Search NASA⌕ Search

SEARCH · Search NASA

Results for “Jacobi Method”

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.

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

At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A two-level GPU-accelerated incomplete LU preconditioner for general sparse linear systems

This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures, specifically, those that are equipped with graphic processing units (GPUs). In addition to block-Jacobi, we present general purpose two-level ILU Schur complement-based approaches, where different strategies are presented to solve the coarse-level reduced system. These strategies are combined with modified ILU methods in the construction of the coarse-level operator, in order to effectively remove smooth errors by targeting an algebraically smooth vector. We leverage available GPU-based sparse matrix kernels to accelerate the setup and the solve phases of the proposed ILU preconditioner. We evaluate the efficiency of the proposed methods as a smoother for algebraic multigrid (AMG) and as a preconditioner for Krylov subspace methods on challenging anisotropic diffusion problems and a collection of general sparse matrices.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1999

Performance and Application of Parallel OVERFLOW Codes on Distributed and Shared Memory Platforms

The presentation discusses recent studies on the performance of the two parallel versions of the aerodynamics CFD code, OVERFLOW_MPI and _MLP. Developed at NASA Ames, the serial version, OVERFLOW, is a multidimensional Navier-Stokes flow solver based on overset (Chimera) grid technology. The code has recently been parallelized in two ways. One is based on the explicit message-passing interface (MPI) across processors and uses the _MPI communication package. This approach is primarily suited for distributed memory systems and workstation clusters. The second, termed the multi-level parallel (MLP) method, is simple and uses shared memory for all communications. The _MLP code is suitable on distributed-shared memory systems. For both methods, the message passing takes place across the processors or processes at the advancement of each time step. This procedure is, in effect, the Chimera boundary conditions update, which is done in an explicit "Jacobi" style. In contrast, the update in the serial code is done in more of the "Gauss-Sidel" fashion. The programming efforts for the _MPI code is more complicated than for the _MLP code; the former requires modification of the outer and some inner shells of the serial code, whereas the latter focuses only on the outer shell of the code. The _MPI version offers a great deal of flexibility in distributing grid zones across a specified number of processors in order to achieve load balancing. The approach is capable of partitioning zones across multiple processors or sending each zone and/or cluster of several zones into a single processor. The message passing across the processors consists of Chimera boundary and/or an overlap of "halo" boundary points for each partitioned zone. The MLP version is a new coarse-grain parallel concept at the zonal and intra-zonal levels. A grouping strategy is used to distribute zones into several groups forming sub-processes which will run in parallel. The total volume of grid points in each group are approximately balanced. A proper number of threads are initially allocated to each group, and in subsequent iterations during the run-time, the number of threads are adjusted to achieve load balancing across the processes. Each process exploits the multitasking directives already established in Overflow.

Djomehri, M. Jahed↗
DOE OSTI2014

Block-Iterative Methods for 3D Constant-Coefficient Stencils on GPUs and Multicore CPUs

Block iterative methods are extremely important as smoothers for multigrid methods, as preconditioners for Krylov methods, and as solvers for diagonally dominant linear systems. Developing robust and efficient smoother algorithms suitable for current and evolving GPU and multicore CPU systems is a significant challenge. We address this issue in the case of constant-coefficient stencils arising in the solution of elliptic partial differential equations on structured 3D uniform and adaptively refined block structured grids. Robust, highly parallel implementations of block Jacobi and chaotic block Gauss-Seidel algorithms with exact inversion of the blocks are developed using different parallelization techniques. Experimental results for NVIDIA Fermi/Kepler GPUs and AMD multicore systems are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2012

(3 + 1)-dimensional cylindrical Korteweg-de Vries equation for nonextensive dust acoustic waves: Symbolic computation and exact solutions

