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DOE OSTI2022

HSC-XXL: Baryon budget of the 136 XXL groups and clusters

Abstract We present our determination of the baryon budget for an X-ray-selected XXL sample of 136 galaxy groups and clusters spanning nearly two orders of magnitude in mass (M500 ∼ 1013–1015 M⊙) and the redshift range 0 ≲ z ≲ 1. Our joint analysis is based on the combination of Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) weak-lensing mass measurements, XXL X-ray gas mass measurements, and HSC and Sloan Digital Sky Survey multiband photometry. We carry out a Bayesian analysis of multivariate mass-scaling relations of gas mass, galaxy stellar mass, stellar mass of brightest cluster galaxies (BCGs), and soft-band X-ray luminosity, by taking into account the intrinsic covariance between cluster properties, selection effect, weak-lensing mass calibration, and observational error covariance matrix. The mass-dependent slope of the gas mass–total mass (M500) relation is found to be $1.29_{-0.10}^{+0.16}$, which is steeper than the self-similar prediction of unity, whereas the slope of the stellar mass–total mass relation is shallower than unity; $0.85_{-0.09}^{+0.12}$. The BCG stellar mass weakly depends on cluster mass with a slope of $0.49_{-0.10}^{+0.11}$. The baryon, gas mass, and stellar mass fractions as a function of M500 agree with the results from numerical simulations and previous observations. We successfully constrain the full intrinsic covariance of the baryonic contents. The BCG stellar mass shows the larger intrinsic scatter at a given halo total mass, followed in order by stellar mass and gas mass. We find a significant positive intrinsic correlation coefficient between total (and satellite) stellar mass and BCG stellar mass and no evidence for intrinsic correlation between gas mass and stellar mass. All the baryonic components show no redshift evolution.

Akino, Daichi↗

DOE OSTI2020

On the Solution of ℓ 0 -Constrained Sparse Inverse Covariance Estimation Problems

The sparse inverse covariance matrix is used to model conditional dependencies between variables in a graphical model to fit a multivariate Gaussian distribution. Estimating the matrix from data are well known to be computationally expensive for large-scale problems. Sparsity is employed to handle noise in the data and to promote interpretability of a learning model. Although the use of a convex ℓ 1 regularizer to encourage sparsity is common practice, the combinatorial ℓ 0 penalty often has more favorable statistical properties. In this paper, we directly constrain sparsity by specifying a maximally allowable number of nonzeros, in other words, by imposing an ℓ 0 constraint. Here, we introduce an efficient approximate Newton algorithm using warm starts for solving the nonconvex ℓ 0 -constrained inverse covariance learning problem. Numerical experiments on standard data sets show that the performance of the proposed algorithm is competitive with state-of-the-art methods.

$\ell_0$-Constrained↗

DOE OSTI2024

The 3D Lyman- α forest power spectrum from eBOSS DR16

We measure the three-dimensional power spectrum (P3D) of the transmitted flux in the Lyman-α (Ly α) forest using the complete extended Baryon Oscillation Spectroscopic Survey data release 16 (eBOSS DR16). This sample consists of ~205 000 quasar spectra in the redshift range 2 ≤ z ≤ 4 at an effective redshift z = 2.334. We propose a pair-count spectral estimator in configuration space, weighting each pair by exp( i k ∙ r), for wave vector k and pixel pair separation r, effectively measuring the anisotropic power spectrum without the need for fast Fourier transforms. This accounts for the window matrix in a tractable way, avoiding artefacts found in Fourier-transform based power spectrum estimators due to the sparse sampling transverse to the line of sight of Ly α skewers. We extensively test our pipeline on two sets of mocks: (i) idealized Gaussian random fields with a sparse sampling of Ly α skewers, and (ii) log-normal LyaCoLoRe mocks including realistic noise levels, the eBOSS survey geometry and contaminants. On eBOSS DR16 data, the Kaiser formula with a non-linear correction term obtained from hydrodynamic simulations yields a good fit to the power spectrum data in the range $(0.02 ≤ k ≤ 0.35)$ h Mpc -1 at the 1–2σ level with a covariance matrix derived from LyaCoLoRe mocks. We demonstrate a promising new approach for full-shape cosmological analyses of Ly α forest data from cosmological surveys such as eBOSS, the currently observing Dark Energy Spectroscopic Instrument and future surveys such as the Prime Focus Spectrograph, WEAVE-QSO, and 4MOST.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Modernization efforts for the R -Matrix code SAMMY [Abstract]

