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At least 1,261 records · Page 70

Deep learning insights into cosmological structure formation

The evolution of linear initial conditions present in the early Universe into extended halos of dark matter at late times can be computed using cosmological simulations. However, a theoretical understanding of this complex process remains elusive; in particular, the role of anisotropic information in the initial conditions in establishing the final mass of dark matter halos remains a long-standing puzzle. Here, we build a deep learning framework to investigate this question. We train a three-dimensional convolutional neural network to predict the mass of dark matter halos from the initial conditions, and quantify in full generality the amounts of information in the isotropic and anisotropic aspects of the initial density field about final halo masses. We find that anisotropies add a small, albeit statistically significant amount of information over that contained within spherical averages of the density field about final halo mass. However, the overall scatter in the final mass predictions does not change qualitatively with this additional information, only decreasing from 0.9 dex to 0.7 dex. Given such a small improvement, our results demonstrate that isotropic aspects of the initial density field essentially saturate the relevant information about final halo mass. Therefore, instead of searching for information directly encoded in initial conditions anisotropies, a more promising route to accurate, fast halo mass predictions is to add approximate dynamical information based e.g. on perturbation theory. More broadly, our results indicate that deep learning frameworks can provide a powerful tool for extracting physical insight into cosmological structure formation. Published by the American Physical Society 2024

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness

We report the accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of the modulus of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. Although most of the analysis is general and not specific to neural networks, we validate our theoretical assumptions and results numerically for neural networks by several data sets of images. The numerical results confirm that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. We also observe a clear consistency between test loss and neural network smoothness during the training process. In addition, we demonstrate empirically that the neural network smoothness decreases when the network size increases whereas the smoothness is insensitive to training dataset size.

97 MATHEMATICS AND COMPUTING↗

Surrogate modeling for efficiently, accurately and conservatively estimating measures of risk

We present a surrogate modeling framework for conservatively estimating measures of risk from limited realizations of an expensive physical experiment or computational simulation. Risk measures combine objective probabilities with the subjective values of a decision maker to quantify anticipated outcomes. Given a set of samples, we construct a surrogate model that produces estimates of risk measures that are always greater than their empirical approximations obtained from the training data. These surrogate models limit over-confidence in reliability and safety assessments and produce estimates of risk measures that converge much faster to the true value than purely sample-based estimates. We first detail the construction of conservative surrogate models that can be tailored to a stakeholder’s risk preferences and then present an approach, based on stochastic orders, for constructing surrogate models that are conservative with respect to families of risk measures. Our surrogate models include biases that permit them to conservatively estimate the target risk measures. We provide theoretical results that show that these biases decay at the same rate as the L 2 error in the surrogate model. Numerical demonstrations confirm that risk-adapted surrogate models do indeed overestimate the target risk measures while converging at the expected rate.

42 ENGINEERING↗

A portrait of the Higgs boson by the CMS experiment ten years after the discovery

In July 2012, the ATLAS and CMS collaborations at the CERN Large Hadron Collider announced the observation of a Higgs boson at a mass of around 125 gigaelectronvolts. Ten years later, and with the data corresponding to the production of a 30-times larger number of Higgs bosons, we have learnt much more about the properties of the Higgs boson. The CMS experiment has observed the Higgs boson in numerous fermionic and bosonic decay channels, established its spin–parity quantum numbers, determined its mass and measured its production cross-sections in various modes. Here the CMS Collaboration reports the most up-to-date combination of results on the properties of the Higgs boson, including the most stringent limit on the cross-section for the production of a pair of Higgs bosons, on the basis of data from proton–proton collisions at a centre-of-mass energy of 13 teraelectronvolts. Within the uncertainties, all these observations are compatible with the predictions of the standard model of elementary particle physics. Much evidence points to the fact that the standard model is a low-energy approximation of a more comprehensive theory. Several of the standard model issues originate in the sector of Higgs boson physics. An order of magnitude larger number of Higgs bosons, expected to be examined over the next 15 years, will help deepen our understanding of this crucial sector.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Stationary Bragg reflection of laser light in inhomogeneous absorbing plasmas inside inertial confinement fusion Hohlraums

