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Results for “Structure exploitation iterative methods”

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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DOE OSTI2014

Solving security constrained optimal power flow problems by a structure exploiting interior point method

In this paper we present a new approach to solve the DC (n - 1) security constrained optimal power flow (SCOPF) problem by a structure exploiting interior point solver. Our approach is based on a reformulation of the linearised SCOPF model, in which most matrices that need to be factorized are constant. Hence, most factorizations and a large number of back-solve operations only need to be performed once. However, assembling the Schur complement matrix remains expensive in this scheme. To reduce the effort, we suggest using a preconditioned iterative method to solve the corresponding linear system. We suggest two main schemes to pick a good and robust preconditioner based on combining different “active” contingency scenarios of the SCOPF model. These new schemes are implemented within the object-oriented parallel solver (OOPS), a structure-exploiting primal-dual interior-point implementation. We give results on several SCOPF test problems. The largest example contains 500 buses. We compare the results from the original interior point method (IPM) implementation in OOPS and our new reformulation.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2017

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

We consider the nonlinear eigenvalue problem M ( λ ) x = 0, where M ( λ ) is a large parameter-dependent matrix. In several applications, M ( λ ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169–195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. In conclusion, the efficiency and properties of the algorithm are illustrated with two large-scale problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Breaking Computational Barriers: Real-time Analysis and Optimization with Large-scale Nonlinear Models via Model Reduction

Model reduction for dynamical systems is a promising approach for reducing the computational cost of large-scale physics-based simulations to enable high-fidelity models to be used in many- query (e.g., Bayesian inference) and near-real-time (e.g., fast-turnaround simulation) contexts. While model reduction works well for specialized problems such as linear time-invariant systems, it is much more difficult to obtain accurate, stable, and efficient reduced-order models (ROMs) for systems with general nonlinearities. This report describes several advances that enable nonlinear reduced-order models (ROMs) to be deployed in a variety of time-critical settings. First, we present an error bound for the Gauss-Newton with Approximated Tensors (GNAT) nonlinear model reduction technique. This bound allows the state-space error for the GNAT method to be quantified when applied with the backward Euler time-integration scheme. Second, we present a methodology for preserving classical Lagrangian structure in nonlinear model reduction. This technique guarantees that important properties--such as energy conservation and symplectic time-evolution maps--are preserved when performing model reduction for models described by a Lagrangian formalism (e.g., molecular dynamics, structural dynamics). Third, we present a novel technique for decreasing the temporal complexity --defined as the number of Newton-like iterations performed over the course of the simulation--by exploiting time-domain data. Fourth, we describe a novel method for refining projection-based reduced-order models a posteriori using a goal-oriented framework similar to mesh-adaptive h -refinement in finite elements. The technique allows the ROM to generate arbitrarily accurate solutions, thereby providing the ROM with a 'failsafe' mechanism in the event of insufficient training data. Finally, we present the reduced-order model error surrogate (ROMES) method for statistically quantifying reduced- order-model errors. This enables ROMs to be rigorously incorporated in uncertainty-quantification settings, as the error model can be treated as a source of epistemic uncertainty. This work was completed as part of a Truman Fellowship appointment. We note that much additional work was performed as part of the Fellowship. One salient project is the development of the Trilinos-based model-reduction software module Razor , which is currently bundled with the Albany PDE code and currently allows nonlinear reduced-order models to be constructed for any application supported in Albany. Other important projects include the following: 1. ROMES-equipped ROMs for Bayesian inference: K. Carlberg, M. Drohmann, F. Lu (Lawrence Berkeley National Laboratory), M. Morzfeld (Lawrence Berkeley National Laboratory). 2. ROM-enabled Krylov-subspace recycling: K. Carlberg, V. Forstall (University of Maryland), P. Tsuji, R. Tuminaro. 3. A pseudo balanced POD method using only dual snapshots: K. Carlberg, M. Sarovar. 4. An analysis of discrete v. continuous optimality in nonlinear model reduction: K. Carlberg, M. Barone, H. Antil (George Mason University). Journal articles for these projects are in progress at the time of this writing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

PolyCheck: Dynamic Verification of Iteration Space Transformations on Affine Programs

