Search NASA⌕ Search

SEARCH · Search NASA

Results for “sparse grid interpolation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

A Sparse-Grid Probabilistic Scheme for Approximation of the Runaway Probability of Electrons in Fusion Tokamak Simulation

Runaway electrons (RE) generated during magnetic disruptions present a major threat to the safe operation of plasma nuclear fusion reactors. A critical aspect of understanding RE dynamics is to calculate the runaway probability, i.e., the probability that an electron in the phase space will runaway on, or before, a prescribed time. Such probability can be obtained by solving the adjoint equation of the underlying Fokker-Planck equation that controls the electron dynamics. In this effort, we present a sparse-grid probabilistic scheme for computing the runaway probability. The key ingredient of our approach is to represent the solution of the adjoint equation as a conditional expectation, such that discretizing the differential operator reduces to the approximation of a set of integrals. Adaptive sparse grid interpolation is utilized to approximate the map from the phase space to the runaway probability. The main novelties of this effort are the integration of the sparse-grid method into the probabilistic numerical scheme for computing escape probability, and the application of the proposed method in computing RE probabilities. Two numerical examples are given to illustrate that the proposed method can achieve O(Δt) convergence, and that the local anisotropic adaptive refinement strategy (M. Stoyanov, Adaptive sparse grid construction in a context of local anisotropy and multiple hierarchical parents. In: Sparse Grids and Applications-Miami 2016, Springer, Berlin, 2018, pp. 175–199) can effectively handle the sharp transition layer between the runaway and non-runaway regions.

Yang, Minglei↗
DOE OSTI2017

A two-stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media

A new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability. For pressure estimation, it combines the dimension-adaptive stochastic collocation method on Smolyak sparse grids and the Kronrod–Patterson–Hermite nested quadrature formulas. For saturation estimation, an additional stage is developed, in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples, to address the low regularity problem of the saturation. Numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two-stage adaptive stochastic collocation method on nested sparse grids.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. Here, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Advanced surrogate model for electron-scale turbulence in tokamak pedestals

We derive an advanced surrogate model for predicting turbulent transport at the edge of tokamaks driven by electron temperature gradient (ETG) modes. Our derivation is based on a recently developed sensitivity-driven sparse grid interpolation approach for uncertainty quantification and sensitivity analysis at scale, which informs the set of parameters that define the surrogate model as a scaling law. Our model reveals that ETG-driven electron heat flux is influenced by the safety factor q, electron beta β e and normalized electron Debye length λ D , in addition to well-established parameters such as the electron temperature and density gradients. To assess the trustworthiness of our model's predictions beyond training, we compute prediction intervals using bootstrapping. The surrogate model's predictive power is tested across a wide range of parameter values, including within-distribution testing parameters (to verify our model) as well as out-of-bounds and out-of-distribution testing (to validate the proposed model). Overall, validation efforts show that our model competes well with, or can even outperform, existing scaling laws in predicting ETG-driven transport.

fusion plasma↗
DOE OSTI2022

A general framework for quantifying uncertainty at scale

Abstract In many fields of science, comprehensive and realistic computational models are available nowadays. Often, the respective numerical calculations call for the use of powerful supercomputers, and therefore only a limited number of cases can be investigated explicitly. This prevents straightforward approaches to important tasks like uncertainty quantification and sensitivity analysis. This challenge can be overcome via our recently developed sensitivity-driven dimension-adaptive sparse grid interpolation strategy. The method exploits, via adaptivity, the structure of the underlying model (such as lower intrinsic dimensionality and anisotropic coupling of the uncertain inputs) to enable efficient and accurate uncertainty quantification and sensitivity analysis at scale. Here, we demonstrate the efficiency of this adaptive approach in the context of fusion research, in a realistic, computationally expensive scenario of turbulent transport in a magnetic confinement tokamak device with eight uncertain parameters, reducing the effort by at least two orders of magnitude. In addition, we show that this refinement method intrinsically provides an accurate surrogate model that is nine orders of magnitude cheaper than the high-fidelity model.

