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DOE OSTI2021

The impacts of convex piecewise linear cost formulations on AC optimal power flow

Despite strong connections through shared application areas, research efforts on power market optimization (e.g., unit commitment) and power network optimization (e.g., optimal power flow) remain largely independent. A notable illustration of this is the treatment of power generation cost functions, where nonlinear network optimization has largely used polynomial representations and market optimization has adopted piecewise linear encodings. This work combines state-of-the-art results from both lines of research to understand the best mathematical formulations of the nonlinear AC optimal power flow problem with piecewise linear generation cost functions. An extensive numerical analysis of non-convex models, linear approximations, and convex relaxations across fifty-four realistic test cases illustrates that nonlinear optimization methods are surprisingly sensitive to the mathematical formulation of piecewise linear functions. The results indicate that a poor formulation choice can slow down algorithm performance by a factor of ten, increasing the runtime from seconds to minutes. Furthermore, these results provide valuable insights into the best formulations of nonlinear optimal power flow problems with piecewise linear cost functions, an important step towards building a new generation of energy markets that incorporate the nonlinear AC power flow model.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2021

Verification of a quasi-linear model for gyrokinetic turbulent transport

The verification and calibration of a new quasi-linear transport model with a large database of gyrokinetic turbulence simulations is presented in this paper. In a previous paper (Staebler et al 2020 Plasma Phys. Control. Fusion 63 015013), a model for the saturated spectrum of electric potential fluctuations was developed based on the properties of the non-linear 3D spectrum. In this paper, a modification to the overall multiplicative factor in this model is found to be necessary to improve the fit to scans of the temperature and density gradients. The error in the fit of the quasi-linear fluxes of electron and ion energy fluxes is significantly better than for previous saturation models. The spectral shift model for the impact of equilibrium E × B velocity shear (Staebler et al 2013 Phys. Rev. Lett. 110 055003) and the zonal flow mixing model for electron-scale turbulence (Staebler et al 2016 Phys. Plasmas 23 062518) are both revised to be compatible with this new model. Furthermore, the models for the loss of bounce averaging and electron collisions in the TGLF reduced linear equations (Staebler et al 2005 Phys. Plasmas 12 102508) are also changed to improve the linear eigenmodes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Multirate linearly-implicit GARK schemes

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Linear model decision trees as surrogates in optimization of engineering applications

Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. Importantly, we observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

42 ENGINEERING↗

DOE OSTI2025

Hyperplane decision trees as piecewise linear surrogate models for chemical process design

Recent trends in chemical engineering research point towards an increasing reliance on data-driven modeling approaches. Neural networks, for instance, have proven to be accurate when data is plentiful and high-dimensional, but in many cases, they require computationally-intensive training procedures. Here, in this work, we describe hyperplane decision trees (HT) as a highly expressive and low-compute machine learning model architecture. These models are locally linear and have linear decision boundaries, resulting in a piecewise linear model of the data. This property allows them to be converted into mixed-integer linear constraints which can be globally optimized. Our open-source PyTorch implementation of this method is a fast, flexible, and accessible way to build accurate piecewise linear models of data.

Decision trees↗

DOE OSTI2022

A high accuracy/resolution spectral element/Fourier–Galerkin method for the simulation of shoaling non-linear internal waves and turbulence in long domains with variable bathymetry

A high-order hybrid continuous-Galerkin numerical method, designed for the simulation of non-linear, non -hydrostatic internal waves and turbulence in long computational domains with complex bathymetry, is presented. The spatial discretization in the non-periodic wave-propagating directions, utilizes the nodal spectral element method. Such a high-order element-based discretization allows the highly accurate representation of complex domain geometry along with the flexibility of concentrating resolution in areas of interest. Under the assumption of the normal-to-isobath propagation of non-linear internal waves, a third periodic direction is incorporated via a Fourier-Galerkin discretization. The distinct non-hydrostatic nature of non-linear internal waves and, any instabilities and turbulence therein, necessitates the numerically challenging solution of the pressure Poisson problem. A defining feature of this work is the application of a domain decomposition approach, combined with block-Jacobi/deflation-based preconditioning to the pressure Poisson problem. Such a combined approach is particularly suitable for the long high aspect-ratio complex domains of interest and enables the efficient high-accuracy reproduction of the non-hydrostatic dynamics of non-linear internal waves. Implementation details are also described in the context of the stability of the solver and its parallelization strategy. A series of benchmarks of increasing complexity demonstrate the robustness of the flow solver. The benchmarks culminate with the three-dimensional simulation of a convectively breaking mode-one non-linear internal wave over a realistic South-China-Sea bathymetric transect and background current/stratification profiles.