The R-Matrix code SAMMY is a widely used nuclear data evaluation code focused on the resolved range, which includes corrections for experimental effects. The code is still mostly written in Fortran 77, and uses a memory management system suitable for the time of its initial writing (1984). A modernization effort is under way to bring the code in-line with modern software development practices. A continuous-integration testing framework was added, automating the large existing set of test cases. It is run on every commit. The memory management was updated to current standard practices suitable for modern software analysis tools. The code can be obtained from https://code.ornl.gov/RNSD/SAMMY. The resonance parameters and covariance information are now stored in C++ objects shared by SAMMY and AMPX, the processing code that generates nuclear data libraries for SCALE. This allows for easier maintenance and access to the resonance parameters inside and outside of SAMMY. This feature is already used by accessing and changing parameters in memory in the Bayesian Monte Carlo Evaluation Framework for Cross Sections Nuclear Data and Integral Benchmark Experiments project, Further plans include the switch to the ENDF reading and writing routines in AMPX, as these routines are more robust, easier to maintain, and support more features. Of note here is support for the new GNDS format. Previously it wasn’t easy to share the full covariance matrix for evaluations containing more than one isotope due to limitations on the ENDF format; this is now supported in GNDS. The data are currently available in a binary SAMMY format and can be exported to GNDS to make them more widely available and sharable. The next step will be to use the same resonance processing code at 0K in AMPX and SAMMY as one of the available Reich-Moore R-Matrix formalism. The first step toward this goal is to isolate the reconstruction into a module that takes resonance parameters as its input and does not depend on SAMMY global parameters. This goal has been achieved and it should now be possible to more easily change the resonance formalism and add enhancements as the Phenomenological R-Matrix parameterization of direct, doorway, and compound nuclear reactions discussed elsewhere on this conference. This concerted modernization and enhancement effort provides multiple advantages to the nuclear data community. It will allow parameter optimization using enhanced formalisms, including experimental effects, that better match complex experimental data. Then those evaluated parameters can immediately be passed off to AMPX to be reconstructed with the exact same cross section model and be put into a data library for subsequent testing using SCALE and the Valid Benchmark suite or other suitable benchmark suites.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS2018

A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements

We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to incur no bias from the choice of the prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model and verify our derivations and bias-free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations in which knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.

Ramanathan, Anand K.↗

DOE OSTI2023

Exact Gaussian processes for massive datasets via non-stationary sparsity-discovering kernels

Abstract A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. Its success is largely attributed to the GP’s analytical tractability, robustness, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $$O(N^3)$$ O ( N 3 ) in computation and $$O(N^2)$$ O ( N 2 ) in storage. All existing methods addressing this issue utilize some form of approximation—usually considering subsets of the full dataset or finding representative pseudo-points that render the covariance matrix well-structured and sparse. These approximate methods can lead to inaccuracies in function approximations and often limit the user’s flexibility in designing expressive kernels. Instead of inducing sparsity via data-point geometry and structure, we propose to take advantage of naturally-occurring sparsity by allowing the kernel to discover—instead of induce—sparse structure. The premise of this paper is that the data sets and physical processes modeled by GPs often exhibit natural or implicit sparsities, but commonly-used kernels do not allow us to exploit such sparsity. The core concept of exact, and at the same time sparse GPs relies on kernel definitions that provide enough flexibility to learn and encode not only non-zero but also zero covariances. This principle of ultra-flexible, compactly-supported, and non-stationary kernels, combined with HPC and constrained optimization, lets us scale exact GPs well beyond 5 million data points.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Development and application of two-step uncertainty propagation and sensitivity analysis methodology for fast reactor safety analysis

Uncertainty quantification (UQ) in nuclear reactors for transients is directly linked with safety assessment through the cross-sections uncertainties, provided as a covariance matrix, which are propagated through the reactor system to output of interest pertaining to reactor safety, such as peak temperatures in fuel/clad/coolant. Using a two-step approach, uncertainties are first quantified and propagated from basic input variables (such as reaction cross-sections) to intermediate quantities (such as reactivity feedback coefficients) through lattice level calculations. Uncertainties of intermediate quantities (from the first step) are then propagated through the system transient calculations, in the second step, to obtain uncertainties on reactor safety output parameters of interest. The scope of this work consists of Uncertainty Quantification & Propagation of nuclear data uncertainties that are highly correlated through unprotected transient overpower and unprotected loss of flow to assess their impact on core safety parameters. This two-step approach in the presence of covariance renders the sensitivity analysis very challenging. In fact, usually the sensitivity analysis is restricted to each step, which limits its application since the sensitivities between the system output quantities and the basic input variables are difficult to obtain. Here, in this work, we address this issue by proposing a simple, general methodology to combine the sensitivity indices obtained in each step by assuming the model behavior being linear. For the first step Generalized Perturbation theory based indices are used while in the second step the recently studied Johnson indices. The uncertainty quantification and sensitivity methodologies discussed here are demonstrated on a generic LFR design which is based on the 500 MWth demonstration Lead-cooled fast reactor (DLFR) using oxide fuel, developed by Westinghouse Electric Company (WEC).