Farcaş, Ionuţ-Gabriel (ORCID:0000000238841580)↗
DOE OSTI2018

Adaptive Sparse Grid Construction in a Context of Local Anisotropy and Multiple Hierarchical Parents

We consider general strategy for hierarchical multidimensional interpolation based on sparse grids, where the interpolation nodes and locally supported basis functions are constructed from tensors of a one dimensional hierarchical rule. We consider four different hierarchies that are tailored towards general functions, high or low order polynomial approximation, or functions that satisfy homogeneous boundary conditions. The main advantage of the locally supported basis is the ability to choose a set of functions based on the observed behavior of the target function. We present an alternative to the classical surplus refinement techniques, where we exploit local anisotropy and refine using functions with not strictly decreasing support. The more flexible refinement strategy improves stability and reduces the total number of expensive simulations, resulting in significant computational saving. We demonstrate the advantages of the different hierarchies and refinement techniques by application to series of simple functions as well as a system of ordinary differential equations given by the Kermack-McKendrick SIR model.

Stoyanov, Miroslav K.↗
DOE OSTI2016

Calibration of an agricultural-hydrological model (RZWQM2) using surrogate global optimization

Robust calibration of an agricultural-hydrological model is critical for simulating crop yield and water quality and making reasonable agricultural management. However, calibration of the agricultural-hydrological system models is challenging because of model complexity, the existence of strong parameter correlation, and significant computational requirements. Therefore, only a limited number of simulations can be allowed in any attempt to find a near-optimal solution within an affordable time, which greatly restricts the successful application of the model. The goal of this study is to locate the optimal solution of the Root Zone Water Quality Model (RZWQM2) given a limited simulation time, so as to improve the model simulation and help make rational and effective agricultural-hydrological decisions. To this end, we propose a computationally efficient global optimization procedure using sparse-grid based surrogates. We first used advanced sparse grid (SG) interpolation to construct a surrogate system of the actual RZWQM2, and then we calibrate the surrogate model using the global optimization algorithm, Quantum-behaved Particle Swarm Optimization (QPSO). As the surrogate model is a polynomial with fast evaluation, it can be efficiently evaluated with a sufficiently large number of times during the optimization, which facilitates the global search. We calibrate seven model parameters against five years of yield, drain flow, and NO 3 -N loss data from a subsurface-drained corn-soybean field in Iowa. Results indicate that an accurate surrogate model can be created for the RZWQM2 with a relatively small number of SG points (i.e., RZWQM2 runs). Compared to the conventional QPSO algorithm, our surrogate-based optimization method can achieve a smaller objective function value and better calibration performance using a fewer number of expensive RZWQM2 executions, which greatly improves computational efficiency.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

Calibration of the E3SM Land Model Using Surrogate‐Based Global Optimization

Abstract Calibration of the Energy Exascale Earth System Model (E3SM), land model (ELMv0) is challenging because of its model complexity, strong model nonlinearity, and significant computational requirements. Therefore, only a limited number of simulations can be allowed in any attempt to find a near‐optimal solution within an affordable time. The goal of this study is to calibrate some of the ELMv0 parameters to improve model projection of carbon fluxes. We propose a computationally efficient global optimization procedure using sparse‐grid based surrogates. We first use advanced sparse grid (SG) interpolation to construct a surrogate system of the ELMv0, and then calibrate the surrogate model in the optimization process. As the surrogate model is a polynomial whose evaluation is fast, it can be efficiently evaluated a sufficiently large number of times in the optimization, which facilitates the global search. We calibrate eight parameters against five years of net ecosystem exchange, total leaf area index, and latent heat flux data from the U.S. Missouri Ozark flux tower. The calibrated model is then used for predicting the three variables in the following 4 years. The results indicate that an accurate surrogate model can be created for the ELMv0 with a relatively small number of SG points, i.e., a few ELMv0 simulations that can be fully parallel. And, the application of the optimized parameters leads to a better model performance and a higher predictive capability than the default parameter values in the ELMv0.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection

This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing approximation. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Nucleon-Pion Spectroscopy from Sparsened Correlators

Neutrino oscillation experiments require accurate reconstructions of neutrino energies, which depend in part on a theoretical understanding of the axial $N\rightarrow\Delta$ transition form factors. A lattice QCD study of this transition will require construction of all hadronic states with energies up to $m_\Delta$, which at the physical point includes $N\pi$ and $N\pi\pi$. Building interpolating operators from sparse grids at the source and sink is a versatile method that allows construction of a wide range of diagram topologies and has successfully been used in other multi-hadron calculations. We will discuss application of this method to nucleon-pion systems and present preliminary results.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Towards Lattice QCD Calculations of Pion Production

