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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.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Iterative methods in GPU-resident linear solvers for nonlinear constrained optimization

Linear solvers are major computational bottlenecks in a wide range of decision support and optimization computations. The challenges become even more pronounced on heterogeneous hardware, where traditional sparse numerical linear algebra methods are often inefficient. For example, methods for solving ill-conditioned linear systems have relied on conditional branching, which degrades performance on hardware accelerators such as graphical processing units (GPUs). To improve the efficiency of solving ill-conditioned systems, our computational strategy separates computations that are efficient on GPUs from those that need to run on traditional central processing units (CPUs). Our strategy maximizes the reuse of expensive CPU computations. Iterative methods, which thus far have not been broadly used for ill-conditioned linear systems, play an important role in our approach. In particular, we extend ideas from Arioli et al., (2007) to implement iterative refinement using inexact LU factors and flexible generalized minimal residual (FGMRES), with the aim of efficient performance on GPUs. In conclusion, we focus on solutions that are effective within broader application contexts, and discuss how early performance tests could be improved to be more predictive of the performance in a realistic environment.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Algebraic Multigrid with Filtering: An Efficient Preconditioner for Interior Point Methods in Large-Scale Contact Mechanics Optimization

Large-scale contact mechanics simulations are crucial in many engineering fields such as structural design and manufacturing. In the frictionless case, contact can be modeled by minimizing an energy functional; however, these problems are often nonlinear, nonconvex, and increasingly difficult to solve as mesh resolution increases. In this work, we employ a Newton-based interior-point (IP) filter line-search method, an effective approach for large-scale constrained optimization. While this method converges rapidly, each iteration requires solving a large saddle-point linear system that becomes ill-conditioned as the optimization process converges, largely due to IP treatment of the contact constraints. Such ill-conditioning can hinder solver scalability and increase iteration counts with mesh refinement. Here, to address this, we introduce a novel preconditioner, algebraic multigrid with filtering (AMGF), tailored to the Schur complement of the saddle-point system. Building on the classical AMG solver, commonly used for elasticity, we augment it with a specialized subspace correction that filters near null space components introduced by contact interface constraints. Through theoretical analysis and numerical experiments on a range of linear and nonlinear contact problems, we demonstrate that the proposed solver achieves mesh independent convergence and maintains robustness against the ill-conditioning that notoriously plagues IP methods. These results indicate that AMGF makes contact mechanics simulations more tractable and broadens the applicability of Newton-based IP methods in challenging engineering scenarios. More broadly, AMGF is well suited for problems, optimization or otherwise, where solver performance is limited by a low-dimensional subspace, such as those arising from localized constraints, interface conditions, or model heterogeneities. This makes the method widely applicable beyond contact mechanics and constrained optimization.

Mathematics and Computing↗
DOE OSTI2023

CP decomposition for tensors via alternating least squares with QR decomposition

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Information theory optimization of signals from small-angle scattering measurements

Small-angle X-ray scattering (SAXS) of particles in solution informs on the conformational states and assemblies of biological macromolecules (bioSAXS) outside of cryo- and solid-state conditions. In bioSAXS, the SAXS measurement under dilute conditions is resolution limited, and through an inverse Fourier transform, the measured SAXS intensities directly relate to the physical space occupied by the particles via the P (r)-distribution. Yet, this inverse transform of SAXS data has been historically cast as an ill-posed, ill-conditioned problem requiring an indirect approach. Here, we show that through the applications of matrix and information theories, the inverse transform of SAXS intensity data is a well-conditioned problem. The so-called ill-conditioning of the inverse problem is directly related to the Shannon number. By exploiting the oversampling enabled by modern detectors, a direct inverse Fourier transform of the SAXS data is possible, provided the recovered information does not exceed the Shannon number. The Shannon limit corresponds to the maximum number of significant singular values that can be recovered in a SAXS experiment, suggesting this relationship is a fundamental property of band-limited inverse integral transform problems. This correspondence reduces the complexity of the inverse problem to the Shannon limit and maximum dimension. We propose a hybrid scoring function using an information theory framework that assesses both the quality of the model-data fit as well as the quality of the recovered P (r)-distribution. The hybrid score utilizes the Akaike information criteria and Durbin-Watson statistic that considers parameter-model complexity, i.e., degrees of freedom, and the randomness of the model-data residuals. The described tests and findings extend the boundaries for bioSAXS by completing the information theory formalism initiated by Peter B. Moore to enable a quantitative measure of resolution in SAXS, robustly determine maximum dimension, and more precisely define the best parameter model appropriately representing the observed scattering data.