Laser-produced plasma in inertial confinement fusion (ICF) Hohlraums are marked with density non-uniformity whose length scale can go down to micrometers. This scale is of the order of the laser wavelength. The WKB approximation, which is classically used in radiation-hydrodynamic codes to compute the laser trajectory, cannot correctly take into account such small-scale inhomogeneity of the plasma. Going beyond this approximation, we predict a novel mechanism for the laser reflection. Here we show that an electromagnetic plane wave with wave number k resonates with the k B =2 k Fourier component of a multimode perturbation of the background density and generates a reflected wave. It is the first time that this reflection is considered for stationary inhomogeneous ICF plasmas, and the energy absorption is taken into account. This mechanism, which is a form of Bragg reflection, can occur away from the critical surface and generate a drift of the location of the laser absorption. Furthermore, this absorption will be periodically modulated with a k B wave number. The stationary Bragg reflection can explain ongoing discrepancies between experimental and numerical data about laser trajectory and absorption in ICF Hohlraums.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Hamiltonian switching control of noisy bipartite qubit systems

Abstract We develop a Hamiltonian switching ansatz for bipartite control that is inspired by the quantum approximate optimization algorithm, to mitigate environmental noise on qubits. We demonstrate the control for a central spin coupled to bath spins via isotropic Heisenberg interactions, and then make physical applications to the protection of quantum gates performed on superconducting transmon qubits coupling to environmental two-level-systems (TLSs) through dipole-dipole interactions, as well as on such qubits coupled to both TLSs and a Lindblad bath. The control field is classical and acts only on the system qubits. We use reinforcement learning with policy gradient to optimize the Hamiltonian switching control protocols, using a fidelity objective for specific target quantum gates. We use this approach to demonstrate effective suppression of both coherent and dissipative noise, with numerical studies achieving target gate implementations with fidelities over 0.9999 (four nines) in the majority of our test cases and showing improvement beyond this to values of 0.999 999 999 (nine nines) upon a subsequent optimization by GRadient Ascent Pulse Engineering (GRAPE). We analyze how the control depth, total evolution time, number of environmental TLS, and choice of optimization method affect the fidelity achieved by the optimal protocols and reveal some critical behaviors of bipartite control of quantum gates.

Physics↗

Geologic Characterization of the South Georgia Rift Basin for Source Proximal CO2 Storage

The project Geologic Characterization of the South Georgia Rift Basin for Source Proximal CO2 Storage is one of 9 site characterization projects that were implemented as part of ARRA (American Recovery and Reinvestment Act). Data from this project was used to improve resolution of data in NATCARB in the area of study. Data related to this study has already been incorporated in NATCARB Atlas. The South Carolina Research Foundation and partners evaluated the feasibility of CCS in the Jurassic/ Triassic (J / TR) saline formations of the buried Mesozoic South Georgia Rift (SGR) Basin that extends from South Carolina into Georgia. The J / TR sequence, based on preliminary assessment of limited geologic and geophysical data, appears to have both the appropriate areal extent and multiple horizons to permanently and safely store CO2 The presence of several igneous rock layers within the sequence may potentially provide adequate seals to prevent upward CO2 migration into the Coastal Plain aquifer systems. Approximately 81 kilometers of 2-D seismic reflection data were collected by Bay Geophysical, Inc. to explore a portion of the SGR located in southern Georgia. The 81 kilometers were divided into two lines approximately 40.5 kilometers each, with Line 1 intersecting Georgia well GGS 3457. Line 2 intersects Line 1 at the southern portion of Line 1 to maximize the extent of coverage away from GGS-3457 (a deep well drilled in the 1980s for oil and gas exploration). This well had a set of usable logs, including gamma and neutron logs that provided promising results related to CO2 storage. Results showed sandstone with porosity values greater than 10 percent and a thickness of 120 meters. The design of the seismic shot was to extrapolate information away from the well and to better define the extent of the SGR and the necessary reservoir and caprock for a successful CO2 injection. A numerical simulation model of CO2 Injection and migration was developed based on the geology log for the GGS-3457 well. The simulation model was used to investigate the feasibility of injecting 30 million metric tons of CO2 into SGR J / TA sediments and integrity of the diabase layers as seals to prevent CO2 migration.

