SEARCH · Search NASA

Results for “iterative linear solution”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 217 records · Page 12

All sources NASA NTRS DOE OSTI

DOE OSTI2015

FIRST PRECISION LIGHT CURVE ANALYSIS OF THE NEGLECTED EXTREME MASS RATIO SOLAR-TYPE BINARY HR BOOTIS

HR Bootis is a neglected binary that is found to be a solar-type (G2V) extreme mass ratio binary (EMRB). It was discovered by Hanley and Shapley in 1940. Surprisingly, little has been published in the intervening years. In 1999 it was characterized by a 0.31587 day orbital period. Since that time it has been observed by various observers who have determined ∼20 timings of minimum light over the past ∼15,000 orbits. Our observations in 2012 represent the first precision curves in the BVR{sub c}I{sub c} Johnson–Cousins wavelength bands. The light curves have rather low amplitudes, averaging some 0.5 magnitudes, yet they exhibit total eclipses, which is typical of the rare group of solar-type EMRBs. An improved linear ephemeris was computed along with a quadratic ephemeris showing a decaying orbit, which indicates magnetic breaking may be occurring. The light curve solution reveals that HR Boo is a contact system with a somewhat low 21% Roche-lobe fill-out but a mass ratio of q = 4.09 (0.2444), which defines it as an EMRB. Two spots, both hot, were allowed to iterate to fit the light curve asymmetries. Their radii are 32° and 16°. Both are high-latitude polar spots indicative of strong magnetic activity. The shallow contact yet nearly equal component temperatures makes it an unusual addition to this group.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

Security-Constrained Unit Commitment with Linearized AC Optimal Power Flow.

We propose a mathematical programming-based approach to optimize the security-constrained unit commitment problem with a full AC transmission network representation. Our approach is based on our previously introduced successive linear programming (SLP) approach to solving the non-linear, nonconvex AC optimal power flow (ACOPF) problem. By linearizing the ACOPF, we are able to leverage powerful commercial mixed-integer solvers to iteratively optimize the combined unit commitment plus ACOPF model. We demonstrate our approach on six-bus, IEEE RTS-96, and IEEE 118-bus test systems. We perform a comparative analysis of the relative impacts of singlebus, DC, and AC transmission network models on the unit commitment and dispatch solutions and their associated costs.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2018

Hybrid iterative approach for simulation of radio-frequency fields in plasma

In this work, a novel iterative approach for solving discretized linear wave equations in a frequency domain, which combines time evolution with iterative relaxation schemes, is presented. In this hybrid approach, each iteration cycle consists of evolution of electromagnetic (EM) fields in time over a specified number of field periods followed by several iterative relaxations. Provided that there is sufficient dissipation, both the time evolution and the iterative relaxations contribute to the convergence of the EM fields to the solution of the formulated full wave boundary value problem. Time evolution rapidly distributes EM fields, propagating with group velocity, over the simulation domain, while the iterative relaxations smooth the fields, reducing the numerical errors such that iteration cycles converge to a steady state solution, approximating the solution of the formulated problem. This approach is intended for large scale simulations which are beyond the capabilities of direct solvers presently used for solving wave equations in the frequency domain. Lastly, the technique is demonstrated for solving wave equations on a regular grid using a cold plasma dielectric model with collisions for 2D modeling of EM fields in tokamak in an electron cyclotron frequency range.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

Efficient algorithm for locating and sizing series compensation devices in large power transmission grids: I. Model implementation

We explore optimization methods for planning the placement, sizing and operations of Flexible Alternating Current Transmission System (FACTS) devices installed to relieve transmission grid congestion. We limit our selection of FACTS devices to Series Compensation (SC) devices that can be represented by modification of the inductance of transmission lines. Our master optimization problem minimizes the l 1 norm of the inductance modification subject to the usual line thermal-limit constraints. We develop heuristics that reduce this non-convex optimization to a succession of Linear Programs (LP) which are accelerated further using cutting plane methods. The algorithm solves an instance of the MatPower Polish Grid model (3299 lines and 2746 nodes) in 40 seconds per iteration on a standard laptop—a speed up that allows the sizing and placement of a family of SC devices to correct a large set of anticipated congestions. We observe that our algorithm finds feasible solutions that are always sparse, i.e., SC devices are placed on only a few lines. In a companion manuscript, we demonstrate our approach on realistically-sized networks that suffer congestion from a range of causes including generator retirement. In this manuscript, we focus on the development of our approach, investigate its structure on a small test system subject to congestion from uniform load growth, and demonstrate computational efficiency on a realistically-sized network.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2014

Efficient Algorithm for Locating and Sizing Series Compensation Devices in Large Transmission Grids: Model Implementation (PART 1)

25 ENERGY STORAGE↗

DOE OSTI2021

Transient Efficiency Flexibility and Reliability Optimization of Coal-Fired Power Plants: Model-Predictive Control Library Development (Report)

