High-Order Finite Element Second Moment Methods for Linear Transport
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73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
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A Second Moment Method for k -Eigenvalue Acceleration with Continuous Diffusion and Discontinuous Transport Discretizations
The second moment method is a linear acceleration technique that couples the transport equation to a diffusion equation with transport-dependent additive closures. The resulting low-order diffusion equation can be discretized independent of the transport discretization, unlike diffusion synthetic acceleration, and is symmetric positive definite, unlike quasidiffusion. While this method has been shown to be comparable to quasidiffusion in iterative performance for fixed source and time-dependent problems, it is largely unexplored as an eigenvalue problem acceleration scheme due to the belief that the resulting inhomogeneous source makes the problem ill posed. Recently, a preliminary feasibility study was performed on the second moment method for eigenvalue problems. The results suggested comparable performance to quasidiffusion and more robust performance than diffusion synthetic acceleration. This work extends the initial study to more realistic reactor problems using state-of-the-art discretization techniques. Finally, the results in this paper show that the second moment method is more computationally efficient than its alternatives on complex reactor problems with unstructured meshes.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Consistent Second Moment Methods with Scalable Linear Solvers for Radiation Transport
Second moment methods (SMMs) are developed that are consistent with the discontinuous Galerkin spatial discretization of the discrete ordinates (or S\(_N\)) transport equations. The low-order (LO) diffusion system of equations is discretized with fully consistent P\(_1\), local discontinuous Galerkin (LDG), and interior penalty (IP) methods. A discrete residual approach is used to derive SMM correction terms that make each of the LO systems consistent with the high-order discretization. We show that the consistent methods are more accurate and have better solution quality than independently discretized LO systems, that they preserve the diffusion limit, and that the LDG and IP consistent SMMs can be scalably solved in parallel on a challenging, multimaterial benchmark problem.
97 MATHEMATICS AND COMPUTING↗
A hybrid Monte Carlo-deterministic second moment method with efficient variance reduction
In this work, we present a hybrid method that combines Monte Carlo with deterministic finite element methods to solve a linear Boltzmann transport equation. Our hybrid method runs orders of magnitude faster than Monte Carlo, without sacrificing accuracy, for a proxy problem from radiative transfer that contains both optically-thick and optically-thin material. We believe that this is the first demonstration of a hybrid Second Moment Method in more than one spatial dimension, the first to consider more than one material, and the first to use variance reduction. Our variance reduction approach arises from an asymptotic analysis in which we show that the magnitude of the scattering source grows without bound. We transform the problem to compute the deviation of the radiation intensity from isotropy. The magnitude of the source in the transformed problem is bounded, and the quality of the hybrid method solution is dramatically improved by a substantial reduction in the variance.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
A Hybrid Second Moment Method for Thermal Radiative Transfer
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Nuclear science and engineering, Astronomy and Ast↗
A Hybrid Monte Carlo-Deterministic Second Moment Method for Thermal Radiative Transfer
Explore the source record for details and available documents.
