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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Parameter Reduction of Composite Load Model Using Active Subspace Method

Over the past decades, the increasing penetration of distributed energy resources (DERs) has dramatically changed the power load composition in the distribution networks. The traditional static and dynamic load models can hardly capture the dynamic behavior of modern loads especially for fault-induced delayed voltage recovery (FIDVR) events. Thus, a more comprehensive composite load model with combination of static load, different types of induction motors, single-phase A/C motor, electronic load and DERs has been proposed by Western Electricity Coordinating Council (WECC). However, due to the large number of parameters and model complexity, the WECC composite load model (WECC CMLD) raises new challenges to power system studies. To overcome these challenges, in this paper, a cutting-edge parameter reduction (PR) approach for WECC CMLD based on active subspace method (ASM) is proposed. Firstly, the WECC CMLD is parameterized in a discrete-time manner for the application of the proposed method. Then, parameter sensitivities are calculated by discovering the active subspace, which is a lower-dimensional linear subspace of the parameter space of WECC CMLD in which the dynamic response is most sensitive. The interdependency among parameters can be taken into consideration by our approach. Finally, the numerical experiments validate the effectiveness and advantages of the proposed approach for WECC CMLD model.

active subspace↗
DOE OSTI2020

Subspace Methods in Multi-Parameter Seismic Full Waveform Inversion

In full waveform inversion (FWI) high-resolution subsurface model parameters are sought. FWI is normally treated as a nonlinear least-squares inverse problem, in which the minimum of the corresponding misfit function is found by updating the model parameters. When multiple elastic or acoustic properties are solved for, simple gradient methods tend to confuse parameter classes. This is referred to as parameter cross-talk; it leads to incorrect model solutions, poor convergence and strong dependence on the scaling of the different parameter types. Determining step lengths in a subspace domain, rather than directly in terms of gradients of different parameters, is a potentially valuable approach to address this problem. The particular subspace used can be defined over a span of different sets of data or different parameter classes, provided it involves a small number of vectors compared to those contained in the whole model space. Additionally, in a subspace method, the basis vectors are defined first, and a local minimum is found in the space spanned by these. We examine the application of the subspace method within acoustic FWI in determining simultaneously updates for velocity and density. We first discuss the choice of basis vectors to construct the spanned space, from linear updates by distinguishing only the contributions of different parameter classes towards nonlinear updates by adding the contributions of higher-order perturbations of each parameter class. The numerical character of FWI solutions generated via subspace methods involving different basis vectors is then analyzed and compared with traditional FWI methods. The subspace methods can provide better reconstructions of the model, especially for the velocity, as well as improved convergence rates, while the computational costs are still comparable with the traditional FWI methods.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Krylov Subspace Methods for Quantum Dynamics with Time-Dependent Generators

Krylov subspace methods in quantum dynamics identify the minimal subspace in which a process unfolds. To date, their use is restricted to time evolutions governed by time-independent generators. Here, we introduce a generalization valid for driven quantum systems governed by a time-dependent Hamiltonian that maps the evolution to a diffusion problem in a one-dimensional lattice with nearest-neighbor hopping probabilities that are inhomogeneous and time dependent. This representation is used to establish a novel class of fundamental limits to the quantum speed of evolution and operator growth. We also discuss generalizations of the algorithm, adapted to discretized time evolutions and periodic Hamiltonians, with applications to many-body systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

A methodology for generating reduced-order models for large-scale buildings using the Krylov subspace method

Developing a computationally efficient but accurate building energy simulation (BES) model is important for many purposes. Model order reduction (MOR) methods are attractive and much more reliable than identification approaches, since it directly extract a lower-dimensional model from a detailed physics-based model without any pre-simulations. However, because of computational and data storage requirements, there are challenges of applying these methods to a large-scale building. To overcome the problem, this work introduces the Krylov subspace method to the building science field. Technical issues of applying the method to building applications are addressed and a suitable algorithm that overcomes those challenges is presented. Furthermore, to demonstrate the reliability of the algorithm, comparisons between the resulted reduced-order model (ROM) and a high-fidelity model from a commercial BES software for a 60-zone case study building are provided. The ROM was a factor of 100 faster than the high fidelity model but with high accuracy.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

Preliminary Theoretical Analysis of Mixed Precision Krylov Subspace Methods (Q3 Report)

