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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Practical Aspects of Krylov Subspace Iterative Methods in CFD

Implementation issues associated with the application of Krylov subspace iterative methods, such as Newton-GMRES, are presented within the framework of practical computational fluid dynamic (CFD) applications. This paper categorizes, evaluates, and contrasts the major ingredients (function evaluations, matrix-vector products, and preconditioners) of Newton-GMRES Krylov subspace methods in terms of their effect on the local linear and global nonlinear convergence, memory requirements, and accuracy. The discussion focuses on Newton-GMRES in both a structured multi-zone incompressible Navier-Stokes solver and an unstructured mesh finite-volume Navier-Stokes solver. Approximate versus exact matrix-vector products, effective preconditioners, and other pertinent issues are addressed.

Pulliam, Thomas H.↗
NASA NTRS1996

Multigrid and Krylov Subspace Methods for Transport Equations: Absorption Case

In this paper we look at Krylov subspace methods for solving the transport equations in a slab geometry. The spatial discretization scheme used is a finite element method called Modified Linear Discontinuous scheme (MLD). We investigate the convergence rates for a number of Krylov subspace methods for this problem and compare with the results of a spatial multigrid scheme.

Oliveira, S.↗
NASA NTRS2009

Subspace Iteration Method for Complex Eigenvalue Problems with Nonsymmetric Matrices in Aeroelastic System

Modern airplane design is a multidisciplinary task which combines several disciplines such as structures, aerodynamics, flight controls, and sometimes heat transfer. Historically, analytical and experimental investigations concerning the interaction of the elastic airframe with aerodynamic and in retia loads have been conducted during the design phase to determine the existence of aeroelastic instabilities, so called flutter .With the advent and increased usage of flight control systems, there is also a likelihood of instabilities caused by the interaction of the flight control system and the aeroelastic response of the airplane, known as aeroservoelastic instabilities. An in -house code MPASES (Ref. 1), modified from PASES (Ref. 2), is a general purpose digital computer program for the analysis of the closed-loop stability problem. This program used subroutines given in the International Mathematical and Statistical Library (IMSL) (Ref. 3) to compute all of the real and/or complex conjugate pairs of eigenvalues of the Hessenberg matrix. For high fidelity configuration, these aeroelastic system matrices are large and compute all eigenvalues will be time consuming. A subspace iteration method (Ref. 4) for complex eigenvalues problems with nonsymmetric matrices has been formulated and incorporated into the modified program for aeroservoelastic stability (MPASES code). Subspace iteration method only solve for the lowest p eigenvalues and corresponding eigenvectors for aeroelastic and aeroservoelastic analysis. In general, the selection of p is ranging from 10 for wing flutter analysis to 50 for an entire aircraft flutter analysis. The application of this newly incorporated code is an experiment known as the Aerostructures Test Wing (ATW) which was designed by the National Aeronautic and Space Administration (NASA) Dryden Flight Research Center, Edwards, California to research aeroelastic instabilities. Specifically, this experiment was used to study an instability known as flutter. ATW was a small-scale airplane wing comprised of an airfoil and wing tip boom. This wing was formulated based on a NACA-65A004 airfoil shape with a 3.28 aspect ratio. The wing had a span of 18 inch with root chord length of 13.2 inch and tip chord length of 8.7 inch. The total area of this wing was 197 square inch. The wing tip boom was a 1 inch diameter hollow tube of length 21.5 inch. The total weight of the wing was 2.66 lbs.

Pak, Chan-gi↗
NASA NTRS2017

Reachability Subspace Exploration Using Continuation Methods

Reachability manifold computation suffers from the curse of dimensionality and for large state spaces is computationally intractable. This paper examines the use of continuation methods to address this issue by formulating the reachability sub-space manifold calculation into a number of initial valued problems. As a result of computing the reachability manifold for a subspace of interest, an exponential improvement in computational cost occurs. This concept is applied to a position subspace reachability problem of a spacecraft in a Keplerian orbit under maximum thrust constraints. Future work includes a comparison of the proposed method with computing reachability manifolds using viscosity solutions of the Hamilton Jacobi Bellman partial differential equation.

orbital mechanics↗
NASA NTRS2017

Subspace-based Background Subtraction Applied to Aeroacoustic Wind Tunnel Testing

A subspace-based form of background subtraction is presented and applied to aeroacoustic wind tunnel data. A variant of this method has seen use in other fields such as climatology and medical imaging. The technique is based on an eigenvalue decomposition of the background noise cross-spectral matrix. Simulated results indicate similar performance to conventional background subtraction when the subtracted spectra are weaker than the true contaminating background levels. Superior performance is observed when the subtracted spectra are stronger than the true contaminating background levels, and when background data do not match between measurements. Experimental results show limited success in recovering signal behavior for data in which conventional background subtraction fails. The results also demonstrate the subspace subtraction technique's ability to maintain a physical coherence relationship in the modified cross-spectral matrix. Deconvolution results from microphone phased array data indicate that array integration methods are largely insensitive to subtraction type, and that background subtraction with appropriate background data is an effective alternative to diagonal removal.

