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DOE OSTI2022

Expedited Real Time Processing for the NETL HYPER Cyber-Physical System (Final Report)

The primary objective of the proposed project was to provide the National Energy Technology Laboratory’s Hybrid Performance (Hyper) Facility the needed numerical methods algorithm(s), software development and implementation support to enact real time cyber-physical systems (CPSs) that simulate process dynamics on the order of five milliseconds or smaller. The proposed paths forward comprised three distinct approaches to hasten transient simulations. They fell under the numerical methods categorical options of: 1) optimizing key parameters within the facility’s pre-existing real time processing scheme; 2) introducing an “informed” processing approach wherein a priori computations expedite real time attempts; and 3) implementing alternatives to the employed explicit-implicit blended finite difference (spatio-temporal) approach. Each of these three classes were proposed independently as options for improvement, yet in some cases one class of approach could have complemented another.

20 FOSSIL-FUELED POWER PLANTS↗

DOE OSTI2022

KSSOLV 2.0: An efficient MATLAB toolbox for solving the Kohn-Sham equations with plane-wave basis set

KSSOLV (Kohn-Sham Solver) is a MATLAB toolbox for performing Kohn-Sham density functional theory (DFT) calculations with a plane-wave basis set. KSSOLV 2.0 preserves the design features of the original KSSOLV software to allow users and developers to easily set up a problem and perform ground-state calculations as well as to prototype and test new algorithms. Furthermore, it includes new functionalities such as new iterative diagonalization algorithms, k-point sampling for electron band structures, geometry optimization and advanced algorithms for performing DFT calculations with local, semi-local, and hybrid exchange-correlation functionals. It can be used to study the electronic structures of both molecules and solids. So, we describe these new capabilities in this work through a few use cases. We also demonstrate the numerical accuracy and computational efficiency of KSSOLV on a variety of examples.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

ZEUS: An Efficient GPU Optimization Method Integrating PSO, BFGS, and Automatic Differentiation

We introduce a novel, efficient computational method, ZEUS, for numerical optimization, and provide an open-source implementation. It has four key ingredients: (1) particle swarm optimization (PSO), (2) the use of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, (3) automatic differentiation (AD), and (4) GPUs. Our approach addresses the computational challenges inherent in high-dimensional, non-convex optimization problems. In the first phase of the algorithm, we get a potentially good set of starting points using PSO. Thereafter, we run BFGS independently in parallel from these starting points. BFGS is one of the best-performing algorithms for numerical optimization. However, it requires the gradient of the function being optimized. ZEUS integrates automatic differentiation into BFGS thus avoiding the need for the user to calculate derivatives explicitly. The use of GPUs allows ZEUS to speed up the calculations substantially. We carry out systematic studies to explore the trade-offs between the number of PSO iterations taken, starting points, and BFGS iteration depth. We show that a handful of iterations of PSO can improve global convergence when combined with BFGS. We also present performance studies using common test functions. The source code can be found at https://github.com/fnal-numerics/global-optimizer-gpu.

Soos, Dominik [Old Dominion U.]↗

DOE OSTI2019

The Bloch equation for spin dynamics in electron storage rings: Computational and theoretical aspects

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the [Formula: see text] option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

Physics↗

DOE OSTI2022

Differential holography

Differential Holography technology measures the amplitude and/or phase of, e.g., an incident linearly polarized spatially coherent quasi-monochromatic optical field by optically computing the first derivative of the field and linearly mapping it to an irradiance signal detectable by an image sensor. This information recorded on the image sensor is then recovered by a simple algorithm. In some embodiments, an input field is split into two or more beams to independently compute the horizontal and vertical derivatives (using amplitude gradient filters in orthogonal orientations) for detection on one image sensor in separate regions of interest (ROIs) or on multiple image sensors. A third unfiltered beam recorded in a third ROI directly measures amplitude variations in the input field to numerically remove its contribution as noise before recovering the original wavefront using a numerical in algorithm. When combined, the measured amplitude and phase constitute a holographic recording of the incident optical field.