Neutrino oscillation experiments require accurate reconstructions of neutrino energies, which depend in part on a theoretical understanding of the axial N→Δ transition form factors. A lattice QCD study of this transition will require construction of all hadronic states with energies up to mΔ, which at the physical point includes Nπ and Nππ. Building interpolating operators from sparse grids at the source and sink is a versatile method that allows construction of a wide range of diagram topologies and has successfully been used in other multi-hadron calculations. We will discuss application of this method to nucleon-pion systems and present preliminary results.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

A New Precision Measurement of the Small-scale Line-of-sight Power Spectrum of the Ly α Forest

We present a new measurement of the Lyα forest power spectrum at 1.8 < z < 3.4 using 74 Keck/HIRES and VLT/UVES high-resolution, high-signal-to-noise-ratio quasar spectra. We developed a custom pipeline to measure the power spectrum and its uncertainty, which fully accounts for finite resolution and noise and corrects for the bias induced by masking missing data, damped Lyα absorption systems, and metal absorption lines. Our measurement results in unprecedented precision on the small-scale modes $k\gt 0.02\,{\rm{s}}\,{\mathrm{km}}^{-1}$, inaccessible to previous SDSS/BOSS analyses. It is well known that these high-k modes are highly sensitive to the thermal state of the intergalactic medium, but contamination by narrow metal lines is a significant concern. We quantify the effect of metals on the small-scale power and find a modest effect on modes with $k\lt 0.1\,{\rm{s}}\,{\mathrm{km}}^{-1}$. As a result, by masking metals and restricting to $k\lt 0.1\,{\rm{s}}\,{\mathrm{km}}^{-1}$, their impact is completely mitigated. We present an end-to-end Bayesian forward-modeling framework whereby mock spectra with the same noise, resolution, and masking as our data are generated from Lyα forest simulations. These mock spectra are used to build a custom emulator, enabling us to interpolate between a sparse grid of models and perform Markov chain Monte Carlo fits. Our results agree well with BOSS on scales $k\lt 0.02\,{\rm{s}}\,{\mathrm{km}}^{-1}$, where the measurements overlap. The combination of the percent-level low-k precision of BOSS with our 5%–15% high-k measurements results in a powerful new data set for precisely constraining the thermal history of the intergalactic medium, cosmological parameters, and the nature of dark matter. Lastly, the power spectra and their covariance matrices are provided as electronic tables.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2018

On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains

Here, this work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subset of the real line and can be extended to unbounded domains with the introduction of a weight function w:R → [0,1]. Due to a simple recursive formulation in one dimension, such abscissas provide a foundation for high-dimensional approximation methods such as sparse grid collocation, deterministic least squares and compressed sensing. Just as in the unweighted case of interpolation on a compact domain, we use results from potential theory to prove that the Lebesgue constant for the Leja points grows subexponentially with the number of interpolation nodes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A scalable framework for efficient coupling of thermal and microstructural simulations in additive manufacturing

Predicting microstructure evolution in metal additive manufacturing (AM) is important for process optimization, but spatiotemporal scale disparities between thermal transport and microstructure evolution create significant challenges for efficient data transfer between simulation codes. To address this, we present Stork, a scalable framework for coupling thermal and microstructural simulations. Stork uses a sparse data representation to identify and store active solidification sub-volumes, enabling highly parallel quad-linear interpolation from coarse thermal grids to fine microstructure grids without large intermediate storage. We demonstrate the framework by coupling the semi-analytic heat transfer code 3DThesis with the time-parallel cellular automata code Toucan. This approach achieves over two orders of magnitude reduction in data generation time and file size compared to prior workflows. Numerical studies show that quad-linear interpolation preserves grain morphology and crystallographic texture in laser powder bed fusion (LPBF) simulations for coarsening ratios up to 16. Overall, Stork provides a scalable pathway for high-throughput, component-scale AM simulations on modern high-performance computing systems.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Quantifying uncertainty in the process-structure relationship for Al–Cu solidification

Phase field method (PFM) is a simulation tool to predict microstructural evolution during solidification and is helpful to establish the process-structure relationship for alloys. The robustness of the relationship however is affected by model-form and parameter uncertainties in PFM. In this paper, the uncertainty associated with the thermodynamic and process parameters of PFM is studied and quantified. Surrogate modeling is used to interpolate four quantities of interests (QoIs), including dendritic perimeter, area, primary arm length, and solute segregation, as functions of thermodynamic and process parameters. A sparse grid approach is applied to mitigate the curse-of-dimensionality computational burden in uncertainty quantification. Polynomial chaos expansion is employed to obtain the probability density functions of the QoIs. The effect of parameter uncertainty on the Al–Cu dendritic growth during solidification simulation are investigated. Here, the results show that the dendritic morphology varies significantly with respect to the interface mobility and the initial temperature.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Bayesian Assimilation of Multiscale Precipitation Data and Sparse Ground Gauge Observations in Mountainous Areas