Nuclear science and engineering, Astronomy and Ast↗
OpenSn: A massively parallel, open-source simulation environment for discrete ordinates radiation transport
OpenSn is an open-source, massively parallel deterministic radiation transport code for solving the discrete-ordinates ( S N ) form of the Boltzmann transport equation on unstructured, arbitrary polyhedral meshes. It supports high-fidelity simulations involving steady-state, eigenvalue, and adjoint problems for neutral particles (e.g., neutrons, photons, multi-particles), using the multigroup approximation in energy. OpenSn combines angular discretization via discrete ordinates with a discontinuous Galerkin finite element method (DGFEM) in space, enabling accurate resolution of transport physics on arbitrary polyhedral cells, included locally refined spatial grids. It includes multiple angular quadrature types, including locally refined angular quadratures. Written in modern C++ with a Python API, OpenSn runs efficiently on platforms ranging from laptops to supercomputers. The transport sweep algorithm is implemented using a task-based, directed-acyclic-graph (DAG) approach for each angle and supports asynchronous parallelism across thousands of MPI ranks. Group-set aggregation improves compute intensity, and synthetic acceleration techniques (e.g., diffusion synthetic acceleration, second-moment method) enhance solver convergence. OpenSn has been verified on reactor physics problems and demonstrated excellent weak and strong scaling performance on more than 32,768 processes, making it a versatile and robust platform for large-scale transport simulations in complex geometries.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Direct sensitivity analysis on the parameterization of crystal plasticity models
Various methods for calibrating crystal plasticity finite element (CPFE) models lead to non-unique input parameter values, which subsequently introduce uncertainty in the predicted mechanical response. Sensitivity analysis (SA) conducted on crystal plasticity models is used to identify how variability in these parameters contribute to output uncertainty. Traditional SA on CPFE parameters uses simplified surrogate models to save computational time. However, the accuracy of the surrogate models depends on the quantity of training data used, and any modeling error can propagate into the SA results, potentially affecting their reliability. In this work, the elementary effects test (EET) method, a global SA technique using direct CPFE simulations was employed, and the results obtained were compared with the First Order Second Moment (FOSM) method. ExaConstit, an open-source GPU-enabled CPFE code, was used to perform the simulations and direct SA. The EET method was accurately able to capture the non-linear effects of all the input parameters on the output and is a valuable approach for reliably attributing parameter sensitivities in CPFE models. Based on the results, efficient strategies to perform future parameter calibration and SA are discussed. Additionally, the SA trends observed in different single crystal orientations closely mirrored the activity of the slip systems.
Elementary Effects Test↗
An adaptive moments-based interface reconstruction using intersection of the cell with one half-plane, two half-planes and a circle
We present a new adaptive moment-of-fluid (A-MOF) interface reconstruction method. It uses the zeroth, first, and second moments of the fragment of material inside a cell of the mesh to construct a shape that approximates the respective material fragment. The new method requires information about the material moments only for the cell under consideration. The adaptive method chooses between shapes obtained by the intersection of the cell with one half-plane, two half-planes, or a circle. The A-MOF method allows to exactly reproduce several convex shapes: corners, filaments, and their concave cell-complements; as well as pieces of the circles and its cell-compliments. Interface reconstruction is formulated as a local (for each cell), non-linear, equality constrained optimization problem, which does not require additional communication and allows for an efficient parallel implementation. In conclusion, we present an extensive set of test problems, both for interface reconstruction on a single cell, and for reconstruction of a variety of shapes on the entire mesh.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Power Density Distributions in Electron Beams
Measuring and controlling the power density distribution of electron beams used for welding is critical for producing repeatable welds and for transferring welding parameters between different machines. On any given machine, the power density distribution is controlled by defocusing the beam relative to its sharpest focused condition. However, measuring the power density distribution can be difficult due to the intense nature of welding beams and is further complicated by imperfect electron optics that can distort the beam, making it difficult to quantify. The enhanced modified Faraday cup (EMFC) diagnostic method was used here for beam analysis that utilizes computed tomography to reconstruct the beam’s power density distribution. These results were compared to the International Standards Organization (ISO) method for characterizing laser beams using a second-moment D4σ calculation. For ideal Gaussian-shaped beams, both methods would give the same result. However, for imperfect beams, the calculated D4σ diameter was shown to be about 25% larger relative to the FWe2 diameter measured by the EMFC due to the heavier weighting of data in the tails of the beam by D4σ. Finally, although both methods produce repeatable welds, it is important to understand the differences in the reported beam diameters, divergence angles, and beam parameter products when transferring parameters between machines.
