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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis

Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a “minimax” problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Finally, several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Monolithic Multigrid for a Reduced-Quadrature Discretization of Poroelasticity

Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Further, numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Nonoverlapping block smoothers for the Stokes equations

Overlapping block smoothers efficiently damp the error contributions from highly oscillatory components within multigrid methods for the Stokes equations but they are computationally expensive. This paper is concentrated on the development and analysis of new block smoothers for the Stokes equations that are discretized on staggered grids. These smoothers are nonoverlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, using multigrid methods to efficiently solve more difficult problems such as the Stokes equations leads to computationally more expensive smoothers, for example, overlapping block smoothers. Nonoverlapping smoothers are less expensive, but have been considered less efficient in the literature. In this paper, we develop new nonoverlapping smoothers, the so-called triad-wise smoothers, and show their efficiency within multigrid methods to solve the Stokes equations. In addition, we compare overlapping and nonoverlapping smoothers by measuring their computational costs and analyzing their behavior by the use of local Fourier analysis.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Low-order preconditioning of the Stokes equations

A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor–Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the Q 1 iso Q 2 /Q 1 discretization of the Stokes operator as a preconditioner for the Q 2 /Q 1 discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess–Sarazin relaxation schemes and to achieve robust convergence. Furthermore, these results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the Q 2 /Q 1 system, our ultimate motivation is to apply algebraic multigrid within solvers for Q 2 /Q 1 systems via the Q 1 iso Q 2 /Q 1 discretization, which will be considered in a companion paper.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Overlapping Schwarz Methods Are Not Anisotropy‐Robust Multigrid Smoothers

We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. Here, a natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter 𝒪⁡(𝜖 −1/2 ) for anisotropy ratio 𝜖 ∈(0,1] .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Mass Transport in Membrane Systems: Flow Regime Identification by Fourier Analysis

The numerical calculation of local mass distributions in membrane systems by computational fluid dynamics (CFD) offers indispensable benefits. However, the concept to calculate such distributions in response to separate variations of operation conditions (OCs) makes it difficult to address overall, flow-physics-related questions, which require the consideration of the collective interaction of OCs. It is shown that such understanding-related relationships can be obtained by the analytical solution of the advection–diffusion equation considered. A Fourier series model (FSM) is presented, which provides exact solutions of an advection–diffusion equation for a wide range of OCs. On this basis, a new zeroth-order model is developed, which is very simple and as accurate as the complete FSM for all conditions of practical relevance. Advection-dominated blocked and diffusion-dominated unblocked flow regimes are identified (depending on a Péclet number which compares the flow geometry with a length scale imposed by the flow), which implies relevant requirements for the use of lab results for pilot- and full-scale applications. Analyses reveal the equivalence of variations of OCs, which offers a variety of options to accomplish desired flow regime changes.

Heinz, Stefan (ORCID:0000000248712416)↗
DOE OSTI2023

Impedance-Based Detection of NO 2 Using Ni-MOF-74: Influence of Competitive Gas Adsorption

Chemically robust, low-power sensors are needed for the direct electrical detection of toxic gases. Metal–organic frameworks (MOFs) offer exceptional chemical and structural tunability to meet this challenge, though further understanding is needed regarding how coadsorbed gases influence or interfere with the electrical response. To probe the influence of competitive gases on trace NO 2 detection in a simulated flue gas stream, a combined structure–property study integrating synchrotron powder diffraction and pair distribution function analyses was undertaken, to elucidate how structural changes associated with gas binding inside Ni-MOF-74 pores correlate with the electrical response from Ni-MOF-74-based sensors. Data were evaluated for 16 gas combinations of N 2 , NO 2 , SO 2 , CO 2 , and H 2 O at 50 °C. Fourier difference maps from a rigid-body Rietveld analysis showed that additional electron density localized around the Ni-MOF-74 lattice correlated with large decreases in Ni-MOF-74 film resistance of up to a factor of 6 × 10 3 , observed only when NO 2 was present. These changes in resistance were significantly amplified by the presence of competing gases, except for CO 2 . Without NO 2 , H 2 O rapidly (<120 s) produced small (1–3×) decreases in resistance, though this effect could be differentiated from the slower adsorption of NO 2 by the evaluation of the MOF’s capacitance. Furthermore, samples exposed to H 2 O displayed a significant shift in lattice parameters toward a larger lattice and more diffuse charge density in the MOF pore. Evaluating the Ni-MOF-74 impedance in real time, NO 2 adsorption was associated with two electrically distinct processes, the faster of which was inhibited by competitive adsorption of CO 2 . Together, this work points to the unique interaction of NO 2 and other specific gases (e.g., H 2 O, SO 2 ) with the MOF’s surface, leading to orders of magnitude decrease in MOF resistance and enhanced NO2 detection. Finally, understanding and leveraging these coadsorbed gases will further improve the gas detection properties of MOF materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Auto-Curation of Seismic Event Data for Signal Denoising