Rambo, Robert P. [Science and Technology Facilitie↗
DOE OSTI2025

Estimating QSVT angles for matrix inversion with large condition numbers

Quantum Singular Value Transformation (QSVT) is a state-of-the-art, near-optimal quantum algorithm that can be used for matrix inversion. The QSVT circuit is parameterized by a sequence of angles that must be pre-calculated classically, with the number of angles increasing as the matrix condition number grows. Computing QSVT angles for ill-conditioned problems is a numerically challenging task. Here, we propose a numerical technique for estimating QSVT angles for large condition numbers. This technique allows one to avoid expensive numerical computations of QSVT angles and to emulate QSVT circuits for solving ill-conditioned problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Robust Iterative Method for Symmetric Quantum Signal Processing in All Parameter Regimes

Here, this paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function α cos(1000x) (polynomial degree ≈ 1433) takes 6 iterations to converge to machine precision when α = 0.9, and the number of iterations only increases to 18 iterations when α = 1 – 10 -9 with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Efficient CP Rounding Using Alternating Least Squares with QR Decomposition

The CANDECOMP/PARAFAC (CP) decomposition is widely used for analyzing multidimensional data, and the alternating least squares (CP-ALS) algorithm is a common method for its computation. CP rounding is the problem of computing a lower-rank CP decomposition of an input already in a higher-rank CP format. While the normal equations (NE) approach in CP-ALS is efficient for the CP rounding problem and frequently used, it becomes unstable in the presence of ill-conditioned subproblems. This paper presents a new QR-based CP-ALS method for CP rounding that preserves both numerical stability and computational efficiency. Here, our experiments show that the proposed method offers significant speedup over a previous QR-based approach and the Tensor Toolbox's NE-based implementation, particularly for higher-order tensors. Furthermore, our approach demonstrates a marked reduction in error for ill-conditioned problems, with error reductions several orders of magnitude smaller compared to the NE-based method, while achieving faster convergence and more accurate solutions. By using a more numerically stable approach, we can solve more problems in reduced working precision, which enables further reduction in time to solution.

CANDECOMP/PARAFAC↗
DOE OSTI2024

Challenges in Training PINNs: A Loss Landscape Perspective

This paper explores challenges in training Physics Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-FGS, showing the superiority of Adam+L-BFGS, and introduce a novel secondorder optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.

Rathore, Pratik↗
DOE OSTI2025

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

Gauss-Newton↗
DOE OSTI2022

Quantum microgrid state estimation

This paper investigates the feasibility and efficiency of quantum-circuit-based algorithms for microgrid state estimation. Here, our new contributions include: (1) a general quantum state estimation (GQSE) formulation is devised for swing-bus-contained microgrids through the quantized Gaussian–Newton iteration, (2) a preconditioned quantum linear solver (PQLS) is developed for tackling the ill-conditioned GQSE with limited quantum resources, and (3) an enhanced quantum state estimation (EQSE) algorithm is further established for hierarchical-control-based microgrids with exogenous disturbances. Extensive case studies demonstrate the correctness of GQSE, PQLS and EQSE in two typical microgrids. The robustness and convergence performance of EQSE are also verified.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Accelerating gradient descent and Adam via fractional gradients

Here we propose a class of novel fractional-order optimization algorithms. We define a fractional-order gradient via the Caputo fractional derivatives that generalizes integer-order gradient. We refer it to as the Caputo fractional-based gradient, and develop an efficient implementation to compute it. A general class of fractional-order optimization methods is then obtained by replacing integer-order gradients with the Caputo fractional-based gradients. To give concrete algorithms, we consider gradient descent (GD) and Adam, and extend them to the Caputo fractional GD (CfGD) and the Caputo fractional Adam (CfAdam). We demonstrate the superiority of CfGD and CfAdam on several large scale optimization problems that arise from scientific machine learning applications, such as ill-conditioned least squares problem on real-world data and the training of neural networks involving non-convex objective functions. Numerical examples show that both CfGD and CfAdam result in acceleration over GD and Adam, respectively. We also derive error bounds of CfGD for quadratic functions, which further indicate that CfGD could mitigate the dependence on the condition number in the rate of convergence and results in significant acceleration over GD.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Linear solvers for power grid optimization problems: A review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite, and they become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL T decompositions. Here, we benchmark five well known direct linear solver packages on CPU- and GPU-based hardware, using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases. For completeness of the comparison we include results for MA57, which is one of the most efficient and reliable CPU solvers for this class of problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Quantum computing and preconditioners for hydrological linear systems