Deflation↗

DOE OSTI2023

Linear instabilities in the hot-ion regime in a high-field spherical tokamak

Abstract The present operating high-field compact spherical tokamak ST40 is an important step toward an ST-based fusion reactor (Gryaznevich and Asunta 2017 Fusion Eng. Des. 123 177; McNamara 2023 Nucl. Fusion 63 054002). Temperature and density profiles and their uncertainties in recent hot ion ST40 plasmas with central ion temperatures exceeding 8.6 keV, have been inferred using an integrated analysis of several diagnostics including line-of-sight integration and volume average measurements, as well as limited profile information from a charge-exchange-recombination spectrometer. A linear gyrokinetic stability analysis has been carried out to identify the most unstable micro-instabilities in these hot ion plasmas. In one of these plasmas, it is found that linear growth rates of both ion- and electron-scale ( k θ ρ s ⩾ 0.2 where k θ is the poloidal wavenumber and ρ s is the ion gyro-radius with sound speed) modes decrease from the edge toward the core of the plasma (i.e. from ρ = 0.8 to 0.3 where ρ is the square root of the normalized magnetic flux), while the change of linear growth rates at k θ ρ s < 0.2 is non-monotonic. In particular, at ρ = 0.3 no unstable mode was found at k θ ρ s > 5 , and about an order of magnitude reduction in the maximum linear growth rate in the ion-scale wavenumber range of 0.2 ⩽ k θ ρ s < 1 is seen at ρ = 0.3 compared with the ρ = 0.8 location. It is found through parametric scans that while the ion temperature gradient (ITG) mode/trapped electron mode (TEM) and kinetic ballooning mode (KBM)/kinetic shear Alfvén (KSA) mode are more important in the plasma core, ubiquitous mode (UM) is found to be the important ion-scale instability at ρ ⩾ 0.6 . At ρ = 0.8, two instabilities are found at the electron scale, with one being the typical electrostatic electron temperature gradient (ETG) mode at ρ e scale and beyond and the other being an unidentified low-real-frequency instability (at the intermediate scale between electron and ion gyroradii) which can propagate in the ion direction and is driven by ETG. The existence of UM at the ion scale and the low-real-frequency instability and ETG mode at the electron scale at ρ = 0.8 is supported by an extensive linear gyrokinetic stability analysis of another ST40 hot ion plasma in a wider parametric range, taking into account the uncertainties in the experimental profiles.

ubiquitous mode↗

DOE OSTI2025

Randomized Adiabatic Quantum Linear Solver Algorithm with Optimal Complexity Scaling and Detailed Running Costs

Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı et al. [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number 𝜅 of the linear system and the target error 𝜖. Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling 𝑂⁡(𝜅⁢log (1/𝜖))—an exponential improvement in 𝜖, and a shaving of a log 𝜅 scaling factor in 𝜅. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837⁢𝜅 at 𝜖 = 10 −10 for Hermitian matrices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

NREL's Improved Linearity Testing of Photovoltaic Reference Cells

Photovoltaic devices are characterized under standard testing conditions that include a defined reference spectrum and total irradiance. International standards for reference cell calibrations require that reported reference cell response (typically Isc) vs. total irradiance must be linear. How can linearity be efficiently determined? In 2006 NREL developed a test bed, based on the "two-lamp method" that provided a low cost, but low accuracy method for determining whether cell response was linear with irradiance. This paper describes very simple changes to NREL's historical method [1] for determining linearity that yield greatly improved results. It also describes a method that can be used to quantify and correct for non-linearity.