deconvolution↗
DOE OSTI2025

Subspace-Driven Learning for Anomaly Detection in Process Transients

Nuclear power plant (NPP) monitoring and diagnostic centers are actively investigating and implementing automated anomaly detection algorithms to help plants catch anomalies sooner, thereby preventing or reducing the duration of unexpected shutdowns. Current machine learning-based anomaly detection methods are expected to be highly effective during stable, full-power operations because NPPs typically operate as baseload power generators, meaning there are extensive operating data available from plant equipment. However, it is expected that anomaly detection methods will face significant challenges during transient conditions (i.e., when power output falls below full power) because plants only occasionally operate at these lower power levels, generating sparse transient operational data, and resulting in false alarms or missed detections. Here, to address this issue, transfer learning is used, which for this problem leverages knowledge (in the form of learned features) from stable, full-power operations to improve detection accuracy during transient conditions, even with limited data. In this effort, a novel subspace approach is developed to transfer a subset of the data features from full power operation to transients. This approach is validated through experiments using synthetic data and was found to outperform two baseline transfer learning approaches in anomaly detection performance across a range of amounts of transient data used in the training process.

46 - INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AN↗
DOE OSTI2025

Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work

Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Johnson, Charles A.↗
DOE OSTI2024

Static Subspace Approximation for Random Phase Approximation Correlation Energies: Implementation and Performance

Developing theoretical understanding of complex reactions and processes at interfaces requires using methods that go beyond semilocal density functional theory to accurately describe the interactions between solvent, reactants and substrates. Methods based on many-body perturbation theory, such as the random phase approximation (RPA), have previously been limited due to their computational complexity. However, this is now a surmountable barrier due to the advances in computational power available, in particular through modern GPU-based supercomputers. In this work, we describe the implementation of RPA calculations within BerkeleyGW and show its favorable computational performance on large complex systems relevant for catalysis and electrochemistry applications. Our implementation builds off of the static subspace approximation which, by employing a compressed representation of the frequency dependent polarizability, enables the evaluation of the RPA correlation energy with significant acceleration and systematically controllable accuracy. We find that the computational cost of calculating the RPA correlation energy scales only linearly with system size for systems containing up to 50 thousand bands, and is expected to scale quadratically thereafter. We also show excellent strong scaling results across several supercomputers, demonstrating the performance and portability of this implementation.

algorithmic development↗
DOE OSTI2025

Contextual subspace variational quantum eigensolver calculation of the dissociation curve of molecular nitrogen on a superconducting quantum computer

Abstract We present an experimental demonstration of the Contextual Subspace Variational Quantum Eigensolver on superconducting hardware. Calculating the potential energy curve of molecular nitrogen proves challenging for many conventional quantum chemistry techniques, since static correlation dominates in the dissociation limit. Our quantum simulations retain good agreement with the Full Configuration Interaction energy, outperforming all benchmarked single-reference wavefunction techniques in capturing the bond-breaking appropriately. Moreover, our methodology is competitive with multiconfigurational approaches but at a saving of quantum resource, meaning larger active spaces can be treated for a fixed qubit allowance. To achieve this result, we deploy an error mitigation/suppression strategy comprised of Dynamical Decoupling, Measurement-Error Mitigation and Zero-Noise Extrapolation. Circuit parallelization also provides passive noise-averaging and improves the effective shot yield to reduce the measurement overhead. Furthermore, we introduce a modified adaptive ansatz construction algorithm that incorporates hardware awareness into our variational circuits, minimizing the transpilation cost for the target qubit topology.

Physics↗
DOE OSTI2024

Fragment-based initialization for quantum subspace methods

Here, we present a novel quantum-classical algorithm called LAS-QKSD for multireference systems, by combining a classical localized active space (LAS) fragment-based multireference algorithm with the quantum Krylov subspace diagonalization (QKSD) method for quantum computers. The algorithm uses wave function information from a LAS self-consistent field (LASSCF) calculation to prepare an initial state with better overlap with the target ground state than the Hartree-Fock state. This is coupled with the use of QKSD to ultimately converge to the exact energy, providing faster convergence than starting from the Hartree-Fock state. Fragmentation has the two-fold benefit of fewer configurations on the classical side of the algorithm as well as fewer state preparation gates on the quantum side. First, we compare the LAS-QKSD method to the classical LASSCF method and to QKSD with a Hartree-Fock initial state. We then examine ways to load the LASSCF wave function using direct initialization and a QKSD-motivated spectral filtering approach. Finally, using a bimetallic complex, we show that the LAS-QKSD method is an efficient alternative to highly expensive complete active space SCF (CASSCF) calculations on strongly correlated systems.

