. This paper explores and offers preliminary answers to the above questions, presenting a researched treatment of the philosophical nature of the CA problem and the null hypothesis choice that achieves the greatest consistency with all of the different aspects of operational CA conduct. It then profiles all of the different proposed risk assessment metrics enumerated in the earlier paragraph against an extremely large database of conjunction events at both the 550km and 700km altitudes. The combination of the philosophical exploration of the CA problem and the results of the profiling activity articulates what a risk assessment metric will need to demonstrate, in terms of both innate construction and performance, in order to be a true competitor to the Pc.

conjunction assessment↗

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗

Probabilistic constrained Bayesian inversion for transpiration cooling

Abstract To enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant required to ensure a safe operation. To study this, we introduce an approach that determines the posterior probability distribution of the Reynolds number using an inverse problem and constraining the maximum temperature of the system under parameter uncertainties. Mathematically, this chance inequality constraint is dealt with by a generalized polynomial chaos expansion of the system. The posterior distribution will be evaluated by different Markov chain Monte Carlo based methods. A novel method for the constrained case is proposed and tested among others on two‐dimensional transpiration cooling models.

97 MATHEMATICS AND COMPUTING↗

Machine learning-assisted identification of potential sources of bias in measurements of prompt-fission neutron spectra

Unrecognized sources of uncertainty (USU) can bias the reported mean and/or covariance of experimental nuclear data. These biases, in turn, can propagate through evaluated nuclear data to application simulations or may poorly inform nuclear theory that is fitted to the experimental data. Such unknown sources of bias must be tied to the inherent physical constituents of the measurements such as the characteristics of a detector response or a background reduction technique. Here, in this article, a sparse Bayesian learning model is used to support experts in their efforts to identify and characterize USU in experimental prompt fission neutron spectra (PFNS) for spontaneous fissioning of 252 Cf by linking observed biases to features of the measurement system. Three different bias components were found. The first acts as a verification case for the algorithm as it identifies a bias coming from a well-known source related to the use of 6 Li in the neutron detection system. The second two cases demonstrate how this method can benefit the evaluation of experimental nuclear data by identifying, quantifying, and relating unknown biases to potential causes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Application of a Geochemically Informed Leak Detection (GILD) Model to CO 2 Injection Sites on the United States Gulf Coast

The Gulf Coast region possesses great potential for CO 2 enhanced oil recovery (EOR) and CO 2 storage. A geochemically informed leak detection (GILD) model has been applied to CO 2 injection sites on the Gulf Coast with considerations of measurement variability. The Jasper aquifer in Montgomery County, Texas, was chosen to demonstrate the method. Based on background data from wells in the area, combinations of mineral and fluid compositions were used to create 23 scenarios for the geochemical model. The output from the geochemical model was used to identify sensitive monitoring species, and response functions were generated for these as a function of the CO 2 leakage concentration. The sources of measurement variability for background conditions were characterized from the Jasper aquifer background data, and then normalized using the coefficient of variation of each species across the monitoring wells. Bayesian belief network (BBN) models were constructed, and measurement variability of different levels were added to compare leak detection probabilities. Increasing measurement variability decreased the power to detect a leak of a given size. For a moderately high CO 2 concentration of 0.2 mol/kg, the probability of detecting this leakage effect using pH as the monitoring variable in an aquifer with calcite decreases from 98% (no measurement variability) to 61% (medium variability) to 33% (high variability). The loss in power of the sampling protocol with increasing measurement variability is similar in magnitude when Ca 2+ or HCO 3 - is used as the monitoring parameter, but only for aquifers with calcite.

58 GEOSCIENCES↗

Exploratory analysis of machine learning techniques in the Nevada geothermal play fairway analysis

Play fairway analysis (PFA) is commonly used to generate geothermal potential maps and guide exploration studies, with a particular focus on locating and characterizing blind geothermal systems. This study evaluates the application of machine learning techniques to PFA in the Great Basin region of Nevada. Following the evaluation of various techniques, we identified two approaches to PFA that produced promising results, 1) supervised Bayesian probabilistic neural networks to generate geothermal potential maps with confidence intervals, and 2) unsupervised principal component analysis paired with k-means clustering to generate both cluster maps to help identify spatial patterns, as well as new combined feature inputs. We applied these techniques to perform a comparative analysis between two principal sets of geological and geophysical features related to permeability and heat and a set of positive (known geothermal resources) and negative training sites (known drill sites with unsuitable geothermal conditions). We found that these methods constrain previously unrecognized feature controls on geothermal favorability, many of which are spatially organized within the extent of cluster groups and the major structural-hydrologic domains of the study area. Furthermore, we utilized exploratory unsupervised modeling to highlight spatial relationships between input data and predictive output results of our supervised modeling. As a result, we demonstrate how our models compare to the previous Nevada PFA and how the rapid insights these machine learning techniques offer may support future assessments of both known and undiscovered blind geothermal systems in the Great Basin region of Nevada and beyond.

15 GEOTHERMAL ENERGY↗

"Sintering" Models and In-Situ Experiments: Data Assimilation for Microstructure Prediction in SLS Additive Manufacturing of Nylon Components

Selective laser sintering methods are workhorses for additively manufacturing polymer-based components. The ease of rapid prototyping also means it is easy to produce illicit components. It is necessary to have a data-calibrated in-situ physical model of the build process in order to predict expected and defective microstructure characteristics that inform component provenance. Toward this end, sintering models are calibrated and characteristics such as component defects are explored. This is accomplished by assimilating multiple data streams, imaging analysis, and computational model predictions in an adaptive Bayesian parameter estimation algorithm. From these data sources, along with a phase-field model, bulk porosity distributions are inferred. Model parameters are constrained to physically-relevant search directions by sensitivity analysis, and then matched to predictions using adaptive sampling. Using this feedback loop, data-constrained estimates of sintering model parameters along with uncertainty bounds are obtained.

3D printing, Additive Manufacturing, polymer, sint↗

Reweighting the quark Sivers function with STAR jet data

The Bayesian reweighting procedure is applied for the first time to a TMD distribution, the quark Sivers function extracted from SIDIS data. By exploiting the recent published single spin asymmetry data for the inclusive jet production in p^\uparrow p p ↑ p collisions from the STAR collaboration at RHIC, we show how such a procedure allows to incorporate the information contained in the new data set, without the need of re-fitting, and to explore a much wider x x region compared to SIDIS measurements. The reweighting method is also extended to the case of asymmetric errors, and the results show a significant improvement on the knowledge of the quark Sivers function.

Flore, Carlo↗

Sequential Bayesian Experimental Design for Calibration of Expensive Simulation Models

Simulation models of critical systems often have parameters that need to be calibrated using observed data. For expensive simulation models, calibration is done using an emulator of the simulation model built on simulation output at different parameter settings. Using intelligent and adaptive selection of parameters to build the emulator can drastically improve the efficiency of the calibration process. The article proposes a sequential framework with a novel criterion for parameter selection that targets learning the posterior density of the parameters. The emergent behavior from this criterion is that exploration happens by selecting parameters in uncertain posterior regions while simultaneously exploitation happens by selecting parameters in regions of high posterior density. Furthermore, the advantages of the proposed method are illustrated using several simulation experiments and a nuclear physics reaction model.

97 MATHEMATICS AND COMPUTING↗