42 - ENGINEERING↗

DOE OSTI2021

Tractable Learning in Underexcited Power Grids

Estimating the structure of physical flow networks, such as power grids, is critical to secure delivery of energy. This article discusses statistical structure estimation in power grids in the “underexcited” regime, where a subset of internal nodes has zero injection fluctuations. Prior estimation algorithms based on nodal voltages fail for such grids as the voltage covariance matrix is not invertible. Here, we propose a novel topology learning algorithm for learning underexcited general networks. Our algorithm uses physics-informed conservation laws to first identify the zero-injection buses and their neighbors, and then estimates the remaining edges in the grid. We prove the asymptotic correctness of our algorithm for grids with nonadjacent internal zero-injection nodes. More important, we theoretically analyze our algorithm’s efficacy under noisy measurements, and determine bounds on maximum noise under which asymptotically correct recovery is guaranteed. Our approach is validated through simulations with voltage samples generated on test distribution grids with real injection data and nonlinear power flow models.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2023

Empirical probability and machine learning analysis of m , n = 2, 1 tearing mode onset parameter dependence in DIII-D H-mode scenarios

m, n = 2, 1 tearing mode onset empirical probability and machine learning analyses of a multiscenario DIII-D database of over 14 000 H-mode discharges show that the normalized plasma beta, the rotation profile, and the magnetic equilibrium shape have the strongest impact on the 2,1 tearing mode stability, in qualitative agreement with neoclassical tearing modes (m and n are the poloidal and toroidal mode numbers, respectively). In addition, 2,1 tearing modes are most likely to destabilize when n > 1 tearing modes are already present in the core plasma. The covariance matrix of tearing sensitive plasma parameters takes a nearly block-diagonal form, with the blocks incorporating thermodynamic, current and safety factor profile, separatrix shape, and plasma flow parameters, respectively. This suggests a number of paths to improved stability at fixed pressure and edge safety factor primarily by preserving a minimum of 1 kHz differential rotation, increasing the minimum safety factor above unity, using upper single null magnetic configuration, and reducing the core impurity radiation. In addition, lower triangularity, lower elongation, and lower pedestal pressure may also help to improve stability. Furthermore, the electron and ion temperature, collisionality, resistivity, internal inductance, and the parallel current gradient appear to only weakly correlate with the 2,1 tearing mode onsets in this database.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets

We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Furthermore, our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Scalable computations for nonstationary Gaussian processes

Nonstationary Gaussian process models can capture complex spatially varying dependence structures in spatial datasets. However, the large number of observations in modern datasets makes fitting such models computationally intractable with conventional dense linear algebra. In addition, derivative-free or even first-order optimization methods can be very slow to converge when estimating many spatially varying parameters. In this paper, we present a computational framework which couples an algebraic block diagonal plus low-rank covariance matrix approximation with stochastic trace estimation to facilitate the efficient use of second-order solvers for maximum likelihood estimation of Gaussian process models with many parameters. We demonstrate the effectiveness of these methods by simultaneously fitting 192 parameters in the popular nonstationary model of Paciorek and Schervish using 107,600 sea surface temperature anomaly measurements.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Robust PCA-Deep Belief Network Surrogate Model for Distribution System Topology Identification with DERs

With the expansion of distribution networks and increased penetration of distributed energy resources (DERs), it is becoming increasingly important to obtain accurate distribution network topology in real-time. In this paper, a robust principal component analysis coupled deep belief network (PCA-DBN) surrogate model is proposed for distribution system topology identification. It integrates the benefits of robust feature extraction from PCA to deal with data quality issues and filter out noise, and the strength of DBN in capturing the nonlinear relationship between voltage amplitudes and the binary states of switchable connections. This also significantly reduces the DBN training complexity without loss of accuracy. It is shown that the widely used standard deviation of voltage drop and the voltage covariance matrix features yield less accuracy as compared to that of the voltage amplitudes in presence of high penetration of DERs and ZIP loads. Comparison results with other alternatives, such as the random forest (RF), multi-output regression (MOR) and the traditional DBN methods demonstrate that the proposed method can achieve a much higher topology identification accuracy while maintaining robustness to missing data and measurement noise under various penetration levels of DERs.