High-level compiler transformations, especially loop transformations, are widely recognized as critical optimizations to restructure programs to improve data locality and expose parallelism. Guaranteeing the correctness of program transformations is essential, and to date three main approaches have been developed: proof of equivalence of affine programs, matching the execution traces of programs, and checking bit-by-bit equivalence of the outputs of the programs. Each technique suffers from limitations in either the kind of transformations supported, space complexity, or the sensitivity to the testing dataset. In this paper, we take a novel approach addressing all three limitations to provide an automatic bug checker to verify any iteration reordering transformations on affine programs, including non-affine transformations, with space consumption proportional to the original program data, and robust to arbitrary datasets of a given size. We achieve this by exploiting the structure of affine program control- and data-flow to generate at compile-time lightweight checker code to be executed within the transformed program. Experimental results assess the correctness and effectiveness of our method, and its increased coverage over previous approaches.

affine programs↗
DOE OSTI2017

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Multigrid Optimization for Large-Scale Ptychographic Phase Retrieval

Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and improved reconstructions. Ptychographic microscopes allow for large fields to be imaged at high resolution at the cost of additional computational expense. Herein, we propose a multigrid-based optimization framework to reduce the computational burdens of large-scale ptychographic phase retrieval. Our proposed method exploits the inherent hierarchical structures in ptychography through tailored restriction and prolongation operators for the object and data domains. Our numerical results show that our proposed scheme accelerates the convergence of its underlying solver and outperforms the ptychographical iterative engine, a workhorse in the optics community.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Online data-enabled predictive control

We develop an online data-enabled predictive (ODeePC) control method for optimal control of unknown systems, building on the recently proposed DeePC (Coulson et al., 2019). Our proposed ODeePC method leverages a primal-dual algorithm with real-time measurement feedback to iteratively compute the corresponding real-time optimal control policy as system conditions change. The proposed ODeePC conceptual-wise resembles standard adaptive system identification and model predictive control (MPC), but it provides a new alternative for the standard methods. ODeePC is enabled by computationally efficient methods that exploit the special structure of the Hankel matrices in the context of DeePC with Fast Fourier Transform (FFT) and primal-dual algorithm We provide theoretical guarantees regarding the asymptotic behavior of ODeePC, and we demonstrate its performance through numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Phased: Phase-Aware Submodularity-Based Energy Disaggregation

Energy disaggregation is the task of discerning the energy consumption of individual appliances from aggregated measurements, which holds promise for understanding and reducing energy usage. In this paper, we propose PHASED, an optimization approach for energy disaggregation that has two key features: PHASED (i) exploits the structure of power distribution systems to make use of readily available measurements that are neglected by existing methods, and (ii) poses the problem as a minimization of a difference of sub-modular functions. We leverage this form by applying a discrete optimization variant of the majorization-minimization algorithm to iteratively minimize a sequence of global upper bounds of the cost function to obtain high-quality approximate solutions. PHASED improves the disaggregation accuracy of state-of-the-art models by up to 61% and achieves better prediction on heavy load appliances.

ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION↗
DOE OSTI2022

Time-discretization of a plasma-neutral MHD model with a semi-implicit leapfrog algorithm

The semi-implicit leapfrog time-discretization is a workhorse algorithm for initial-value MHD codes to bridge between vastly separated time scales. Inclusion of atomic interactions with neutrals breaks the functional structure of the MHD equations that exploited by the leapfrog. In this work, we address how to best integrate atomic physics into the semi-implicit leapfrog. Following the Crank-Nicolson method, one approach is to time-center the atomic interactions in the linear solver and use a Newton method to include the nonlinear contributions. Alternatively, another family of methods are based on operator-splitting the terms associated with the atomic interactions using a Strang-splitting technique. These methods naturally break equations into constituent ODE and PDE parts and preserve the structure exploited by the semi-implicit leapfrog. We study the accuracy and efficiency of these methods through a battery of 0D and 1D cases and show that a second-order-in-time Douglas-Rachford inspired coupling between the ODE and PDE advances is effective in reducing the time-discretization error to be comparable to that of Crank-Nicolson with Newton iteration of the nonlinear terms. Splitting ODE and PDE parts results in independent matrix solves for each field which reduces the computational cost considerably and provides parallelization over species relative to Crank-Nicolson.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