41 EE - Solar Energy Technologies Office (EE-4S)↗

DOE OSTI2020

Non-linear flow modes of identified particles in Pb-Pb collisions at $ \sqrt{s_{\mathrm{NN}}} $ = 5.02 TeV

The pT-differential non-linear flow modes, v 4,22 , v 5,32 , v 6,33 and v 6,222 for π ± , K ± , K$^0_S$ , $p + \bar{p}, Λ + \bar{Λ}$ and &straightphi;-meson have been measured for the first time at $\sqrt{s_{NN}}$ = 5.02 TeV in Pb-Pb collisions with the ALICE detector at the Large Hadron Collider. The results were obtained with a multi-particle technique, correlating the identified hadrons with reference charged particles from a different pseudorapidity region. These non-linear observables probe the contribution from the second and third order initial spatial anisotropy coefficients to higher flow harmonics. All the characteristic features observed in previous p T -differential anisotropic flow measurements for various particle species are also present in the non-linear flow modes, i.e. increase of magnitude with increasing centrality percentile, mass ordering at low p T and particle type grouping in the intermediate p T range. Hydrodynamical calculations (iEBE-VISHNU) that use different initial conditions and values of shear and bulk viscosity to entropy density ratios are confronted with the data at low transverse momenta. These calculations exhibit a better agreement with the anisotropic flow coefficients than the non-linear flow modes. These observations indicate that non-linear flow modes can provide additional discriminatory power in the study of initial conditions as well as new stringent constraints to hydrodynamical calculations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Gradient-based constrained optimization using a database of linear reduced-order models

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Role of Polymer Architecture in CO 2 Capture from Air Using Supported Poly(alkylenimine)s: Linear vs Branched Polymers

Direct air capture (DAC) of CO 2 coupled with geologic storage is a promising climate change mitigation strategy, with some applications employing amines supported on porous solids as CO 2 sorbents. While branched poly(ethylenimine) (PEI) is the standard benchmark amine material, it suffers from limited oxidative stability. Poly(propylenimine) (PPI), as an alternative, has previously demonstrated improved resistance to degradation under harsh oxidative conditions. Linear and branched PEI are commercially available, though at different molecular weights, while PPI is not commercially available. For this reason, a comparative study of all four polymers (linear PEI, branched PEI, linear PPI, branched PPI) has not been reported for DAC. In this study, we synthesize and compare low-molecular-weight (∼800 g/mol) linear (L) and branched (B) PEI and PPI supported on a model support, SBA-15 silica. These materials are evaluated for CO 2 adsorption under dry, DAC-relevant conditions (400 ppm of CO 2 , 30 °C). LPPI exhibited the highest amine efficiency at all loadings, reaching a maximum of 0.14 mmol CO 2 /mmol N, outperforming BPEI, while LPEI consistently showed the lowest uptake capacity. Temperature-programmed desorption reveals that the structure of the amine polymer impacts the CO 2 binding strength, with branched polymers displaying higher desorption energies of 102−111 kJ/mol. In situ infrared spectroscopy experiments show that all sorbents preferentially capture CO 2 as ammonium carbamate. Isobaric CO 2 uptake studies further underscore the influence of polymer mobility and support pore crowding on performance, while demonstrating the sorbents’ performance at elevated temperatures and CO 2 concentrations. All materials demonstrated good stability over 25 adsorption−desorption cycles using thermal regeneration in an inert gas purge, with only BPPI displaying a 10−11% decrease in capacity/amine efficiency during cycling, possibly due to the loss of low molecular weight, oligomeric amines. This is the first side-by-side comparison of the CO 2 sorption properties of linear and branched PEI and PPI with similar molecular weights. These findings highlight the significant role of polymer architecture in CO 2 capture efficiency and inform future designs of durable, high-performance DAC sorbents.

adsorbents↗

DOE OSTI2023

Universal linear intensity transformations using spatially incoherent diffractive processors