Ravizza, Frank↗

DOE OSTI2024

Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms

Non-thermal advanced fuel fusion trades the requirement of a large amount of recirculating tritium in the system for that of large recirculating power. Phase space engineering technologies utilizing externally injected electromagnetic fields can be applied to meet the challenge of maintaining non-thermal particle distributions at a reasonable cost. The physical processes of the phase space engineering are studied from a theoretical and algorithmic perspective. It is emphasized that the operational space of phase space engineering is limited by the underpinning symplectic dynamics of charged particles. The phase space incompressibility according to the Liouville theorem is just one of many constraints, and Gromov's non-squeezing theorem determines the minimum footprint of the charged particles on every conjugate phase space plane. In this sense and level of sophistication, the mathematical abstraction of phase space engineering is symplectic topology. To simulate the processes of phase space engineering, such as the Maxwell demon and electromagnetic energy extraction, and to accurately calculate the minimum footprints of charged particles, recently developed structure-preserving geometric algorithms can be used. The family of algorithms conserves exactly, on discretized spacetime, symplecticity and thus incompressibility, non-squeezability, and symplectic capacities. The algorithms apply to the dynamics of charged particles under the influence of external electromagnetic fields as well as the charged particle–electromagnetic field system governed by the Vlasov–Maxwell equations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Computational projects with the Landau–Zener problem in the quantum mechanics classroom

The Landau–Zener problem, where a minimum energy separation is passed with constant rate in a two-state quantum-mechanical system, is an excellent model quantum system for a computational project. It requires a low-level computational effort, but has a number of complex numerical and algorithmic issues that can be resolved through dedicated work. It can be used to teach computational concepts, such as accuracy, discretization, and extrapolation, and it reinforces quantum concepts of time-evolution via a time-ordered product and of extrapolation to infinite time via time-dependent perturbation theory. Additionally, we discuss the concept of compression algorithms, which are employed in many advanced quantum computing strategies, and easy to illustrate with the Landau–Zener problem.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Simulation-Based Inference for Neutrino Interaction Model Parameter Tuning

High-energy physics experiments studying neutrinos rely heavily on simulations of their interactions with atomic nuclei. Limitations in the theoretical understanding of these interactions typically necessitate ad hoc tuning of simulation model parameters to data. Traditional tuning methods for neutrino experiments have largely relied on simple algorithms for numerical optimization. While adequate for the modest goals of initial efforts, the complexity of future neutrino tuning campaigns is expected to increase substantially, and new approaches will be needed to make progress. In this paper, we examine the application of simulation-based inference (SBI) to the neutrino interaction model tuning for the first time. Using a previous tuning study performed by the MicroBooNE experiment as a test case, we find that our SBI algorithm can correctly infer the tuned parameter values when confronted with a mock data set generated according to the MicroBooNE procedure. This initial proof-of-principle illustrates a promising new technique for next-generation simulation tuning campaigns for the neutrino experimental community.

Tame-Narvaez, Karla Maria [Fermilab]↗

DOE OSTI2025

Simulation-based inference for neutrino interaction model parameter tuning

Tame-Narvaez, Karla [Fermilab] (ORCID:000000022249↗

DOE OSTI2025

Randomized Algorithms for Linear Solvers

Recently, randomized algorithms in numerical linear algebra, specifically those centered around random sketching, have gained traction in primarily theoretical research due to their potential to significantly reduce problem dimensionality at the cost of an O(1) multiplicative distortion factor. It has been assumed that this sketching can be done efficiently, but thorough investigation into how precisely to do it has been neglected. Moreover, the theory-based community has argued for sketching’s ability to reduce computational cost via complexity analysis, but has not researched how it affects the stability of the algorithms. At Sandia, efficient linear solvers that scale well on modern HPC architectures while maintaining stability are imperative for practical applications. In this LDRD, we developed a random sketching strategy that is substantially faster than existing ones, and demonstrate its superior performance in practice on a NVIDIA H100 GPU. Moreover, we show how this can be used to significantly outperform existing linear least squares solvers while improving the solver’s stability as well. Additionally, we demonstrate how this sketching strategy can be used to make a fast, stable QR factorization that can subsequently be used in s-step and block Krylov solvers. Finally, we incorporate a sketching-based block orthogonalization scheme into s-step GMRES, which is stable and faster than existing approaches on the Perlmutter supercomputer.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. Here we also show at the end an application in the framework of plasma physics.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Extended Barton–Bandis model for rock joints under cyclic loading: Formulation and implicit algorithm