By combining the effects of bounded cylindrical geometry, azimuthal and axial perturbations, the nonlinear dust acoustic waves (DAWs) in an unmagnetized plasma consisting of negatively charged dust grains, nonextensive ions, and nonextensive electrons are studied in this paper. Using the reductive perturbation method, a (3 + 1)-dimensional variable-coefficient cylindrical Korteweg-de Vries (KdV) equation describing the nonlinear propagation of DAWs is derived. Via the homogeneous balance principle, improved F-expansion technique and symbolic computation, the exact traveling and solitary wave solutions of the KdV equation are presented in terms of Jacobi elliptic functions. Moreover, the effects of the plasma parameters on the solitary wave structures are discussed in detail. The obtained results could help in providing a good fit between theoretical analysis and real applications in space physics and future laboratory plasma experiments where long-range interactions are present.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Implementation of a Red-Black SOR CMFD Solver in MPACT

As part of the Consortium for Advanced Simulation of Light Water Reactors (CASL), the Virtual Environment for Reactor Applications (VERA) is being developed to provide high-fidelity multi-physics simulations of nuclear reactor cores. The MPACT code being developed collaboratively by Oak Ridge National Laboratory and the University of Michigan includes the primary deterministic neutron transport solver available within VERA. MPACT employs the 2D/1D method to solve three-dimensional problems, using 2D method of characteristics (MOC) radial transport solvers with 1D NEM-SP3 axial transport solvers, which are coupled using axial and radial transverse leakages. A 3D coarse mesh finite difference (CMFD) solver is used to provide accelerated convergence that is more stable. The Portable, Extensible Tool kit for Scientific Computation (PETSc) generalized minimal residual (GMRES) iterative solver with a block Jacobi preconditioner has been used to solve the CMFD linear systems. This work focused on the addition of a red-black successive over-relaxation (RBSOR) solver to reduce overall runtime by reducing time spent solving the CMFD linear systems. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Fast Numerical Solution of the Plasma Response Matrix for Real-time Ideal MHD Control

To help effectuate near real-time feedback control of ideal MHD instabilities in tokamak geometries, a parallelized version of A.H. Glasser’s DCON (Direct Criterion of Newcomb) code is developed. To motivate the numerical implementation, we first solve DCON’s δW formulation with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD Riccati equation. We then describe our adaptation of DCON with numerical methods natural to solutions of the Riccati equation, parallelizing it to enable its operation in near real-time. We replace DCON’s serial integration of perturbed modes—which satisfy a singular Euler- Lagrange equation—with a domain-decomposed integration of state transition matrices. Output is shown to match results from DCON with high accuracy, and with computation time < 1s. Such computational speed may enable active feedback ideal MHD stability control, especially in plasmas whose ideal MHD equilibria evolve with inductive timescale $\tau$ ≳ 1s—as in ITER. Further potential applications of this theory are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Two Dimensional Topology Optimization of Heat Exchangers with the Density and Level-Set Methods

We design heat exchangers using two topology optimization approaches: the density, i.e. volume fraction and level set methods. Our goal is to maximize the heat exchange between two fluids in separate channels while constraining the pressure drop across each channel. The heat exchanger is modeled with a coupled thermal-flow formulation. The flow is governed by an isothermal and incompressible Stokes-Brinkman equation and the heat transfer is governed by a convection-diffusion equation with high Peclet number. We solve one set of Stokes-Brinkman equations per fluid. Each Brinkman term in the flow equation serves to model the other phase as a solid, thereby preventing mixing. We first represent the solid and fluid phases using a volume fraction variable and apply a SIMP-like penalization in the Brinkman term to drive the optimization to a discrete design. The cost and constraint function derivatives are automatically calculated with the library pyadjoint and the optimization is performed by the Method of Moving Asymptotes. In a second optimization formulation, we use the level set approach to define the interface that separates the two fluids. Pyadjoint calculates the shape derivatives of the cost and constraint functions and the Hamilton-Jacobi advects the interface, allowing for topological changes. We present results in two dimensions and discuss the advantages and disadvantages of each approach.

42 ENGINEERING↗
NASA NTRS1986

A guidance law for the aeroassisted plane change maneuver in the presence of atmospheric uncertainties

A stochastic feedback control law for a space vehicle performing an aeroassisted plane-change maneuver is developed. The stochastic control law is designed to minimize the energy loss while taking into consideration the uncertainty in the atmospheric density. The solution is based on expansion of the stochastic Hamilton-Jacobi-Bellman equation (or dynamic programming) about a zeroth-order known integrable solution. The resulting guidance law is expressed as a series expansion in the noise power spectral densities. A numerical example indicates the potential improvement of this method.