2-D seismic↗

Arctic observations and numerical simulations of surface wind effects on Multi-Angle Snowflake Camera measurements

Abstract. Ground-based measurements of frozen precipitation are heavily influenced by interactions of surface winds with gauge-shield geometry. The Multi-Angle Snowflake Camera (MASC), which photographs hydrometeors in free-fall from three different angles while simultaneously measuring their fall speed, has been used in the field at multiple midlatitude and polar locations both with and without wind shielding. Here, we present an analysis of Arctic field observations – with and without a Belfort double Alter shield – and compare the results to computational fluid dynamics (CFD) simulations of the airflow and corresponding particle trajectories around the unshielded MASC. MASC-measured fall speeds compare well with Ka-band Atmospheric Radiation Measurement (ARM) Zenith Radar (KAZR) mean Doppler velocities only when winds are light (≤5ms-1) and the MASC is shielded. MASC-measured fall speeds that do not match KAZR-measured velocities tend to fall below a threshold value that increases approximately linearly with wind speed but is generally <0.5ms-1. For those events with wind speeds ≤1.5ms-1, hydrometeors fall with an orientation angle mode of 12∘ from the horizontal plane, and large, low-density aggregates are as much as 5 times more likely to be observed. Simulations in the absence of a wind shield show a separation of flow at the upstream side of the instrument, with an upward velocity component just above the aperture, which decreases the mean particle fall speed by 55 % (74 %) for a wind speed of 5 m s−1 (10 m s−1). We conclude that accurate MASC observations of the microphysical, orientation, and fall speed characteristics of snow particles require shielding by a double wind fence and restriction of analysis to events where winds are light (≤5ms-1). Hydrometeors do not generally fall in still air, so adjustments to these properties' distributions within natural turbulence remain to be determined.

54 ENVIRONMENTAL SCIENCES↗

Physics–Informed Neural Networks of the Saint–Venant Equations for Downscaling a Large–Scale River Model

Large-scale river models are being refined over coastal regions to improve the scientific understanding of coastal processes, hazards and responses to climate change. However, coarse mesh resolutions and approximations in physical representations of tidal rivers limit the performance of such models at resolving the complex flow dynamics near the river-ocean interface, resulting in inaccurate simulations of flood inundation. In this research, we propose a machine learning (ML) framework based on the state-of-the-art physics-informed neural network (PINN) to simulate the downscaled flow at the subgrid scale. First, we demonstrate that PINN is able to assimilate observations of various types and solve the one-dimensional (1-D) Saint-Venant equations (SVE) directly. We perform the flow simulations over a floodplain and along an open channel in several synthetic case studies. The PINN performance is evaluated against analytical solutions and numerical models. Our results indicate that the PINN solutions of water depth have satisfactory accuracy with limited observations assimilated. In the case of flood wave propagation induced by storm surge and tide, a new neural network architecture is proposed based on Fourier feature embeddings that seamlessly encodes the periodic tidal boundary condition in the PINN's formulation. Furthermore, we show that the PINN-based downscaling can produce more reasonable subgrid solutions of the along-channel water depth by assimilating observational data. The PINN solution outperforms the simple linear interpolation in resolving the topography and dynamic flow regimes at the subgrid scale. This study provides a promising path towards improving emulation capabilities in large-scale models to characterize fine-scale coastal processes.