This document pertains to the reporting requirements of DOE contract FE-0031767. The document covers the development of a model predictive control (MPC) library for implementing MPC for a general dynamic system. The library is implemented in a standardized manner in Matlab/Simulink, where core functions on model prediction, linearization and formulation and solution of a quadratic programming (QP) optimization problem is done in the core library - independent of the specific application. The user can provide the application-specific dynamic model in continuous and discrete time, to rapidly implement and test the MPC performance in a desktop simulation. The MPC optimization objective and constraints are also easily configured via an Excel file to allow iterative refinement as needed. Finally, the MPC library enables a rapid deployment to a target environment through auto C-code generation and containerization. The MPC library works seamlessly with the model based estimation (MBE) library to obtain the overall output feedback control solution. In this program, the reduced order model (ROM) of a coal-fired power plant (CFPP) is used to implement and test the MPC solution.

01 COAL, LIGNITE, AND PEAT↗

DOE OSTI2017

Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Numerical solution of 2D wet steam flow with non-equilibrium condensation and real thermodynamics

An approach to modeling of wet steam flow with non-equilibrium condensation phenomenon is presented. The first part of our flow model is homogeneous Euler system of transport equations for mass, momentum and total energy of wet steam (mixture). The additional second part describes liquid phase via non-homogeneous system of transport equations for moments of droplets number distribution function and relies on corrected classical nucleation theory. Moment equations are closed by linearization of droplet growth rate model. All necessary relations for thermodynamic properties of steam are provided by IAPWS set of equations. However, properties of condensate are simply modeled by liquid saturation data. Two real equations of state are implemented. Recently developed CFD formulation for entropy (does not require iteration process) and so-called IAPWS special gas equation for Helmholtz energy (one iteration loop is necessary). Flow model is validated on converging-diverging supersonic nozzle with Barschdorff geometry. Simulations were performed by in-house CFD code based on finite volume method and stiff character of equations was solved by symmetrical time operator splitting. Achieved results satisfactorily agreed with experimental data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

The iterative thermal emission method: A more implicit modification of IMC

For over 40 years, the Implicit Monte Carlo (IMC) method has been used to solve challenging problems in thermal radiative transfer. These problems typically contain regions that are optically thick and diffusive, as a consequence of the high degree of “pseudo-scattering” introduced to model the absorption and reemission of photons from a tightly-coupled, radiating material. IMC has several well-known features that could be improved: a) it can be prohibitively computationally expensive, b) it introduces statistical noise into the material and radiation temperatures, which may be problematic in multiphysics simulations, and c) under certain conditions, solutions can be nonphysical, in that they violate a maximum principle, where IMC-calculated temperatures can be greater than the maximum temperature used to drive the problem. We have developed a variant of IMC called iterative thermal emission IMC, which is designed to have a reduced parameter space in which the maximum principle is violated. ITE IMC is a more implicit version of IMC in that it uses the information obtained from a series of IMC photon histories to improve the estimate for the end of time step material temperature during a time step. A better estimate of the end of time step material temperature allows for a more implicit estimate of other temperature-dependent quantities: opacity, heat capacity, Fleck factor (probability that a photon absorbed during a time step is not reemitted) and the Planckian emission source. We have verified the ITE IMC method against 0-D and 1-D analytic solutions and problems from the literature. These results are compared with traditional IMC. We perform an infinite medium stability analysis of ITE IMC and show that it is slightly more numerically stable than traditional IMC. We find that significantly larger time steps can be used with ITE IMC without violating the maximum principle, especially in problems with non-linear material properties. The ITE IMC method does however yield solutions with larger variance because each sub-step uses a different Fleck factor (even at equilibrium)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Multiscale simulations for multi-continuum Richards equations

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2011

Modelling of a diode laser with a resonant grating of quantum wells and an external mirror

A three-dimensional numerical model of a diode laser with a resonant grating of quantum wells (QWs) and an external mirror is developed and used to calculate diode laser pulses that are long compared to the time of reaching a stationary regime and are short enough to neglect heating of the medium. The consistent solutions of the Helmholtz field equation and the system of diffusion equations for inversion in each QW are found. A source of charge carriers can be both an electron beam and a pump laser beam. The calculations yielded the longitudinal and radial profiles of the generated field, as well as its wavelength and power. The effective threshold pump current is determined. In the created iteration algorithm, the calculation time linearly increases with the number of QWs, which allows one to find the characteristics of lasers with a large number of QWs. The output powers and beam divergence angles of a cylindrical laser are calculated for different cavity lengths and pump spot radii. After calculating the fundamental mode characteristics, high-order modes were additionally calculated on the background of the frozen carrier distributions in the QW grating. It is shown that all the competing modes remain below the excitation threshold for the pump powers used in the experiment. The calculated and experimental data for the case of pumping by a nanosecond electron beam are qualitatively compared.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2013

Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients [Adaptive Algebraic Multigrid for Sequence of Problems with Slowly Varying Random Coefficients]