36 MATERIALS SCIENCE↗
Beyond the two-point correlation: Constraining primordial non-Gaussianity with density-perturbation moments
Constraining primordial non-Gaussianity (PNG) on the large-scale cosmic structure (LSS) is an important step in understanding properties of the early Universe, specifically in distinguishing between different inflationary models. Measuring PNG relies on evaluating the scale-dependent correlations in the density field. New summary statistics beyond the two- and three-point correlation functions in configuration space and their Fourier-space counterparts, the power- and bispectrum may provide increased sensitivity. We introduce a new method for extracting the PNG signal imprinted on the LSS by using the first three Gaussian moments of the normalized correlation in density perturbations, evaluated on varying distance scales. We aim to assess this method’s sensitivity to local PNG, parameterized by f NL . We performed spherical convolutions on a range of scales on dark-matter-halo simulations to measure the scale-dependent correlations in the density field. From these, we computed the first three moments and compared them to a model expectation vector, parameterized to the second power in f NL . Our method provides about 21% improvement in sensitivity to f NL with respect to using the two-point correlation function alone. Notably, we find that the second moment alone carries nearly as much constraining power as the mean, highlighting the potential of higher order statistics. Given its simplicity and efficiency, this framework is well suited for application to current and upcoming large-scale surveys such as the Dark Energy Spectroscopic Instrument (DESI).
early universe↗
Second Order Closures for the Radiative Transfer Equation: Some Are Unstable
The largest existing simulations of cosmic reionization model radiative transfer with moment methods that require a closure relation. The two most commonly used closure relations are M1 and OTVET; both close the moment hierarchy at the first moment. We explore the properties of a higher, second-order closure. We show that direct generalizations of M1 and OTVET to one higher order are physically unstable - i.e., the closure equations themselves result in unstable solutions, not just their numerical implementation. In fact, a generalization of OTVET to any order higher than the first one is unstable. We are also able to show that any local (i.e., depending only on the local moments of the radiation field, like M1) second-order closure that depends only on the radiation intensity and radiation flux, but does not explicitly depend on the radiation pressure, is physically unstable. This result restricts the choice of possible second-order closure relations.
Gnedin, Nickolay Y. [Fermilab; Chicago U., Astron.↗
Two datasets are better than one: method of double moments for 3D reconstruction in cryo-EM
Cryo-electron microscopy is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments, which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.
Kam’s method↗
Deriving cloud droplet number concentration from surface-based remote sensors with an emphasis on lidar measurements
Abstract. Given the importance of constraining cloud droplet number concentrations (Nd) in low-level clouds, we explore two methods for retrieving Nd from surface-based remote sensing that emphasize the information content in lidar measurements. Because Nd is the zeroth moment of the droplet size distribution (DSD), and all remote sensing approaches respond to DSD moments that are at least 2 orders of magnitude greater than the zeroth moment, deriving Nd from remote sensing measurements has significant uncertainty. At minimum, such algorithms require the extrapolation of information from two other measurements that respond to different moments of the DSD. Lidar, for instance, is sensitive to the second moment (cross-sectional area) of the DSD, while other measures from microwave sensors respond to higher-order moments. We develop methods using a simple lidar forward model that demonstrates that the depth to the maximum in lidar-attenuated backscatter (Rmax) is strongly sensitive to Nd when some measure of the liquid water content vertical profile is given or assumed. Knowledge of Rmax to within 5 m can constrain Nd to within several tens of percent. However, operational lidar networks provide vertical resolutions of > 15 m, making a direct calculation of Nd from Rmax very uncertain. Therefore, we develop a Bayesian optimal estimation algorithm that brings additional information to the inversion such as lidar-derived extinction and radar reflectivity near the cloud top. This statistical approach provides reasonable characterizations of Nd and effective radius (re) to within approximately a factor of 2 and 30 %, respectively. By comparing surface-derived cloud properties with MODIS satellite and aircraft data collected during the MARCUS and CAPRICORN II campaigns, we demonstrate the utility of the methodology.