Denoising contaminated seismic signals for later processing is a fundamental problem in seismic signals analysis. Neural network approaches have shown success denoising local signals when trained on short-time Fourier transform spectrograms. One challenge of this approach is the onerous process of hand-labeling event signals for training. By leveraging the SCALODEEP seismic event detector, we develop an automated set of techniques for labeling event data. Despite region specific challenges, training the neural network denoiser on machine curated events shows comparable performance to the neural network trained on hand curated events. We showcase our technique with two experiments, one using Utah regional data and one using regional data from the Korean peninsula.

58 GEOSCIENCES↗
DOE OSTI2022

Revealing the Drag Instability in One-fluid Nonideal Magnetohydrodynamic Simulations of a 1D Isothermal C-shock

C-type shocks are believed to be ubiquitous in turbulent molecular clouds thanks to ambipolar diffusion. We investigate whether the drag instability in 1D isothermal C-shocks, inferred from the local linear theory of Gu & Chen, can appear in nonideal magnetohydrodynamic simulations. Two C-shock models (with narrow and broad steady-state shock widths) are considered to represent the typical environment of star-forming clouds. The ionization-recombination equilibrium is adopted for the one-fluid approach. In the 1D simulation, the inflow gas is continuously perturbed by a sinusoidal density fluctuation with a constant frequency. The perturbations clearly grow after entering the C-shock region until they start being damped at the transition to the post-shock region. We show that the profiles of a predominant Fourier mode extracted locally from the simulated growing perturbation match those of the growing mode derived from the linear analysis. Moreover, the local growth rate and wave frequency derived from the predominant mode generally agree with those from the linear theory. Therefore, we confirm the presence of the drag instability in simulated 1D isothermal C-shocks. We also explore the nonlinear behavior of the instability by imposing larger-amplitude perturbations to the simulation. We find that the drag instability is subject to wave steepening, leading to saturated perturbation growth. Issues concerning local analysis, nonlinear effects, one-fluid approach, and astrophysical applications are discussed.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Illustrated formalisms for total scattering data: a guide for new practitioners

The total scattering method is the simultaneous study of both the real- and reciprocal-space representations of diffraction data. While conventional Bragg-scattering analysis (employing methods such as Rietveld refinement) provides insight into the average structure of the material, pair distribution function (PDF) analysis allows for a more focused study of the local atomic arrangement of a material. Generically speaking, a PDF is generated by Fourier transforming the total measured reciprocal-space diffraction data (Bragg and diffuse) into a real-space representation. However, the details of the transformation employed and, by consequence, the resultant appearance and weighting of the real-space representation of the system can vary between different research communities. As the worldwide total scattering community continues to grow, these subtle differences in nomenclature and data representation have led to conflicting and confusing descriptions of how the PDF is defined and calculated. This paper provides a consistent derivation of many of these different forms of the PDF and the transformations required to bridge between them. Some general considerations and advice for total scattering practitioners in selecting and defining the appropriate choice of PDF in their own research are presented. This contribution aims to benefit people starting in the field or trying to compare their results with those of other researchers.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Primordial non-Gaussianity from the completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey II: measurements in Fourier space with optimal weights