54 ENVIRONMENTAL SCIENCES↗

Source term method for binary neutron stars initial data

The initial condition problem for a binary neutron star system requires a Poisson equation solver for the velocity potential with a Neumann-like boundary condition on the surface of the star. Difficulties that arise in this boundary value problem are: (a) the boundary is not known a priori, but constitutes part of the solution of the problem; (b) various terms become singular at the boundary. In this work, we present a new method to solve the fluid Poisson equation for irrotational/spinning binary neutron stars. The advantage of the new method is that it does not require complex fluid surface fitted coordinates and it can be implemented in a Cartesian grid, which is a standard choice in numerical relativity calculations. This is accomplished by employing the source term method proposed by Towers, where the boundary condition is treated as a jump condition and is incorporated as additional source terms in the Poisson equation, which is then solved iteratively. The issue of singular terms caused by vanishing density on the surface is resolved with an additional separation that shifts the computation boundary to the interior of the star. We present two-dimensional tests to show the convergence of the source term method, and we further apply this solver to a realistic three-dimensional binary neutron star problem. By comparing our solution with the one coming from the initial data solver cocal, we demonstrate agreement to approximately 1%. We report our method can be used in other problems with non-smooth solutions like in magnetized neutron stars.

79 ASTRONOMY AND ASTROPHYSICS↗

Evaluating large scale aqueous organic redox flow battery performance with a hybrid numerical and machine learning framework

Aqueous organic redox flow battery (AORFB) is a promising cost-competitive technology for large-scale energy storage. Among existing work, the dihydroxyphenazine (DHP)-based AORFB has demonstrated high energy density and low-capacity degradation in 10 cm$^2$ cells during lab tests. However, its commercial-scale performance in more complex environments remains unknown, posing a barrier to commercialization. To address this gap, this work presents a comprehensive performance evaluation of a 780 cm$^2$ DHP-based AORFB by combining a physics-based numerical model, machine learning (ML)-based surrogate models, and ML-derived sensitivity quantification. Specifically, we first select 12 key battery parameters that include 10 physicochemical and 2 operation quantities, then select 6 performance metrics that include energy efficiency (EE), discharging capacity, charging energy, and power losses due to concentration, activation, and ohmic over-potentials. With such selection, 12800 combinations of the 12 parameters are subsequently generated using the Latin Hypercube Sampling method. Such combinations, together with 38 pre-defined State of Charge, are then integrated to a validated AORFB model developed in COMSOL to compute the performance metrics. With both input parameters and performance metrics, 60 deep neural network (DNN) surrogate models are then trained to approximate the relationship between the 10 physicochemical quantities and 6 performance metrics at each flow rate and current density. Sensitivity scores are then calculated based on the DNN models. Two additional sensitivity analysis tools, i.e., MARS, and SHAP, are also used to cross-validate the sensitivity scores from the DNN. The results demonstrate that 1) the standard potential ranks first in controlling EE and charging energy, 2) the membrane conductivity is most critical for power loss and EE, and 3) specific area and reaction rate control activation power loss.

25 ENERGY STORAGE↗

Survey of baffle plate samples, specimen machining, and hydrogen measurements

This report outlines preliminary results on the survey and characterization of irradiated baffle plate, baffle former, and flux thimble tube specimens as part of the ongoing investigation into irradiation-induced embrittlement in austenitic stainless steels at ambient (room temperature) conditions. The specimens originated from commercial pressurized water reactor (PWR) components, covering displacement damage doses ranging from approximately 0.065 to 75 dpa. Initial scanning electron microscopy (SEM) surveys revealed that specimen surfaces exhibited fine machining marks and in-service-formed oxide layers on the side surfaces of the analyzed specimens. The oxide layers revealed specific features resembling localized "pitting-like" corrosion. An additional set of specimens representing in-service oxidation is being preserved for future microstructure analysis work. As believed, this will provide additional insights into long-term material degradation in PRWs. Specimen machining challenges emerged due to a complex failure of the electric discharge machining (EDM) system located in hot area. Despite partial restoration, issues persist with the EDM’s secondary power supply, necessitating the exploration of alternative EDM or computer numerical control (CNC) machining approaches to facilitate tensile specimen preparation. Currently, low speed saw cutting is in progress to prepare specimens for hydrogen measurements and general microstructure analysis. The near-term goals include completing tensile specimen machining for advanced mechanical testing and characterizing fracture mechanisms, and stress-corrosion cracking testing, ultimately aiming to identify and mitigate the ambient-condition intergranular cracking through targeted post-irradiation annealing strategies.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Estimating Residual Gas Beam Loss in the BNL AGS Booster:Model, Numerical Tools, and Implications for a Polarimeter Chamber