54 ENVIRONMENTAL SCIENCES↗
Efficient Unitary Designs from Random Sums and Permutations
A unitary k-design is an ensemble of unitaries that matches the first k moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ(k poly (n)) many gates. The k dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension (N) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-N expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension N.
algebra↗
Propagation of partially spatially coherent laser beams in instantaneous Kerr media
The propagation of intense, partially spatially coherent laser beams in a medium with instantaneous third-order susceptibility is studied analytically and numerically. For sufficiently high power relative to that required for nonlinear self-focusing, the propagation initially proceeds in two stages. In the first stage, spatial coherence builds up, and in the second stage, the number of speckles reduces. Once the degree of coherence is sufficiently high, whole-beam self-focusing occurs. The beam power is mostly confined within the initial spot radius. Two analytical approaches for describing the evolution of the beam are presented. The method of moments leads to an analytical solution for the rms spot radius that is in excellent agreement with simulations. This method does not require any knowledge of the field statistics beyond the initial conditions and provides no information about the evolution of the individual speckles. The other approach employs a self-similar solution for the second-order coherence function of the field and assumes that the fourth-order coherence function is factorizable and obeys complex circular Gaussian random statistics. The latter method also leads to an analytical expression for the spot radius, but its predictions for the qualitative evolution of the speckles disagree with wave-optics simulations.
lasers↗
Dark Energy Survey Year 3 results: $w$CDM cosmology from simulation-based inference with persistent homology on the sphere
We present cosmological constraints from Dark Energy Survey Year 3 (DES Y3) weak lensing data using persistent homology, a topological data analysis technique that tracks how features like clusters and voids evolve across density thresholds. For the first time, we apply spherical persistent homology to galaxy survey data through the algorithm TopoS2, which is optimized for curved-sky analyses and HEALPix compatibility. Employing a simulation-based inference framework with the Gower Street simulation suite, specifically designed to mimic DES Y3 data properties, we extract topological summary statistics from convergence maps across multiple smoothing scales and redshift bins. After neural network compression of these statistics, we estimate the likelihood function and validate our analysis against baryonic feedback effects, finding minimal biases (under $0.3σ$) in the $Ω_\mathrm{m}-S_8$ plane. Assuming the $w$CDM model, our combined Betti numbers and second moments analysis yields $S_8 = 0.821 \pm 0.018$ and $Ω_\mathrm{m} = 0.304\pm0.037$-constraints 70% tighter than those from cosmic shear two-point statistics in the same parameter plane. Our results demonstrate that topological methods provide a powerful and robust framework for extracting cosmological information, with our spherical methodology readily applicable to upcoming Stage IV wide-field galaxy surveys.
Prat, J. [Nordita; Royal Inst. Tech., Sodertalje; ↗
Robust error calibration for serial crystallography
Serial crystallography is an important technique with unique abilities to resolve enzymatic transition states, minimize radiation damage to sensitive metalloenzymes and perform de novo structure determination from micrometre-sized crystals. This technique requires the merging of data from thousands of crystals, making manual identification of errant crystals unfeasible. cctbx.xfel.merge uses filtering to remove problematic data. However, this process is imperfect, and data reduction must be robust to outliers. We add robustness to cctbx.xfel.merge at the step of uncertainty determination for reflection intensities. This step is a critical point for robustness because it is the first step where the data sets are considered as a whole, as opposed to individual lattices. Robustness is conferred by reformulating the error-calibration procedure to have fewer and less stringent statistical assumptions and incorporating the ability to down-weight low-quality lattices. We then apply this method to five macromolecular XFEL data sets and observe the improvements to each. The appropriateness of the intensity uncertainties is demonstrated through internal consistency. This is performed through theoretical CC 1/2 and I /σ relationships and by weighted second moments, which use Wilson's prior to connect intensity uncertainties with their expected distribution. This work presents new mathematical tools to analyze intensity statistics and demonstrates their effectiveness through the often underappreciated process of uncertainty analysis.
Mittan-Moreau, David W.↗