deep belief network↗

DOE OSTI2020

Combined Neyman–Pearson chi-square: An improved approximation to the Poisson-likelihood chi-square

We describe an approximation to the widely-used Poisson-likelihood chi-square using a linear combination of Neyman’s and Pearson’s chi-squares, namely “combined Neyman–Pearson chi-square” ($X^2_{CNP}$). Through analytical derivations and toy model simulations, we show that leads to a significantly smaller bias on the best-fit model parameters compared to those using either Neyman’s or Pearson’s chi-square. When the computational cost of using the Poisson-likelihood chi-square is high, $X^2_{CNP}$ provides a good alternative given its natural connection to the covariance matrix formalism.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Propagating neutronic uncertainties for FFTF LOFWOS Test #13

The safety evaluation conducted for licensing a Sodium-cooled Fast Reactor (SFR) may require modeling transients with best-estimate calculation tools that must first be validated against real-world measurements. To provide the community with a valuable benchmarking opportunity for validating SFR analysis tools and methods, the International Atomic Energy Agency (IAEA) initiated a coordinated research project (CRP) in 2018 for the analysis of the Fast Flux Test Facility (FFTF) Loss of Flow Without Scram (LOFWOS) Test #13. The impact of nuclear data uncertainties on neutronics parameters was previously investigated based on the COMMARA-2.0 covariance matrix. Since the transient simulation results are very sensitive to certain reactivity coefficients, it was decided to employ rigorous uncertainty propagation methods to quantify the impact of nuclear data uncertainties on the best-estimate predication of FFTF LOFWOS Test #13. The DAKOTA code is used to propagate neutronic uncertainties through SAS4A/SASSYS-1 transient simulations, while taking into account spatial and reaction-wise correlations within these uncertainties. This study shows that the remaining discrepancies observed between the Argonne National Laboratory (ANL) best-estimate results and the experimental measurements can be partly explained by the uncertainty associated with Gas Expansion Module (GEM) worth, which contributes the majority of the overall nuclear data uncertainty on the output from the FFTF LOFWOS Test #13 transient simulation. This study also confirmed the importance of including spatial and reaction-wise correlations of nuclear data uncertainties on feedback coefficients in the uncertainty propagation to avoid under-estimating their impact during the transient simulations.

42 ENGINEERING↗

DOE OSTI2022

Nuclear data uncertainty propagation applied to the versatile test reactor conceptual design

We report the Versatile Test Reactor (VTR) currently under development is a 300 MWth sodium-cooled fast reactor (SFR) fueled with ternary metal alloy fuel, which aims to accelerate the testing of advanced nuclear fuels, materials, instrumentation, and sensors in high flux environments that are necessary to license the next generation of advanced reactor concepts. To support the VTR design process, uncertainties associated with the nuclear data has been propagated through the reactor core neutronics calculation to global parameters of interest, such as the core multiplication factor, kinetic parameters, and various reactivity feedback coefficients, following the sensitivity based uncertainty propagation approach. By folding the sensitivity coefficients, separately computed by the generalized perturbation theory code PERSENT and Monte Carlo code Serpent 2, with the variance-covariance matrices from COMMARA-2.0, we obtain the reaction-wise, isotope-wise, and overall uncertainties for each response of interest due to nuclear data uncertainty. With Serpent 2, the statistical error of the uncertainty is obtained by propagating the statistical error of the sensitivity coefficients through the same process using a newly developed uncertainty propagation method. From both codes, the overall top uncertainty contributors are found to be the cross section of Fe-56 elastic scattering, Na-23 elastic scattering, and U 238 inelastic scattering. The large contributions of the Fe-56 elastic scattering cross sections to global parameters are due to its relatively large relative uncertainty of 5–10% in nuclear data and the large volume of Fe-containing reflector assemblies in the fairly compact VTR core design. Both codes agreed well for the overall uncertainty estimates of all responses of interest, except the delayed neutron fraction, prompt neutron generation time, and the coolant density feedback coefficient, where Serpent 2 yielded a much larger value than PERSENT due to the large statistical error of sensitivity coefficients. The calculated uncertainties are also compared to those associated with other SFR cores. Another outcome of this study is a variance-covariance matrix of reactivity coefficients, which can be used in the subsequent uncertainty propagation to the system level to investigate the impact of identified uncertainties on system responses in the safety analysis.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