ABSTRACT We present measurements of the local primordial non-Gaussianity parameter $f_{\mathrm{NL}}^\mathrm{local} $from the clustering of 343 708 quasars with redshifts 0.8 < z < 2.2 distributed over 4808 deg2 from the final data release (DR16) of the extended Baryon Acoustic Oscillation Spectroscopic Survey (eBOSS), the largest volume spectroscopic survey up to date. Our analysis is performed in Fourier space, using the power spectrum monopole at very large scales to constrain the scale-dependent halo bias. We demonstrate the robustness of our analysis pipeline with EZ-mock catalogues that simulate the eBOSS DR16 target selection. We carefully assess the impact of systematics on our measurement, exploiting a novel neural network scheme for cleaning the DR16 sample as well as test multiple contamination removal methods. We estimate the bias due to the overcorrection of the systematic removal to be sub-dominant compared to the statistical uncertainty. We find fNL = −12 ± 21 (68 per cent confidence) for the main clustering sample including quasars with redshifts between 0.8 and 2.2, after applying redshift weighting techniques, designed for non-Gaussianity measurement from large scales structure, to optimize our analysis, which improve our results by 37 per cent.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Molecular diversity of dissolved organic matter reflects macroecological patterns in river networks

Deciphering dissolved organic matter (DOM) molecular complexity is crucial for understanding ecosystem function. Using the continental-scale Worldwide Hydrobiogeochemistry Observation Network for Dynamic Rivers Systems (WHONDRS) Fourier-transform ion cyclotron resonance mass spectrometry (FTICR-MS) dataset, we reveal fundamental scaling patterns of DOM chemodiversity with watershed characteristics. Analysis of 54 river sites shows local and regional watershed features significantly influence DOM chemodiversity (2500–8718 unique formulae), exhibiting consistent scaling patterns across compound classes and a novel latitudinal gradient (decreasing diversity with increasing latitude). Scaling relationships for DOM composition vary by compound class. Crucially, the scaling parameters (B, baseline chemodiversity; Z, sensitivity) are linearly interrelated. This B–Z relationship is most robust for potentially bio-labile carbohydrates (coefficient of determination R 2 ≈ 0.85), diminishing for recalcitrant, plant-derived molecules (such as lignin), and indicates (potential) biolability-dependent coupling between baseline diversity and environmental responsiveness. These quantitative scaling relationships, with scaling exponents ranging from − 2.1 to 2.2 across compound classes, enable prediction of DOM composition across watersheds, offering a framework to understand ecosystem responses to environmental change. This research bridges biogeochemistry and ecology, providing tools to anticipate molecular transformations across scales.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Power-transfer and fixed-point analysis of sawtooth simulations of a current-carrying stellarator

Power-transfer and fixed-point analysis of previous NIMROD simulations (Roberds et al. , Phys. Plasmas , vol. 23, issue 9, 2016, 092513) improved the understanding of the effect of 3D (non-axisymmetric equilibrium) magnetic fields on sawtooth oscillations in the Compact Toroidal Hybrid (CTH) experiment. Computing the locations of order-1 fixed points, their Greene's residues, and local values for the rotational transform results in a description of CTH sawteeth consistent with Kadomtsev's model. A power-transfer analysis quantifies the distribution of energy among toroidal Fourier modes and their nonlinear interactions. The Lorentz power transfer drives sawtooth growth, and it is unambiguously interpreted as the flow of energy from toroidal mode $n'$ to mode $n$ , catalysed by $\boldsymbol {B}_{n-n'}$ . It has been reported previously that the CTH sawtooth frequency increases with the 3D field strength. This is attributed to an increased growth rate of the internal kink that drives sawtooth oscillations. Here, 3D fields remove energy from the kink, eliminating the possibility that these fields are an additional energy source that drives growth. Instead, 3D fields catalyse energy transfer from large-to-small scales, where magnetic reconnection is stronger. It is proposed that this energy transfer increases the reconnection rate at small scales, which is consistent with the increased growth rate observed at higher 3D field strengths.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

In Situ ATR-FTIR Study of the Cathode–Electrolyte Interphase: Electrolyte Solution Structure, Transition Metal Redox, and Surface Layer Evolution