Beamloss from residual gas scattering is calculated for a storage ring of circumference C = 200m, covering a linear magnetic rigidity ramp from B ρ inj = 1.25Tm to B ρ ext = 9.50Tm over tramp = 1.00s. Three loss mechanisms are considered: nuclear inelastic reactions, single Coulomb scattering beyond the acceptance angle of the machine, and charge-state exchange for heavy ions. Calculations are performed for protons and Au 32+ ions, the latter being the charge state delivered by the EBIS pre-injector into the BNL AGS Booster. At a baseline vacuum of P 0 = 5 × 10 −11 mbar the integrated fractional beam loss over the full ramp is negligible for protons (< 0.001%) and 33.1% for Au 32+ . The impact of a 1m polarimeter section at elevated pressure P s = 5 × 10 −9 mbar is evaluated: the proton loss remains negligible, while the Au 32+ loss rises to 45.1%, establishing the vacuum requirement for the polarimeter chamber. In the present compact approximate model, Au 32+ losses are dominated by charge exchange at injection, while proton losses are dominated by single Coulomb scattering. The design, outgassing budget, pumping strategy, and commissioning sequence for a UHV polarimeter chamber housing silicon detectors and ASICs are described in detail.

43 PARTICLE ACCELERATORS↗

VAN-DAMME: GPU-accelerated and symmetry-assisted quantum optimal control of multi-qubit systems

We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains.

97 MATHEMATICS AND COMPUTING↗

A hybrid numerical and machine learning framework for evaluating the performance of a 780 cm 2 aqueous organic redox flow battery

Aqueous organic redox flow battery (AORFB) is a promising cost-competitive technology for large-scale energy storage. Among existing work, the dihydroxyphenazine (DHP)-based AORFB has demonstrated high energy density and low capacity degradation in 10 cm2 cells during lab tests. However, its commercial-scale performance in more complex environments remains unknown, posing a barrier for commercialization. To address this gap, this work presents a comprehensive performance evaluation of a 780 cm 2 DHP-based AORFB by combining physics-based numerical model, machine learning (ML)-based surrogate models, and ML-derived sensitivity quantification. Specifically, we first select 12 key battery parameters that include 10 physicochemical quantities and 2 operation quantities, then select 6 performance metrics that include energy efficiency (EE), discharging capacity, charging energy, and power losses due to concentration, activation, and ohmic over-potentials. With such selection, 12800 combinations of the 12 parameters are subsequently generated using the Latin Hypercube Sampling method. These combinations, together with 38 pre-defined State of Charge, are then integrated to a validated AORFB model developed in COMSOL to compute the performance metrics. With both input parameters and performance metrics, 60 deep neural network (DNN) surrogate models are then trained to approximate the relationship between the 10 physicochemical quantities and 6 performance metrics at each flow rate and current density. Sensitivity scores are then calculated based on the DNN models. Two additional sensitivity analysis tools, i.e., MARS, and SHAP, are also used to cross-validate the sensitivity scores from the DNN. The results demonstrate that 1) the standard potential ranks the first in controlling EE and charging energy, 2) the membrane conductivity is most critical for power loss and EE, and 3) specific area and reaction rate control activation power loss.