The third quarter of the project was spent performing theoretical finite precision analysis of Krylov subspace method variants that use mixed precision. Our focus here is on the Conjugate Gradient (CG) method and the Lanczos method. We have performed an analysis of maximum attainable accuracy for the classical CG method in which 3 precisions are used: a working precision ε, a precision ε IP for the inner product computations, and a precision ε MV for the matrix-vector products. Our results show that performing inner product computations in lower precision does not affect the attainable accuracy. Further, we have performed a complete error analysis of the s-step Lanczos algorithm. In this case, we show that the numerical behavior of the algorithm can be significantly improved by using extra precision in a small part of the computation. We summarize the main theorems in the remainder of the document. Other activities include attending biweekly xSDK meetings. The subsequent quarter will be spent finalizing these results into technical reports and/or manuscripts for submission to journals, as well as identifying opportunities for future work.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Reduced scaling formulation of CASPT2 analytical gradients using the supporting subspace method

We present a reduced scaling and exact reformulation of state specific complete active space second-order perturbation (CASPT2) analytical gradients in terms of the MP2 and Fock derivatives using the supporting subspace method. This work follows naturally from the supporting subspace formulation of the CASPT2 energy in terms of the MP2 energy using dressed orbitals and Fock builds. For a given active space configuration, the terms corresponding to the MP2-gradient can be evaluated with O(N5) operations, while the rest of the calculations can be computed with O(N3) operations using Fock builds, Fock gradients, and linear algebra. When tensor-hyper-contraction is applied simultaneously, the computational cost can be further reduced to O(N4) for a fixed active space size. The new formulation enables efficient implementation of CASPT2 analytical gradients by leveraging the existing graphical processing unit (GPU)-based MP2 and Fock routines. We present benchmark results that demonstrate the accuracy and performance of the new method. Example applications of the new method in ab initio molecular dynamics simulation and constrained geometry optimization are given.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Co-Active Subspace Methods for the Joint Analysis of Adjacent Computer Models

Active subspace (AS) methods are a valuable tool for understanding the relationship between the inputs and outputs of a Physics simulation. In this article, an elegant generalization of the traditional ASM is developed to assess the co-activity of two computer models. This generalization, which we refer to as a Co-Active Subspace (Co-AS) Method, allows for the joint analysis of two or more computer models allowing for thorough exploration of the alignment (or non-alignment) of the respective gradient spaces. We define co-active directions, co-sensitivity indices, and a scalar “concordance” metric (and complementary “discordance” pseudo-metric) and we demonstrate that these are powerful tools for understanding the behavior of a class of computer models, especially when used to supplement traditional AS analysis. Details for efficient estimation of the Co-AS and an accompanying R package (concordance) are provided. Practical application is demonstrated through analyzing a set of simulated rate stick experiments for PBX 9501, a high explosive, offering insights into complex model dynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

SNS: A Solution-Based Nonlinear Subspace Method for Time-Dependent Model Order Reduction

Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques require the construction of nonlinear term basis, which introduces a computationally expensive offline phase. A novel way of constructing nonlinear term basis within the hyper-reduction process is introduced. In contrast to the traditional hyper-reduction techniques where the collection of nonlinear term snapshots is required, the SNS method avoids collecting the nonlinear term snapshots. Instead, it uses the solution snapshots that are used for building a solution basis, which enables avoiding an extra data compression of nonlinear term snapshots. As a result, the SNS method provides a more efficient offline strategy than the traditional model order reduction techniques, such as the DEIM, GNAT, and ST-GNAT methods. The SNS method is theoretically justified by the conforming subspace condition and the subspace inclusion relation. It is useful for model order reduction of large-scale nonlinear dynamical problems to reduce the offline cost. It is especially useful for ST-GNAT that has shown promising results, such as a good accuracy with a considerable online speed-up for hyperbolic problems in a recent paper by Choi and Carlberg [SIAM J. Sci. Comput., 41 (2019), pp. A26--A58], because ST-GNAT involves an expensive offline cost related to collecting nonlinear term snapshots. Error analysis for the SNS method is presented. Numerical results support that the accuracy of the solution from the SNS method is comparable to the traditional methods and a considerable speed-up (i.e., a factor of two to a hundred) is achieved in the offline phase.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2020

Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

We analyze an expansion of the generalized block Krylov subspace framework of [Electron. Trans. Numer. Anal., 47 (2017), pp. 100--126]. This expansion allows the use of low-rank modifications of the matrix projected onto the block Krylov subspace and contains, as special cases, the block GMRES method and the new block Radau--Arnoldi method. Within this general setting, we present results that extend the interpolation property from the nonblock case to a matrix polynomial interpolation property for the block case, and we relate the eigenvalues of the projected matrix to the latent roots of these matrix polynomials. Some error bounds for these modified block FOM methods for solving linear systems are presented. We then show how cospatial residuals can be preserved in the case of families of shifted linear block systems. This result is used to derive computationally practical restarted algorithms for block Krylov approximations that compute the action of a matrix function on a set of several vectors simultaneously. Finally, we prove some error bounds and present numerical results showing that two modifications of FOM, the block harmonic and the block Radau--Arnoldi methods for matrix functions, can significantly improve the convergence behavior.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A Comparison of Linear Solvers for Resolving Flow in Three-Dimensional Discrete Fracture Networks

We compare various methods for resolving steady flow within three-dimensional discrete fracture networks, including direct methods, Krylov subspace methods with and without preconditioning, and multi-grid methods. We compared the performance of the methods based on compute times and scaling of the solution as a function of the number of grid nodes and log-variance of the hydraulic aperture. The methods are applied to three test cases: (a) variable density of networks with a truncated power-law distribution of fracture lengths, (b) a fixed network composed of monodisperse fracture sizes but varied permeability/aperture heterogeneity, (c) and a network based on field site in Nevada, US. We chose these cases to allow us to study the impact of the mesh size and flow properties, as well as to demonstrate our conclusions on a large-scale, realistic problem (more than 40 million mesh nodes). A direct solution using Cholesky factorization outperformed other methods for every example but was closely followed in performance by some algebraic multigrid (AMG) preconditioned Krylov subspace methods. Among the Krylov methods, conjugate gradients (CG) with an AMG preconditioner performs the best. Generally, Cholesky factorization is recommended, but CG with an AMG preconditioner may be suitable for very large problems beyond 40 million nodes where the entire linear system cannot reside in memory.

58 GEOSCIENCES↗
DOE OSTI2021

Multi-Fidelity Active Subspaces for Wind Farm Uncertainty Quantification

Wind plants operate in stochastic environments characterized by complex turbulent flow dynamics and high-dimensional random variables. A key step in uncertainty quantification studies is sensitivity analysis and dimension reduction that can facilitate the development of surrogate models to be used for forward and inverse propagation or optimization under uncertainty. Prior work has shown active subspaces are an effective tool for identifying important directions in the space of stochastic inputs; however, they have only been applied to single-fidelity wind plant models. In this study, we investigate the efficacy of a multi-fidelity active subspace method for analyzing the uncertainty in wind plant power output. The multi-fidelity active subspace estimator offers the promise of increased accuracy in identifying active subspaces as compared to a single-fidelity estimator for the same computational cost, or a reduction in cost for the same accuracy. This makes the study of uncertainty in larger wind plants and with higher fidelity physics tractable. The multi-fidelity active subspace method is applied to gridded and existing wind plant layouts with single and multiple inflow conditions and its performance for surrogate modeling and uncertainty propagation is compared against a single-fidelity active subspace method. This multi-fidelity approach yields substantial computational speedups of 2x - 3.4x across the test cases along with acceptable accuracy in surrogate modeling and computing statistical moments.

active subspace↗
DOE OSTI2026

Augmenting subspace optimization methods with linear bandits

In this work, we consider the framework of methods for unconstrained minimization that are, in each iteration, restricted to a model that is only a valid approximation to the objective function on some affine subspace containing an incumbent point. These methods are of practical interest in computational settings where derivative information is either expensive or impossible to obtain. Recent attention has been paid in the literature to employing randomized matrix sketching for generating the affine subspaces within this framework. We consider a relatively straightforward, deterministic augmentation of such a generic subspace optimization method. In particular, we consider a sequential optimization framework where actions consist of one-dimensional linear subspaces and rewards consist of (approximations to) the magnitudes of directional derivatives computed in the direction of the action subspace. Reward maximization in this context is consistent with maximizing lower bounds on descent guaranteed by first-order Taylor models. This sequential optimization problem can be analysed through the lens of dynamic regret. We modify an existing linear upper confidence bound (UCB) bandit method and prove sublinear dynamic regret in the subspace optimization setting. We demonstrate the efficacy of employing this linear UCB method in a setting where forward-mode algorithmic differentiation can provide directional derivatives in arbitrary directions and in a derivative-free setting. For the derivative-free setting, we propose SS-POUNDers, an extension of the derivative-free optimization method POUNDers that employs the linear UCB mechanism to identify promising subspaces. Our numerical experiments suggest a preference, in either computational setting, for employing a linear UCB mechanism within a subspace optimization method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Quarter 4 Report: Report on Final Findings and Opportunities for Future Work in the Use of Mixed Precision in Iterative Solvers