In this paper, the Barton–Bandis model for rock joints is extended to cyclic loading conditions, without any new material parameter. Also developed herein is an implicit solution algorithm for the extended Barton–Bandis model, which can also be used for the original Barton–Bandis model for which an implicit algorithm has been unavailable. To this end, we first cast the Barton–Bandis model into an incremental elasto-plastic framework, deriving an expression for the elastic shear stiffness being consistent with the original model formulation. We then extend the model formulation to cyclic loading conditions, incorporating the dependence of shear stress and dilation on the joint position and the shearing direction. The extension is achieved by introducing a few state-dependent variables which can be calculated with the existing material parameters. For robust and accurate utilization of the model, we also develop an implicit algorithm based on return mapping, which is unconditionally stable and guarantees the satisfaction of the strength criterion. We verify that the proposed model formulation and algorithm produce virtually the same results as the original Barton–Bandis model under monotonic shearing conditions. Here, we then validate the extended Barton–Bandis model against experimental data on natural rock joints under cycling loading conditions. The present work thus enables the Barton–Bandis model, which has been exceptionally popular in research and practice, to be applicable to a wider range of problems in rock mechanics and rock engineering.

58 GEOSCIENCES↗

DOE OSTI2023

Sparse matrix‐vector and matrix‐multivector products for the truncated SVD on graphics processors

Summary Many practical algorithms for numerical rank computations implement an iterative procedure that involves repeated multiplications of a vector, or a collection of vectors, with both a sparse matrix and its transpose. Unfortunately, the realization of these sparse products on current high performance libraries often deliver much lower arithmetic throughput when the matrix involved in the product is transposed. In this work, we propose a hybrid sparse matrix layout, named CSRC, that combines the flexibility of some well‐known sparse formats to offer a number of appealing properties: (1) CSRC can be obtained at low cost from the popular CSR (compressed sparse row) format; (2) CSRC has similar storage requirements as CSR; and especially, (3) the implementation of the sparse product kernels delivers high performance for both the direct product and its transposed variant on modern graphics accelerators thanks to a significant reduction of atomic operations compared to a conventional implementation based on CSR. This solution thus renders considerably higher performance when integrated into an iterative algorithm for the truncated singular value decomposition (SVD), such as the randomized SVD or, as demonstrated in the experimental results, the block Golub–Kahan–Lanczos algorithm.

Aliaga, José I.↗

DOE OSTI2025

Determining the Ensemble N -Representability of Reduced Density Matrices

The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy that embeds an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both simple models as well as molecular systems at finite temperature, demonstrating its robustness.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Identification of crystal plasticity model parameters by multi-objective optimization integrating microstructural evolution and mechanical data