MFiX: Fractional-Step Method Implementation

A comprehensive, multiphase computational fluid dynamics (CFD) simulation solves several coupled transport equations including continuity, momentum, species, and energy. Chemical reactions further couple these equations through heats of reaction and rates of formation of products and rates of destruction of reactants. A fractional-step method separates changes attributed to chemical reactions from transport phenomena like convection and diffusion. When the governing equations are split into the transport and reacting components, efficient and independent methodologies can be exploited to solve the different systems. Specifically, discretization of field variable transport equations results in large, sparse matrices which are loosely coupled. These systems are solved in succession using iterative techniques that take advantage of the matrix structure. In contrast, chemical reactions tightly couple field variables locally within the domain (e.g., within a single computational cell) resulting in low dimensional but dense, nonlinear systems that are better solved using direct integration techniques.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Synchronized multiartifact reduction with tomographic reconstruction (SMART-RECON): A statistical model based iterative image reconstruction method to eliminate limited-view artifacts and to mitigate the temporal-average artifacts in time-resolved CT

Purpose: In x-ray computed tomography (CT), a violation of the Tuy data sufficiency condition leads to limited-view artifacts. In some applications, it is desirable to use data corresponding to a narrow temporal window to reconstruct images with reduced temporal-average artifacts. However, the need to reduce temporal-average artifacts in practice may result in a violation of the Tuy condition and thus undesirable limited-view artifacts. In this paper, the authors present a new iterative reconstruction method, synchronized multiartifact reduction with tomographic reconstruction (SMART-RECON), to eliminate limited-view artifacts using data acquired within an ultranarrow temporal window that severely violates the Tuy condition. Methods: In time-resolved contrast enhanced CT acquisitions, image contrast dynamically changes during data acquisition. Each image reconstructed from data acquired in a given temporal window represents one time frame and can be denoted as an image vector. Conventionally, each individual time frame is reconstructed independently. In this paper, all image frames are grouped into a spatial–temporal image matrix and are reconstructed together. Rather than the spatial and/or temporal smoothing regularizers commonly used in iterative image reconstruction, the nuclear norm of the spatial–temporal image matrix is used in SMART-RECON to regularize the reconstruction of all image time frames. This regularizer exploits the low-dimensional structure of the spatial–temporal image matrix to mitigate limited-view artifacts when an ultranarrow temporal window is desired in some applications to reduce temporal-average artifacts. Both numerical simulations in two dimensional image slices with known ground truth and in vivo human subject data acquired in a contrast enhanced cone beam CT exam have been used to validate the proposed SMART-RECON algorithm and to demonstrate the initial performance of the algorithm. Reconstruction errors and temporal fidelity of the reconstructed images were quantified using the relative root mean square error (rRMSE) and the universal quality index (UQI) in numerical simulations. The performance of the SMART-RECON algorithm was compared with that of the prior image constrained compressed sensing (PICCS) reconstruction quantitatively in simulations and qualitatively in human subject exam. Results: In numerical simulations, the 240{sup ∘} short scan angular span was divided into four consecutive 60{sup ∘} angular subsectors. SMART-RECON enables four high temporal fidelity images without limited-view artifacts. The average rRMSE is 16% and UQIs are 0.96 and 0.95 for the two local regions of interest, respectively. In contrast, the corresponding average rRMSE and UQIs are 25%, 0.78, and 0.81, respectively, for the PICCS reconstruction. Note that only one filtered backprojection image can be reconstructed from the same data set with an average rRMSE and UQIs are 45%, 0.71, and 0.79, respectively, to benchmark reconstruction accuracies. For in vivo contrast enhanced cone beam CT data acquired from a short scan angular span of 200{sup ∘}, three 66{sup ∘} angular subsectors were used in SMART-RECON. The results demonstrated clear contrast difference in three SMART-RECON reconstructed image volumes without limited-view artifacts. In contrast, for the same angular sectors, PICCS cannot reconstruct images without limited-view artifacts and with clear contrast difference in three reconstructed image volumes. Conclusions: In time-resolved CT, the proposed SMART-RECON method provides a new method to eliminate limited-view artifacts using data acquired in an ultranarrow temporal window, which corresponds to approximately 60{sup ∘} angular subsectors.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2022