Abstract Under spatially coherent light, a diffractive optical network composed of structured surfaces can be designed to perform any arbitrary complex-valued linear transformation between its input and output fields-of-view (FOVs) if the total number ( N ) of optimizable phase-only diffractive features is ≥~2 N i N o , where N i and N o refer to the number of useful pixels at the input and the output FOVs, respectively. Here we report the design of a spatially incoherent diffractive optical processor that can approximate any arbitrary linear transformation in time-averaged intensity between its input and output FOVs. Under spatially incoherent monochromatic light, the spatially varying intensity point spread function ( H ) of a diffractive network, corresponding to a given, arbitrarily-selected linear intensity transformation, can be written as H ( m , n ; m ′, n ′) = | h ( m , n ; m ′, n ′)| 2 , where h is the spatially coherent point spread function of the same diffractive network, and ( m , n ) and ( m ′, n ′) define the coordinates of the output and input FOVs, respectively. Using numerical simulations and deep learning, supervised through examples of input-output profiles, we demonstrate that a spatially incoherent diffractive network can be trained to all-optically perform any arbitrary linear intensity transformation between its input and output if N ≥ ~2 N i N o . We also report the design of spatially incoherent diffractive networks for linear processing of intensity information at multiple illumination wavelengths, operating simultaneously. Finally, we numerically demonstrate a diffractive network design that performs all-optical classification of handwritten digits under spatially incoherent illumination, achieving a test accuracy of >95%. Spatially incoherent diffractive networks will be broadly useful for designing all-optical visual processors that can work under natural light.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Data-driven linear time advance operators for the acceleration of plasma physics simulation

In this study, we demonstrate the application of data-driven linear operator construction for time advance with a goal of accelerating plasma physics simulation. We apply dynamic mode decomposition (DMD) to data produced by the nonlinear SOLPS-ITER (Scrape-off Layer Plasma Simulator - International Thermonuclear Experimental Reactor) plasma boundary code suite in order to estimate a series of linear operators and monitor their predictive accuracy via online error analysis. We find that this approach defines when these dynamics can be represented by a sequence of approximate linear operators and is essential for providing consistent projections when compared to an unconstrained application. For linear diffusion and advection–diffusion fluid test problems, we construct and apply operators within explicit and implicit time advance schemes, demonstrating that stability can be robustly guaranteed in each case. We further investigate the use of the linear time advance operators within several integration methods including forward Euler, backward Euler, and the matrix exponential. The application of this method to simulation data from SOLPS-ITER, with varying levels of Markov chain Monte Carlo numerical noise, shows that constrained DMD operators yield a capability to identify, extract, and integrate a (slow) subset of the present timescales. Example applications show that for projected speedup factors of [Formula: see text], and [Formula: see text], a mean relative error of 3%, 5%, and 8% and maximum relative error less than 20% are achievable, which appears acceptable for typical SOLPS-ITER steady-state simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Augmenting subspace optimization methods with linear bandits

In this work, we consider the framework of methods for unconstrained minimization that are, in each iteration, restricted to a model that is only a valid approximation to the objective function on some affine subspace containing an incumbent point. These methods are of practical interest in computational settings where derivative information is either expensive or impossible to obtain. Recent attention has been paid in the literature to employing randomized matrix sketching for generating the affine subspaces within this framework. We consider a relatively straightforward, deterministic augmentation of such a generic subspace optimization method. In particular, we consider a sequential optimization framework where actions consist of one-dimensional linear subspaces and rewards consist of (approximations to) the magnitudes of directional derivatives computed in the direction of the action subspace. Reward maximization in this context is consistent with maximizing lower bounds on descent guaranteed by first-order Taylor models. This sequential optimization problem can be analysed through the lens of dynamic regret. We modify an existing linear upper confidence bound (UCB) bandit method and prove sublinear dynamic regret in the subspace optimization setting. We demonstrate the efficacy of employing this linear UCB method in a setting where forward-mode algorithmic differentiation can provide directional derivatives in arbitrary directions and in a derivative-free setting. For the derivative-free setting, we propose SS-POUNDers, an extension of the derivative-free optimization method POUNDers that employs the linear UCB mechanism to identify promising subspaces. Our numerical experiments suggest a preference, in either computational setting, for employing a linear UCB mechanism within a subspace optimization method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Quasi-linear toroidal simulations of resonant magnetic perturbations in eight ITER H-mode scenarios

Abstract Both linear and quasi-linear aspects of the plasma response to the resonant magnetic perturbation (RMP) field are numerically investigated for various H-mode scenarios in ITER, covering the pre-fusion power operation and the fusion power operation phases. Linear response computations for eight ITER scenarios, with varying plasma current and toroidal magnetic field, reveal that the best coil current phasing for controlling the type-I edge localized modes (ELMs) scales roughly linearly with the edge safety factor. The coil phasing is defined as the relative toroidal phase of the coil currents between different rows, for a given toroidal harmonic. Quasi-linear initial value simulation, which is the focus of the present study, shows that application of the n = 3 ( n is the toroidal mode number) RMP field has a minimum side effect on the plasma core momentum confinement but potentially a large effect on the global particle transport. Generally, the RMP field with the best (worst) coil phasing for ELM control produces the strongest (weakest) effect on the plasma edge flow and the overall density. This robustly holds for all eight ITER scenarios. Consequently, in order to minimize the RMP induced side effects while achieving ELM control (suppression) in ITER, a compromise is necessary in choosing the coil current configuration.