Crystal plasticity models evolve a polycrystalline yield surface using meso-scale descriptions of deformation mechanisms. The activation of deformation mechanisms is governed by crystallography and a set of model parameters, which are typically calibrated through the fitting of mechanical data such as stress–strain curves and elastic lattice strains. Microstructural data such as phase fractions and texture evolution are used for verifying crystal plasticity parameters. In this study, we use a multi-objective genetic algorithm to identify hardening parameters from flow stress curves with an option to incorporate texture into the optimization approach. Robust, generalized objective functions are developed and used to identify sets of parameters pertaining to dislocation density-based hardening laws in visco-plastic and elasto-plastic self-consistent (VPSC and EPSC) homogenization models. First, the parameters are identified for pure Nb directly from texture using an objective function based on generalized spherical harmonics. Since texture evolution is driven by the relative contribution of active slip systems, the parameters governing the evolution of slip resistance ratios can be recovered from fitting discrete textures at a series of strains. Next, a comprehensive set of load reversal data for dual phase (DP) 780 steel is used to fit a hardening law and a back-stress law in EPSC. Finally, parameters pertaining to a complex hardening law for the evolution of slip and twinning in pure α-Ti are identified. Remarkably, using texture as an objective in combination with stress–strain objectives constrains the model of Ti to fully reproduce not only stress–strain and texture evolution but also hierarchical twinning measurements as a function of initial grain size and texture. Furthermore, given an appropriate model fit to representative experimental texture evolution, underlying twin volume fractions contributing to texture evolution can be predicted.

42 ENGINEERING↗

DOE OSTI2026

Adaptive Sampling-Based Bi-Fidelity Stochastic Trust Region Method for Stochastic Derivative-Free Optimization

Bi-fidelity stochastic optimization has gained increasing attention as an efficient approach to reduce computational costs by leveraging a low-fidelity (LF) model to optimize an expensive high-fidelity (HF) objective. In this paper, we propose ASTRO-BFDF, an adaptive sampling trust-region method specifically designed for unconstrained bi-fidelity stochastic derivative-free optimization problems. In ASTRO-BFDF, the LF function serves two purposes: (i) to identify better iterates for the HF function when the optimization process indicates a high correlation between them and (ii) to reduce the variance of the HF function estimates using bi-fidelity Monte Carlo (BFMC). The algorithm dynamically determines sample sizes while adaptively choosing between crude Monte Carlo and BFMC to balance the trade-off between optimization and sampling errors. We prove that the iterates generated by ASTRO-BFDF converge to a first-order stationary point almost surely. Additionally, we demonstrate the effectiveness of the proposed algorithm through numerical experiments on synthetic benchmarks and simulation optimization problems involving discrete event systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Modeling the role of local crystallographic correlations in microstructures of Ti-6Al-4V using a correlated structure visco-plastic self-consistent polycrystal plasticity formulation

This paper presents a multi-level crystal plasticity-based simulation framework for modeling mechanical response and microstructure evolution of Ti-6Al-4V with α-lath/lamellar microstructures. Here, the model is a correlated structure visco-plastic self-consistent (CS-VPSC) formulation linking three scales: a single crystals micro-scale, a lath/lamellar colony meso-scale, and a lath/lamellar aggregate macro-scale. A selected hardening law for the evolution of critical resolved shear stress per slip system used in CS-VPSC is phenomenological. However, it adjusts the resistances of basal and prismatic slip systems based on the geometry of slip transfer between adjacent lamellae. Consistent with experimental evidences, the resolved shear stress on the pyramidal slip planes is dependent not only on the stress in the direction of slip but also on the two orthogonal shear stress components and the three normal stress components (non-Schmid effects). Electron backscatter diffraction (EBSD) data in conjunction with a procedure relying on α→β phase transformation is used to construct paired variants of α-laths/lamellae satisfying their local crystallographic correlations. The procedure fits volume fractions of individual laths/lamellae with the experimental EBSD data and selects a distribution of habit planes between adjacent variants with respect to the loading direction. The simulation framework is applied to interpret the deformation behavior in tension and compression along two sample directions of Ti-6Al-4V fabricated via laser powder bed fusion. Moreover, the model is used to simulate texture evolution during rolling of the material to large strains. It is demonstrated that the model is capable of predicting plastic anisotropy/asymmetry and the concomitant texture evolution. While the model reveals a significant effect of habit plane inclination with respect to the loading direction on yield stress, the comparison of the data and model predictions shows that a random distribution of habit planes fits the flow response. It is further inferred that the tension-compression asymmetry arises from the non-Schmid effects.

36 MATERIALS SCIENCE↗