Modeling hydrological fracture networks is a hallmark challenge in computational earth sciences. Accurately predicting critical features of fracture systems, e.g. percolation, can require solving large linear systems far beyond current or future high performance capabilities. Quantum computers can theoretically bypass the memory and speed constraints faced by classical approaches, however several technical issues must first be addressed. Chief amongst these difficulties is that such systems are often ill-conditioned, i.e. small changes in the system can produce large changes in the solution, which can slow down the performance of linear solving algorithms. We test several existing quantum techniques to improve the condition number, but find they are insufficient. We then introduce the inverse Laplacian preconditioner, which improves the scaling of the condition number of the system from O(N) to O($\sqrt{N}$) and admits a quantum implementation. These results are a critical first step in developing a quantum solver for fracture systems, both advancing the state of hydrological modeling and providing a novel real-world application for quantum linear systems algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Application of machine learning techniques for fast MeV x-ray spectra unfolding from filter stack spectrometer data

Recovery of MeV x-ray spectra from detector signals is difficult because the response matrix inversion is ill-conditioned and current methods are too slow for high-repetition-rate experiments. In this work, we make use of neural networks to unfold MeV x-ray spectra from measurements obtained with a filter stack spectrometer at rates of near 40 Hz. The neural network was trained on synthetic data and tested on both synthetic and experimental data, the latter obtained in two separate experiments performed at the Omega EP laser facility. We show here that this unfolding method has good performance on synthetic data and that it is a promising option for experimental data of up to 40 MeV. The accuracy on experimental data is verified by using a simple forward model to compare against measured values.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Disentangling direct from indirect relationships in association networks

Networks are vital tools for understanding and modeling interactions in complex systems in science and engineering, and direct and indirect interactions are pervasive in all types of networks. However, quantitatively disentangling direct and indirect relationships in networks remains a formidable task. Here, we present a framework, called iDIRECT (Inference of Direct and Indirect Relationships with Effective Copula-based Transitivity), for quantitatively inferring direct dependencies in association networks. Using copula-based transitivity, iDIRECT eliminates/ameliorates several challenging mathematical problems, including ill-conditioning, self-looping, and interaction strength overflow. With simulation data as benchmark examples, iDIRECT showed high prediction accuracies. Application of iDIRECT to reconstruct gene regulatory networks in Escherichia coli also revealed considerably higher prediction power than the best-performing approaches in the DREAM5 (Dialogue on Reverse Engineering Assessment and Methods project, #5) Network Inference Challenge. In addition, applying iDIRECT to highly diverse grassland soil microbial communities in response to climate warming showed that the iDIRECT-processed networks were significantly different from the original networks, with considerably fewer nodes, links, and connectivity, but higher relative modularity. Further analysis revealed that the iDIRECT-processed network was more complex under warming than the control and more robust to both random and target species removal ( P < 0.001). As a general approach, iDIRECT has great advantages for network inference, and it should be widely applicable to infer direct relationships in association networks across diverse disciplines in science and engineering.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Lattice QCD estimates of thermal photon production from the QGP

Thermal photons produced in heavy-ion collision experiments are an important observable for understanding quark-gluon plasma (QGP). The thermal photon rate from the QGP at a given temperature can be calculated from the spectral function of the vector current correlator. Extraction of the spectral function from the lattice correlator is known to be an ill-conditioned problem, as there is no unique solution for a spectral function for a given lattice correlator with statistical errors. The vector current correlator, on the other hand, receives a large ultraviolet contribution from the vacuum, which makes the extraction of the thermal photon rate difficult from this channel. We therefore consider the difference between the transverse and longitudinal part of the spectral function, only capturing the thermal contribution to the current correlator, simplifying the reconstruction significantly. The lattice correlator is calculated for light quarks in quenched QCD at T = 470 MeV ( ∼ 1.5 T c ), as well as in 2 + 1 flavor QCD at T = 220 MeV ( ∼ 1.2 T p c ) with m π = 320 MeV . In order to quantify the nonperturbative effects, the lattice correlator is compared with the corresponding NLO + LPM LO estimate of correlator. The reconstruction of the spectral function is performed in several different frameworks, ranging from physics-informed models of the spectral function to more general models in the Backus-Gilbert method and Gaussian process regression. We find that the resulting photon rates agree within errors. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2022

Spectral density reconstruction with Chebyshev polynomials

Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. Here, in the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