Our paper proposes an adaptive strategy for reusing a previously constructed coarse space by algebraic multigrid to construct a two-level solver for a problem with nearby characteristics. Furthermore, a main target application is the solution of the linear problems that appear throughout a sequence of Markov chain Monte Carlo simulations of subsurface flow with uncertain permeability field. We demonstrate the efficacy of the method with extensive set of numerical experiments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method

Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.

bundle method↗

DOE OSTI2018

Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations

The trigonometric Fourier-series method (TFS) is generalized to provide approximate solutions for non-linear point kinetics equations with feedback using varying step sizes. This method can provide a very stable solution against the size of the discrete time step allowing much larger step sizes to be used. Systems of the point kinetics equations are solved using Fourier-series expansion over a partition of the total time interval. The approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high-order derivatives of the solution vector at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. This system is similar to the Vandermonde system. Two successive orders of the partial sums are used to estimate the local truncation error. This error and some other constrains are used to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The process of calculating suitable step sizes should be automatic and inexpensive. Convergence and stability of the proposed method are discussed and a new formula is introduced to maintain stability. The proposed method solves the general linear and non-linear kinetics problems. The method has been applied to five different types of reactivities including step/ramp insertions with temperature feedback. The method is seemed to be valid for larger time intervals than those used in the conventional numerical integration, and is thus useful in reducing computing time. Computational results are found to be consistent with the analysis, they demonstrate that the convergence of the iteration scheme can be accelerated and the resulting computing times can be greatly reduced while maintain computational accuracy.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Hierarchical Krylov and nested Krylov methods for extreme-scale computing

The solution of large, sparse linear systems is typically a dominant phase of computation for simulations based on partial differential equations, which are ubiquitous in scientific and engineering applications. While preconditioned Krylov methods are widely used and provide many advantages for solving sparse linear systems that do not have highly convergent, geometric multigrid solvers or specialized fast solvers, Krylov methods encounter well-known scaling difficulties for over 10,000 processor cores because each iteration requires at least one vector inner product, which in turn requires a global synchronization that scales poorly because of internode latency. To aid in overcoming these difficulties, we have developed hierarchical Krylov methods and nested Krylov methods in the PETSc library that reduce the number of global inner products required across the entire system (where they are expensive), though freely allow vector inner products across smaller subsets of the entire system (where they are inexpensive) or use inner iterations that do not invoke vector inner products at all. Nested Krylov methods are a generalization of inner-outer iterative methods with two or more layers. Hierarchical Krylov methods are a generalization of block Jacobi and overlapping additive Schwarz methods, where each block itself is solved by Krylov methods on smaller blocks. Conceptually, the hierarchy can continue recursively to an arbitrary number of levels of smaller and smaller blocks. As a specific case, we introduce the hierarchical FGMRES method, or h-FGMRES, and we demonstrate the impact of two-level h-FGMRES with a variable preconditioner on the PFLOTRAN subsurface flow application. We also demonstrate the impact of nested FGMRES, BiCGStab and Chebyshev methods. These hierarchical Krylov methods and nested Krylov methods significantly reduced overall PFLOTRAN simulation time on the Cray XK6 when using 10,000 through 224,000 cores through the combined effects of reduced global synchronization due to fewer global inner products and stronger inner hierarchical or nested preconditioners.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Nonlinear Elimination Applied to Radiation Diffusion

We apply a nonlinearly preconditioned, quasi-Newton framework to accelerate the numerical solution of the thermal radiative transfer (TRT) equations. This framework was inspired by the unpublished method that has existed for years in Teton, Lawrence Livermore National Laboratory’s deterministic TRT code. In this paper, we cast this iteration scheme within a formal nonlinear preconditioning framework and compare its performance against other iteration schemes in the framework. With proper choices of iteration controls for the various levels of the solver, we can recover the standard linearized one-step method, a full nonlinear Newton scheme, as well as the method in Teton. In brief, the nonlinear preconditioning TRT scheme formally eliminates the material temperature equation from the nonlinear system in a nonlinear analog of a Schur complement. This nonlinear elimination step involves solving a decoupled nonlinear equation for each spatial degree of freedom and is therefore inexpensive. By applying a quasi-Newton iteration scheme on the new system, we obtain a three-level iteration scheme that is at least as efficient as commonly used TRT schemes. The new method allows full convergence to the nonlinear backward Euler time-discretized system, increasing accuracy and robustness, while using a similar number of linear iterations as the more common linearized one-step methods Eq. (4).

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2013

Optimal Transmission Switching Using the IV-ACOPF Linearization

In this paper we seek to investigate the performance of transmission switching using the iterative linear program approximation to the Current Voltage AC Optimal Power Flow (IV-ACOPF). Several different methods of using this switching are investigated to find a method that addresses the MIP challenges and is both fast and accurate. We consider opening only one line, opening up to five lines, and progressively opening one line at a time. The linear method with switching is much faster than the nonlinear ACOPF and generally finds solutions within 1% of the nonlinear ACOPF.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2020

DESC: A stellarator equilibrium solver

In this paper, the new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f(x) = 0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f(x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