Surrogate Hessian accelerated structural optimization for stochastic electronic structure theories

In this work, we present an efficient energy-based method for structural optimization with stochastic electronic structure theories, such as diffusion quantum Monte Carlo (DMC). This method is based on robust line-search energy minimization in reduced parameter space, exploiting approximate but accurate Hessian information from a surrogate theory, such as density functional theory. The surrogate theory is also used to characterize the potential energy surface, allowing for simple but reliable ways to maximize statistical efficiency while retaining controllable accuracy. We demonstrate the method by finding the minimum DMC energy structures of the selected flake-like aromatic molecules, such as benzene, coronene, and ovalene, represented by 2, 6, and 19 structural parameters, respectively. In each case, the energy minimum is found within two parallel line-search iterations. The method is near-optimal for a line-search technique and suitable for a broad range of applications. It is easily generalized to any electronic structure method where forces and stresses are still under active development and implementation, such as diffusion Monte Carlo, auxiliary-field Monte Carlo, and stochastic configuration interaction, as well as deterministic approaches such as the random-phase approximation. Accurate and efficient means of geometry optimization could shed light on a broad class of materials and molecules, showing high sensitivity of induced properties to structural variables.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Hierarchical Gaussian process-based Bayesian optimization for materials discovery in high entropy alloy spaces

Bayesian optimization (BO) is a powerful and data-efficient method for iterative materials discovery and design, particularly valuable when prior knowledge is limited, underlying functional relationships are complex or unknown, and the cost of querying the materials space is significant. Traditional BO methodologies typically utilize conventional Gaussian Processes (cGPs) to model the relationships between material inputs and properties, as well as correlations within the input space. However, cGP-BO approaches often fall short in multi-objective optimization scenarios, where they are unable to fully exploit correlations between distinct material properties. Leveraging these correlations can significantly enhance the discovery process, as information about one property can inform and improve predictions about others. Here, this study addresses this limitation by employing advanced kernel structures to capture and model multi-dimensional property correlations through multi-task (MTGPs) or deep Gaussian Processes (DGPs), thus accelerating the discovery process. We demonstrate the effectiveness of MTGP-BO and DGP-BO in rapidly and robustly solving complex materials design challenges that occur within the context of complex multi-objective optimization over FCC FeCrNiCoCu high entropy alloy (HEA) spaces, where traditional cGP-BO approaches fail. Furthermore, we highlight how the differential costs associated with querying various material properties can be strategically leveraged to make the materials discovery process more cost-efficient.

36 MATERIALS SCIENCE↗
DOE OSTI2019

Data analysis for synchrotron microcrystal native-SAD phasing

De novo structural evaluation of native biomolecules from single-wavelength anomalous diffraction (SAD) is a challenge because of the weakness of the anomalous scattering. For microcrystals with sizes less than 10 μm, native-SAD phasing at synchrotron microdiffraction beamlines is even more challenging due to difficulties in sample manipulation, diffraction data collection and data analysis. Native-SAD analysis from microcrystals by using X-ray free electron lasers has been demonstrated but has required use of thousands of thousands of microcrystals to achieve the necessary accuracy. Here we show that by exploitation of anomalous microdiffraction signals at a synchrotron beamline, by the use of polyimide wellmounts, and by an iterative crystal and frame rejection method, optimized data may be acquired and assembled which allows microcrystal native-SAD phasing from as few as about 1200 crystals. Finally, beyond microcrystals, our data analysis strategy may work equally well for microdiffraction data from larger crystals.

36 MATERIALS SCIENCE↗
DOE OSTI2016

NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES

The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Weak and strong orders of linear recurring sequences

The concept of the strong order of linear recurring sequence (LRS) is introduced in this paper. Necessary and sufficient conditions for the existence of the strong LRS order are derived. The strong LRS order is exploited for the formalization of the problem of the extension of a sequence from the available fragment (fragments) of that sequence. The definition of the strong LRS order opens new possibilities for formal sequence analysis whenever the weak LRS order of that sequence exists. Computational experiments with discrete iterative maps are used to illustrate the applicability of the strong LRS order in nonlinear system analysis.

97 MATHEMATICS AND COMPUTING↗