Physics↗

DOE OSTI2022

Concept of circular-linear energy recovery accelerator to probe the energy frontier

Energy-frontier accelerators provide powerful tools performing high precision measurements confirming the fundamentals of the physics and broadening new research horizons. Such machines are either driven by circular or linear accelerators. The circular machines, having the centre-of-mass (CM) energy values reaching 200 GeV (for leptons) and above, experience beam energy loss and quality dilution, for example, due to synchrotron radiation, limiting the overall CM energy achievable and requiring a constant energy top-up to compensate the loss and the beam quality dilution. Linear colliders overcome these limitations, while the finite capabilities of generating high average current beams limits the luminosity. This is partially compensated by the quality of the colliding beams. In this work, we suggest a novel design of circular-linear accelerator based on the merging of the “non-emitting”, low-energy storage rings and energy recovery linear accelerators. We suggest using the recently considered dual-axis asymmetric cavities to enable the operation of such a system, and in particular the energy recovery from spent, high-intensity beams. The machine considered, under the scope of the SNOWMASS-2021 initiative, can be potentially used to reach ultimate energy frontiers in high-energy physics as well as to drive next generation light sources. The merging of circular and linear systems, and applications of dual axes cavities, should allow the maintaining of high beam quality, high luminosity, and high energy efficiency, while offering a flexible energy management and opening clear opportunity for reducing the running cost. We note that the numbers shown in the paper are for illustration purpose and can be improved further.

43 PARTICLE ACCELERATORS↗

DOE OSTI2022

Exploiting non-linear scales in galaxy–galaxy lensing and galaxy clustering: A forecast for the dark energy survey

ABSTRACT The combination of galaxy–galaxy lensing (GGL) and galaxy clustering is a powerful probe of low-redshift matter clustering, especially if it is extended to the non-linear regime. To this end, we use an N-body and halo occupation distribution (HOD) emulator method to model the redMaGiC sample of colour-selected passive galaxies in the Dark Energy Survey (DES), adding parameters that describe central galaxy incompleteness, galaxy assembly bias, and a scale-independent multiplicative lensing bias Alens. We use this emulator to forecast cosmological constraints attainable from the GGL surface density profile ΔΣ(rp) and the projected galaxy correlation function wp, gg(rp) in the final (Year 6) DES data set over scales $r_p=0.3\!-\!30.0\, h^{-1} \, \mathrm{Mpc}$. For a $3{{\ \rm per\ cent}}$ prior on Alens we forecast precisions of $1.9{{\ \rm per\ cent}}$, $2.0{{\ \rm per\ cent}}$, and $1.9{{\ \rm per\ cent}}$ on Ωm, σ8, and $S_8 \equiv \sigma _8\Omega _m^{0.5}$, marginalized over all halo occupation distribution (HOD) parameters as well as Alens. Adding scales $r_p=0.3\!-\!3.0\, h^{-1} \, \mathrm{Mpc}$ improves the S8 precision by a factor of ∼1.6 relative to a large scale ($3.0\!-\!30.0\, h^{-1} \, \mathrm{Mpc}$) analysis, equivalent to increasing the survey area by a factor of ∼2.6. Sharpening the Alens prior to $1{{\ \rm per\ cent}}$ further improves the S8 precision to $1.1{{\ \rm per\ cent}}$, and it amplifies the gain from including non-linear scales. Our emulator achieves per cent-level accuracy similar to the projected DES statistical uncertainties, demonstrating the feasibility of a fully non-linear analysis. Obtaining precise parameter constraints from multiple galaxy types and from measurements that span linear and non-linear clustering offers many opportunities for internal cross-checks, which can diagnose systematics and demonstrate the robustness of cosmological results.

79 ASTRONOMY AND ASTROPHYSICS↗