The fourth quarter of the project was spent developing an error analysis of the s-step Lanczos and CG algorithms. Our theoretical bounds and numerical experiments show that the numerical behavior of the algorithm can be significantly improved by using extra precision in a small part of the computation related to the computation and application of the Gram matrix. We have published a technical report which includes all steps of the analysis [8]; a shortened version for journal submission is in preparation. We plan to submit this paper in the following weeks. Activities related to this also include a collaboration with Ichitaro Yamazaki on gathering performance results for these new mixed precision s-step Krylov subspace methods using single/double precision on GPUs. Namely, we would like to obtain performance results that show that the performance overhead of using double the working precision in these select computations is minimal. Other activities include attending biweekly xSDK meetings and presenting a pitch talk on this work to the group on February 25, 2021. In the remainder of the document, we summarize our findings on the potential for mixed precision in classical Krylov subspace methods and s-step Krylov subspace methods, as well as key opportunities for future work.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Data-driven approach to Determining the Fidelity in the Hardware-in-the-loop Systems using Subspace Identification Method

One of the major questions in any Hardware-in-the-loop (HiL) simulation is to understand the fidelity of the HiL simulation itself which is most often indicated qualitatively as either high or low instead of quantifying it. Being cognizant of the level of fidelity forms the crux to assess the validity and credibility of the HiL simulation. In this work, we address this issue by developing a systematic, data-driven approach to assess the fidelity of any HiL simulation, and more specifically, the fidelity of the interface between the simulator and hardware in an HiL simulation. Applying a subspace identification method, a linear system representing the interface is obtained from the time-series data captured between the hardware and simulator in the HiL simulation. Finally, the fidelity is defined based on this linear system properties. The proposed data-driven fidelity quantification framework is illustrated on the IEEE 123 node feeder system running on HYPERSIM and interacting with virtual protection relays.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Towards a Quantum Algorithm for the Incompressible Nonlinear Navier-Stokes Equations

In this work, we present novel concepts for quantum algorithms to solve transient, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the incompressible Navier-Stokes equations as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts solving nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. We propose a new framework based on matrix product states (MPSs) and matrix product operators (MPOs), in addition to the Krylov subspace methods. For example, the solution variables of the Navier-Stokes equations are represented by MPSs, and the linear and nonlinear terms are processed by MPOs. The time evolution of the operators is attained by a fast-forwarding algorithm using Krylov subspace methods. Furthermore, we discuss various techniques for efficient encoding of MPSs, measurement reduction for MPOs, and use of tensor operations to treat multi-variate, multi-physics characteristics of Navier-Stokes.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
DOE OSTI2022

A Subspace-Inclusive Sampling Method for the Computational Design of Compositionally Graded Alloys

Abstract Compositionally graded alloys, a subclass of functionally graded materials (FGMs), utilize localized variations in composition with a single metal part to achieve higher performance than traditional single material parts. In previous work [Kirk, T., Galvan, E., Malak, R., and Arroyave, R., 2018, “Computational Design of Gradient Paths in Additively Manufactured Functionally Graded Materials,” J. Mech. Des., 140, p. 111410. 10.1115/1.4040816], the authors presented a computational design methodology that avoids common issues which limit a gradient alloy’s feasibility, such as deleterious phases, and optimizes for performance objectives. However, the previous methodology only samples the interior of a composition space, meaning designed gradients must include all elements in the space throughout the gradient. Because even small amounts of additional alloying elements can introduce new deleterious phases, this characteristic often neglects potentially simpler solutions to otherwise unsolvable problems and, consequently, discourages the addition of new elements to the state space. The present work improves upon the previous methodology by introducing a sampling method that includes subspaces with fewer elements in the design search. The new method samples within an artificially expanded form of the state space and projects samples outside the true region to the nearest true subspace. This method is evaluated first by observing the sample distribution in each subspace of a 3D, 4D, and 5D state space. Next, a parametric study in a synthetic 3D problem compares the performance of the new sampling scheme to the previous methodology. Lastly, the updated methodology is applied to design a gradient from stainless steel to equiatomic NiTi that has practical uses such as embedded shape memory actuation and for which the previous methodology fails to find a feasible path.

Engineering↗