Mishne, D.↗
DOE OSTI2015

Determining the diffraction properties of a cylindrically bent KAP(001) crystal from 1 to 5 keV

Various crystals are used for the dispersive component of X-ray spectrometers. The crystals are usually bent to meet the desired measurement needs, such as focusing. The bending can change the crystal diffraction properties, thus altering the spectrometer throughput and resolving power. This work concerns measuring the diffraction properties of a potassium acid phthalate (001) [KAP(001)] crystal bent into a circular cylinder segment. The measurement methods using a diode source and a synchrotron source are described. The multi-lamellar model for calculating the diffraction properties of a bent crystal is described. The measurement results are compared to the multi-lamellar model and show qualitative agreement. The measurements show how to make the multi-lamellar calculations a useful estimate. A method is given to make useful estimates of the diffraction properties of the KAP(001) crystal bent into a circular cylinder segment.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1992

Computing interface motion in compressible gas dynamics

An analysis is conducted of the coupling of Osher and Sethian's (1988) 'Hamilton-Jacobi' level set formulation of the equations of motion for propagating interfaces to a system of conservation laws for compressible gas dynamics, giving attention to both the conservative and nonconservative differencing of the level set function. The capabilities of the method are illustrated in view of the results of numerical convergence studies of the compressible Rayleigh-Taylor and Kelvin-Helmholtz instabilities for air-air and air-helium boundaries.

Mulder, W.↗
NASA NTRS2001

A Spectral Algorithm for Solving the Relativistic Vlasov-Maxwell Equations

A spectral method algorithm is developed for the numerical solution of the full six-dimensional Vlasov-Maxwell system of equations. Here, the focus is on the electron distribution function, with positive ions providing a constant background. The algorithm consists of a Jacobi polynomial-spherical harmonic formulation in velocity space and a trigonometric formulation in position space. A transform procedure is used to evaluate nonlinear terms. The algorithm is suitable for performing moderate resolution simulations on currently available supercomputers for both scientific and engineering applications.

Shebalin, John V.↗
NASA NTRS1989

Planetary nebulae as standard candles. III - The distance to M81

The results of a survey for PN in the nearby Sb galaxy M81 are reported, including the identification of 185 PN candidates. A distance to M81 of 3.50 + or - 0.40 Mpc is derived using the methods outlined by Ciardullo, et al. (1989). This value compares very well with values derived using traditional methods. Based on this agreement, it is concluded that PN are as good as, or better than, other standard candles for deriving distances to galaxies beyond 10 Mpc.

Jacoby, George H.↗
NASA NTRS2006

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Helenbrook, B. T.↗
DOE OSTI2022

Preconditioned least‐squares Petrov–Galerkin reduced order models

Abstract In this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

Lindsay, Payton↗
DOE OSTI2022

Randomized Cholesky Preconditioning for Graph Partitioning Applications

A graph is a mathematical representation of a network; we say it consists of a set of vertices, which are connected by edges. Graphs have numerous applications in various fields, as they can model all sorts of connections, processes, or relations. For example, graphs can model intricate transit systems or the human nervous system. However, graphs that are large or complicated become difficult to analyze. This is why there is an increased interest in the area of graph partitioning, reducing the size of the graph into multiple partitions. For example, partitions of a graph representing a social network might help identify clusters of friends or colleagues. Graph partitioning is also a widely used approach to load balancing in parallel computing. The partitioning of a graph is extremely useful to decompose the graph into smaller parts and allow for easier analysis. There are different ways to solve graph partitioning problems. For this work, we focus on a spectral partitioning method which forms a partition based upon the eigenvectors of the graph Laplacian (details presented in Acer, et. al.). This method uses the LOBPCG algorithm to compute these eigenvectors. LOBPCG can be accelerated by an operator called a preconditioner. For this internship, we evaluate a randomized Cholesky (rchol) preconditioner for its effectiveness on graph partitioning problems with LOBPCG. We compare it with two standard preconditioners: Jacobi and Incomplete Cholesky (ichol). This research was conducted from August to December 2021 in conjunction with Sandia National Laboratories.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