25 ENERGY STORAGE↗

Theory of the tertiary instability and the Dimits shift within a scalar model

The Dimits shift is the shift between the threshold of the drift-wave primary instability and the actual onset of turbulent transport in a magnetized plasma. It is generally attributed to the suppression of turbulence by zonal flows, but developing a more detailed understanding calls for consideration of specific reduced models. The modified Terry–Horton system has been proposed by St-Onge ( J. Plasma Phys. , vol. 83, 2017, 905830504) as a minimal model capturing the Dimits shift. Here, we use this model to develop an analytic theory of the Dimits shift and a related theory of the tertiary instability of zonal flows. We show that tertiary modes are localized near extrema of the zonal velocity $U(x)$ , where $x$ is the radial coordinate. By approximating $U(x)$ with a parabola, we derive the tertiary-instability growth rate using two different methods and show that the tertiary instability is essentially the primary drift-wave instability modified by the local $U'' \doteq {\rm d}^2 U/{\rm d} x^2 $ . Then, depending on $U''$ , the tertiary instability can be suppressed or unleashed. The former corresponds to the case when zonal flows are strong enough to suppress turbulence (Dimits regime), while the latter corresponds to the case when zonal flows are unstable and turbulence develops. This understanding is different from the traditional paradigm that turbulence is controlled by the flow shear $| {\rm d} U / {\rm d} x |$ . Our analytic predictions are in agreement with direct numerical simulations of the modified Terry–Horton system.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Progress Report on Model Development for the Transport of Aerosol through Microchannels

This report summarizes the current progress in the development of a phenomenological model of aerosol transport, deposition, and plugging through microchannels. The purpose is to introduce a generic, reliable numerical model for the prediction of aerosol transport, deposition, and plugging in leak paths while accounting for potential plugging formation, to a user community involving researchers, regulators, and industry. In that regard, a Graphical User Interface (GUI) was generated by integrating the individual MATLAB scripts that make up the model and adding additional features to aid the general understanding of the problem to a user. This report focuses on the model development and features included. Further, predictions from the GUI are compared with experimental data for validation. The strength of the GUI is the ability of a user to plug in basic parameters such as the initial pressure conditions and canister model specifications in order to obtain a first principal approximation of vital information such as the blowdown pressure differential, aerosol penetration, and deposition as a function of time. The user can obtain this without the know-how of the underlying MATLAB scripts and thus enables the model to be readily applied by regulators, industry, and shareholders to reduce the uncertainty in the off-site radiological consequences. The report also lists future tasks under this work scope. This includes ongoing efforts to include additional crack geometries (divergent and divergent-convergent slots) to the model; GUI development and improvement based on feedback from users, and integration of aerosol source term data from Sibling Pin tests into the model as we approach realistic canister stress corrosion cracking–induced aerosol release scenarios.

54 ENVIRONMENTAL SCIENCES↗

Dark Energy Survey Year 3 Results: Multi-Probe Modeling Strategy and Validation

This paper details the modeling pipeline and validates the baseline analysis choices of the DES Year 3 joint analysis of galaxy clustering and weak lensing (a so-called "3$\times$2pt" analysis). These analysis choices include the specific combination of cosmological probes, priors on cosmological and systematics parameters, model parameterizations for systematic effects and related approximations, and angular scales where the model assumptions are validated. We run a large number of simulated likelihood analyses using synthetic data vectors to test the robustness of our baseline analysis. We demonstrate that the DES Year 3 modeling pipeline, including the calibrated scale cuts, is sufficiently accurate relative to the constraining power of the DES Year 3 analyses. Our systematics mitigation strategy accounts for astrophysical systematics, such as galaxy bias, intrinsic alignments, source and lens magnification, baryonic effects, and source clustering, as well as for uncertainties in modeling the matter power spectrum, reduced shear, and estimator effects. We further demonstrate excellent agreement between two independently-developed modeling pipelines, and thus rule out any residual uncertainties due to the numerical implementation.

79 ASTRONOMY AND ASTROPHYSICS↗