Estimating the spatial distribution of precipitation is important for understanding ecohydrological processes at catchment scales. However, this estimation is difficult in mountainous areas because ground-based observation stations are often sparsely located and do not represent the spatial variability of precipitation. In this study, we develop a Bayesian assimilation method based on data collected on the Tibetan Plateau from 1980 to 2014 to estimate monthly and daily precipitation. To accomplish this, point-scale ground meteorological observations are combined with large-scale precipitation data such as satellite observations or reanalysis data. First, we remove the terrain effects from ground observations by fitting the precipitation data as functions of elevation, and then we spatially interpolate the residuals to 5-km-resolution grids to obtain monthly and daily precipitation. Additionally, we use Tropical Rainfall Measuring Mission (TRMM) satellite observations and ERA-Interim reanalysis data. Cross-validation methods are used to evaluate our method; the results show that our method not only captures the change in precipitation with terrain but also significantly reduces the associated uncertainty. The improvements are more evident in the main river source areas on the edge of the Tibetan Plateau, where elevation changes dramatically, and in high-altitude areas, where the ground gauges are sparse compared with those in low-altitude areas. Our assimilation method is applicable to other regions and is particularly useful for mountainous watersheds where ground meteorological stations are sparse and precipitation is considerably influenced by terrain.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2016

Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations

We extend the reduced basis (RB) accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [16,17] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin high-fidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the so-called Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimension-independent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinite-dimensional parametric, deterministic forward map and operator admit N-term RB and EIM approximations which converge at rates which depend only on the sparsity of the parametric forward map. In several numerical experiments, the proposed algorithms exhibit dimension-independent convergence rates which equal, at least, the currently known rate estimates for N-term approximation. We propose to accelerate Bayesian estimation by first offline construction of reduced basis surrogates of the Bayesian posterior density. The parsimonious surrogates can then be employed for online data assimilation and for Bayesian estimation. They also open a perspective for optimal experimental design.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Volumetric Rendering on Wavelet-Based Adaptive Grid

Numerical modeling of physical phenomena frequently involves processes across a wide range of spatial and temporal scales. In the last two decades, the advancements in wavelet-based numerical methodologies to solve partial differential equations, combined with the unique properties of wavelet analysis to resolve localized structures of the solution on dynamically adaptive computational meshes, make it feasible to perform large-scale numerical simulations of a variety of physical systems on a dynamically adaptive computational mesh that changes both in space and time. Volumetric visualization of the solution is an essential part of scientific computing, yet the existing volumetric visualization techniques do not take full advantage of multi-resolution wavelet analysis and are not fully tailored for visualization of a compressed solution on the wavelet-based adaptive computational mesh. Our objective is to explore the alternatives for the visualization of time-dependent data on space-time varying adaptive mesh using volume rendering while capitalizing on the available sparse data representation. Two alternative formulations are explored. The first one is based on volumetric ray casting of multi-scale datasets in wavelet space. Rather than working with the wavelets at the finest possible resolution, a partial inverse wavelet transform is performed as a preprocessing step to obtain scaling functions on a uniform grid at a user-prescribed resolution. As a result, a solution in physical space is represented by a superposition of scaling functions on a coarse regular grid and wavelets on an adaptive mesh. An efficient and accurate ray casting algorithm is based just on these coarse scaling functions. Additional details are added during the ray tracing by taking an appropriate number of wavelets into account based on support overlap with the interpolation point, wavelet coefficient magnitude, and other characteristics, such as opacity accumulation (front to back ordering) and deviation from frontal viewing direction. The second approach is based on complementing of wavelet-based adaptive mesh to the traditional Adaptive Mesh Refinement (AMR) mesh. Both algorithms are illustrated and compared to the existing volume visualization software for Rayleigh-Benard thermal convection and electron density data sets in terms of rendering time and visual quality for different data compression of both wavelet-based and AMR adaptive meshes.

Vezolainen, Alexei V.↗