D'Cunha, Ruhee↗
DOE OSTI2026

DP-TwoLevel: two-stage gradient subspace learning for differentially private federated learning

Federated learning (FL) enables collaborative model training across distributed data sources without sharing raw data, but faces fundamental challenges in communication efficiency and privacy. Differentially private (DP) training mitigates information leakage but introduces noise that degrades model performance, especially in high-dimensional settings. We propose DP-TwoLevel, a hierarchical gradient projection method that improves utility under fixed DP constraints by exploiting low-dimensional structure in model updates. Our approach learns a two-level PCA-based representation of gradients and applies DP noise in a reduced-dimensional subspace, thereby lowering the effective noise magnitude while preserving dominant signal components. We evaluate the method across three datasets (MNIST, Fashion-MNIST, CIFAR-10) and three privacy regimes (ϵ∈0.5, 1.0, 2.0). Across nine experimental settings, DP-TwoLevel consistently outperforms DP-FedAvg, achieving an average accuracy improvement of 9.44%, with larger gains observed in lower ϵ(higher-noise) regimes (up to +22.31%). We further analyze scalability across models ranging from 100K to 1.49M parameters and identify a variance-based success criterion: performance remains strong when the projection preserves more than 75% of gradient variance, degrades in a marginal regime (65–75%), and fails below this threshold. Our results demonstrate that structure-aware dimensionality reduction can significantly improve the privacy–utility tradeoff in FL without modifying formal privacy guarantees. We also provide empirical evidence of scaling limitations for global projections and motivate per-layer extensions for larger models.

Kotevska, Olivera [ORNL] (ORCID:0000000316772243)↗
NASA NTRS1988

Krylov subspace methods on supercomputers

A short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers is presented. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three-dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel/vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. A few approaches consisting of implementing efficient forward and backward triangular solutions are described in detail. Polynomial preconditioning as an alternative to standard incomplete factorization techniques is also discussed. Another efficient approach is to reorder the equations so as to improve the structure of the matrix to achieve better parallelism or vectorization. An overview of these and other ideas and their effectiveness or potential for different types of architectures is given.

Saad, Youcef↗
NASA NTRS1989

Model reduction and control of flexible structures using Krylov subspaces

Krylov vectors and the concept of parameter-matching are combined to develop a model reduction algorithm for a damped structural dynamics system. The reduced-order model obtained matches a certain number of low-frequency moments of the full-order system. The major application of the present method is to the control of flexible structures. It is shown that, in the control of flexible structures, there generally exist three types of control energy spillover, namely, the control spillover, the observation spillover, and dynamic spillover. The formulation based on Krylov subspaces can eliminate the control and the observation spillover, while leaving only the dynamic spillover to be considered. Two examples are used to illustrate the efficacy of the Krylov method.

Craig, Roy R., Jr.↗
NASA NTRS1993

A nonconforming multigrid method using conforming subspaces

For second-order elliptic boundary value problems, we develop a nonconforming multigrid method using the coarser-grid correction on the conforming finite element subspaces. The convergence proof with an arbitrary number of smoothing steps for nu-cycle is presented.

Lee, Chang Ock↗
NASA NTRS1996

Krylov Subspace and Multigrid Methods Applied to the Incompressible Navier-Stokes Equations

We consider numerical solution methods for the incompressible Navier-Stokes equations discretized by a finite volume method on staggered grids in general coordinates. We use Krylov subspace and multigrid methods as well as their combinations. Numerical experiments are carried out on a scalar and a vector computer. Robustness and efficiency of these methods are studied. It appears that good methods result from suitable combinations of GCR and multigrid methods.

Vuik, C.↗
NASA NTRS1994

An Exploratory Study of a Subspace Iteration Method as an Alternative to the QR Method for Floquet Eigenanalysis

Floquet eigenanalysis requires a few dominant eigenvalues of the Floquet transition matrix (FTM). Although the QR method is used almost exclusively, it is expensive for such partial eigenanalysis; the operation counts and, thereby, the approximate machine-time grow cubically with the matrix order. Accordingly, for Floquet eigenanalysis, the Arnold-Saad method, a subspace iteration method, is investigated as an alternative to the QR method. The two methods are compared for machine-time efficiency and the residual errors of the corresponding eigenpairs. The Arnolds-Saad method takes much less machine-time than the QR method with comparable computational reliability and offers promise fpr large-scale Floquet eigenanalysis.

Achar, N. S.↗