Understanding Peelle’s Pertinent Puzzle bias in generalized least squares regression through eigenspectrum analysis

Certain correlation structures in the data covariance matrix (DCM) used for generalized least squares (GLS) regression can result in biased estimates, commonly known in the field of nuclear data evaluation as Peele’s Pertinent Puzzle (PPP). This article introduces a generative, forward modeling framework within which the PPP bias is characterized through an eigenspectrum analysis of the DCM. This analysis highlights the root cause of the bias, generalizes the problem beyond the nuclear data field, and provides insight to the problem regimes where it can occur. What follows is an understanding that the bias can show up for any experimental neutron time-of-flight data for which systematic uncertainties have been quantified. Lastly, a discussion of the adaptation of cross validation approaches that require pre-whitening to incorporate the known ‘fix’ to the PPP bias in the GLS estimator.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

BEYONDPLANCK II. CMB mapmaking through Gibbs sampling

We present a Gibbs sampling solution to the mapmaking problem for cosmic microwave background (CMB) measurements that builds on existing destriping methodology. Gibbs sampling breaks the computationally heavy destriping problem into two separate steps: noise filtering and map binning. Considered as two separate steps, both are computationally much cheaper than solving the combined problem. This provides a huge performance benefit as compared to traditional methods and it allows us, for the first time, to bring the destriping baseline length to a single sample. Here, we applied the Gibbs procedure to simulated Planck 30 GHz data. We find that gaps in the time-ordered data are handled efficiently by filling them in with simulated noise as part of the Gibbs process. The Gibbs procedure yields a chain of map samples, from which we are able to compute the posterior mean as a best-estimate map. The variation in the chain provides information on the correlated residual noise, without the need to construct a full noise covariance matrix. However, if only a single maximum-likelihood frequency map estimate is required, we find that traditional conjugate gradient solvers converge much faster than a Gibbs sampler in terms of the total number of iterations. The conceptual advantages of the Gibbs sampling approach lies in statistically well-defined error propagation and systematic error correction. This methodology thus forms the conceptual basis for the mapmaking algorithm employed in the BEYONDPLANCK framework, which implements the first end-to-end Bayesian analysis pipeline for CMB observations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

BeyondPlanck: XII. Cosmological parameter constraints with end-to-end error propagation

We present cosmological parameter constraints estimated using the Bayesian BEYONDPLANCK analysis framework. This method supports seamless end-to-end error propagation from raw time-ordered data onto final cosmological parameters. As a first demonstration of the method, we analyzed time-ordered Planck LFI observations, combined with selected external data (WMAP 33–61 GHz, Planck HFI DR4 353 and 857 GHz, and Haslam 408 MHz) in the form of pixelized maps that are used to break critical astrophysical degeneracies. Overall, all the results are generally in good agreement with previously reported values from Planck 2018 and WMAP, with the largest relative difference for any parameter amounting about 1σ when considering only temperature multipoles between 30 ≤ ℓ ≤ 600. In cases where there are differences, we note that the BEYONDPLANCK results are generally slightly closer to the high-ℓ HFI-dominated Planck 2018 results than previous analyses, suggesting slightly less tension between low and high multipoles. Using low-ℓ polarization information from LFI and WMAP, we find a best-fit value of τ = 0.066 ± 0.013, which is higher than the low value of τ = 0.052 ± 0.008 derived from Planck 2018 and slightly lower than the value of 0.069 ± 0.011 derived from the joint analysis of official LFI and WMAP products. Most importantly, however, we find that the uncertainty derived in the BEYONDPLANCK processing is about 30 % greater than when analyzing the official products, after taking into account the different sky coverage. We argue that this uncertainty is due to a marginalization over a more complete model of instrumental and astrophysical parameters, which results in more reliable and more rigorously defined uncertainties. We find that about 2000 Monte Carlo samples are required to achieve a robust convergence for a low-resolution cosmic microwave background (CMB) covariance matrix with 225 independent modes, and producing these samples takes about eight weeks on a modest computing cluster with 256 cores.

79 ASTRONOMY AND ASTROPHYSICS↗