We present a study of the lithium nickel manganese cobalt oxide (LiNi 0.6 Mn 0.2 Co 0.2 O 2 , NMC622) cathode-electrolyte interphase (CEI) during galvanostatic charging and discharging using in situ attenuated total reflectance Fourier transform infrared (ATR-FTIR) methods to investigate the voltage dependent electrolyte solution structure changes at the interface, transition metal (TM) redox chemistry, and cathode/electrolyte interfacial layer evolution. Furthermore, our in situ cell design provides both reliable electrochemical device testing and strong FTIR vibrational absorption signals near the cathode surface. Specifically, advanced spectral analysis elucidates changes of near-surface Li+ ion (de)solvation by solvent molecules during galvanostatic cycling. Moreover, cathode metal-oxygen vibrational absorptions, sensitive to TM redox behaviors and subsequent local structural variations, were correlated to cathode de-lithiation (and lithiation) and electrolyte solution structure changes. In addition, we have detected the formation and evolution of a CEI surface layer on the NMC622 cathode that contributes to the cell's capacity fade.

25 ENERGY STORAGE↗
DOE OSTI2025

A generalizable machine learning-assisted fast Fourier transform algorithm to simulate the large strain phenomena in polycrystalline materials

Machine learning methods have shown initial promise in constitutive modeling for single crystals or homogenized polycrystals, delivering notable computational efficiency. However, existing machine learning-based constitutive models often lack generalizability, limiting their application across diverse boundary value problems. This study introduces a thermodynamics-informed artificial neural network model to accelerate rate-tangent crystal plasticity fast Fourier transform simulations for cross-scale deformation behaviors of polycrystals under complex loading. Our model integrates microstructural variability and local interactions effectively. To address local effects in each grain, we employ K-means clustering to group Gauss points within the microstructure into clusters assumed to be in similar mechanical states. This approach, based on self-clustering analysis, extends model scope from macroscopic stress response to the granular level, capturing mechanical responses and orientation evolution across grains. This reduces the number of nonlinear problems to solve, with cluster responses propagated throughout each group. The thermodynamics-based artificial neural network-extracted features are further processed using local material state clusters to account for history-dependent deformation and evolving microstructures. Additionally, representative volume element simulations with rate-tangent crystal plasticity fast Fourier transform provide reliable datasets for model training. The proposed model demonstrates high efficiency, accuracy, self-consistency, and enhanced generalizability in predicting strain–stress responses and orientation evolution at both individual grain and aggregate scales under complex loading conditions, such as biaxial tension and arbitrary loading scenarios.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Likelihood-based signal and noise analysis for docking of models into cryo-EM maps

Fast, reliable docking of models into cryo-EM maps requires understanding of the errors in the maps and the models. Likelihood-based approaches to errors have proven to be powerful and adaptable in experimental structural biology, finding applications in both crystallography and cryo-EM. Indeed, previous crystallographic work on the errors in structural models is directly applicable to likelihood targets in cryo-EM. Likelihood targets in Fourier space are derived here to characterize, based on the comparison of half-maps, the direction- and resolution-dependent variation in the strength of both signal and noise in the data. Because the signal depends on local features, the signal and noise are analysed in local regions of the cryo-EM reconstruction. The likelihood analysis extends to prediction of the signal that will be achieved in any docking calculation for a model of specified quality and completeness. A related calculation generalizes a previous measure of the information gained by making the cryo-EM reconstruction.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Evaluating Scalograms for Seismic Event Denoising

Denoising contaminated seismic signals for later processing is a fundamental problem in seismic signals analysis. The most straightforward denoising approach, using spectral filtering, is not effective when noise and seismic signal occupy the same frequency range. Neural network approaches have shown success denoising local signal when trained on short-time Fourier transform spectrograms (Zhu et al 2018; Tibi et al 2021). Scalograms, a wavelet-based transform, achieved ~15% better reconstruction as measured by dynamic time warping on a seismic waveform test set than spectrograms, suggesting their use as an alternative for denoising. We train a deep neural network on a scalogram dataset derived from waveforms recorded by the University of Utah Seismograph Stations network. We find that initial results are no better than a spectrogram approach, with additional overhead imposed by the significantly larger size of scalograms. A robust exploration of neural network hyperparameters and network architecture was not performed, which could be done in follow on work.

58 GEOSCIENCES↗