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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Multigrid Reduction in Time for Chaotic and Hyperbolic Problems (Final Report)

The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1984

On the connection between multigrid and cyclic reduction

A technique is shown whereby it is possible to relate a particular multigrid process to cyclic reduction using purely mathematical arguments. This technique suggest methods for solving Poisson's equation in 1-, 2-, or 3-dimensions with Dirichlet or Neumann boundary conditions. In one dimension the method is exact and, in fact, reduces to cyclic reduction. This provides a valuable reference point for understanding multigrid techniques. The particular multigrid process analyzed is referred to here as Approximate Cyclic Reduction (ACR) and is one of a class known as Multigrid Reduction methods in the literature. It involves one approximation with a known error term. It is possible to relate the error term in this approximation with certain eigenvector components of the error. These are sharply reduced in amplitude by classical relaxation techniques. The approximation can thus be made a very good one.

Merriam, M. L.↗
DOE OSTI2024

Non-Intrusive Parallel-in-Time Solvers for Partial Differential Equations (Final Report)

Many time-dependent problems and simulations are often modeled using Partial Differential Equations. Traditional modeling approaches that use sequential time-stepping are reaching a bottleneck in optimizing efficiency. The Center of Applied Science and Computing at Lawrence Livermore National Laboratory extensively works on parallelizing these algorithms to leverage the increasing computational power from the growing number of processors in computer hardware. In particular, they aim to design non-intrusive algorithms that can generalize to a variety of problems and sizes without requiring additional information from or modifications on the original problems. Multigrid Reduction in Time (MGRIT) is a parallel-in-time algorithm that is designed to be non-intrusive. This project focuses on increasing the efficiency of MGRIT by approximating the coarse-grid operator using machine learning approaches as a means to find the most non-intrusive, or general, solution.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1993

Multigrid and cyclic reduction applied to the Helmholtz equation

We consider the Helmholtz equation with a discontinuous complex parameter and inhomogeneous Dirichlet boundary conditions in a rectangular domain. A variant of the direct method of cyclic reduction (CR) is employed to facilitate the design of improved multigrid (MG) components, resulting in the method of CR-MG. We demonstrate the improved convergence properties of this method.

Brackenridge, Kenneth↗
NASA NTRS2004

Textbook Multigrid Efficiency for the Steady Euler Equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Computations for channel flow and flow over a nonlifting airfoil have computed. Using Gauss-Seidel relaxation ordered in the flow direction, textbook multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier-Stokes equations.

Roberts, Thomas W.↗
DOE OSTI2024

Lifting MGARD: Construction of (pre)wavelets on the interval using polynomial predictors of arbitrary order

MGARD (MultiGrid Adaptive Reduction of Data) is an algorithm for compressing and refactoring scientific data, based on the theory of multigrid methods. The core algorithm is built around stable multilevel decompositions of conforming piecewise linear $C^0$ finite element spaces, enabling accurate error control in various norms and derived quantities of interest. In this work, we extend this construction to arbitrary order Lagrange finite elements $\mathbb{Q}_p$, $p \geq 0$, and propose a reformulation of the algorithm as a lifting scheme with polynomial predictors of arbitrary order. Additionally, a new formulation using a compactly supported wavelet basis is discussed, and an explicit construction of the proposed wavelet transform for uniform dyadic grids is described.

Reshniak, Viktor [Oak Ridge National Laboratory (O↗
DOE OSTI2025

Parallel-in-Time Solution of Hyperbolic PDE Systems via Characteristic-Variable Block Preconditioning

We consider the parallel-in-time solution of both linear and nonlinear hyperbolic partial differential equation (PDE) systems in one spatial dimension. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that intervariable coupling between characteristic variables is weak, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small, while that between the original variables is not. For an ℓ-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of ℓ scalar linear(ized)-advection-like problems, each associated with a different characteristic wave-speed in the underlying linear(ized) PDE. Furthermore, we approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions. For many test problems, the solver converges in just a handful of iterations and with mesh-independent convergence rates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Parallel-in-Time Solution of Scalar Nonlinear Conservation Laws

Here, we consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially nonoscillatory (WENO) reconstructions, and in time with high-order explicit Runge–Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of nondifferentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton’s method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration; however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for nonconservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley–Leverett equations. For many test problems, the solver converges in just a handful of iterations with a convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

TorchBraid: High-Performance Layer-Parallel Training of Deep Neural Networks with MPI and GPU Acceleration

TorchBraid is a high-performance implementation of layer-parallel training for deep neural networks (DNNs) supporting MPI-based parallelism and GPU acceleration. Layer-parallel training has been developed to overcome the serialization inherent in forward and backward propagation of DNNs that limits utilization of computational resources in the strong scaling limit. To achieve this, TorchBraid integrates the PyTorch neural network framework with the state-of-the-art XBraid time-parallel library. Furthermore, this article presents the use and performance of TorchBraid, in addition to solutions for overcoming the algorithmic challenges inherent in combining automatic differentiation with layer-parallel. Results are presented with and without GPU acceleration for the Tiny ImageNet and MNIST image classification data sets, as well as recurrent neural networks. Overall, TorchBraid enables fast training of DNNs, both in a strong and weak scaling context. In addition to the TorchBraid software, several new advances in applying layer-parallel algorithms are detailed. Integration of layer-parallel with data-parallel algorithms is presented for the first time, showing the computational advantages of the combination. Standard deep learning techniques, like batch-normalization, are developed for layer-parallel training. Finally, a new approach combining layer-parallel with spatial coarsening in order to accelerate training for 3D image classification shows roughly a 10× speedup over serial execution.

Layer-parallel↗
DOE OSTI2026

Scientific Data Compression for Large Scale Computational Fluid Dynamics (CFD) Simulations

This Cooperative Research and Development Agreement (CRADA) between Oak Ridge National Laboratory (ORNL) and General Electric (GE) investigated methods for reducing the size of large computational fluid dynamics (CFD) simulation datasets using scientific data compression techniques. The work focused on adapting the MultiGrid Adaptive Reduction of Data (MGARD) compression framework and integrating it with high-performance I/O and visualization tools used in CFD workflows. MGARD uses hierarchical multilevel decomposition to enable error-controlled compression of floating-point scientific data while preserving quantities of interest. During the project, MGARD compression was integrated with the ADIOS I/O framework and visualization tools such as ParaView to enable efficient storage, transfer, and analysis of simulation data. The collaboration also explored approaches for improving compression performance for CFD data defined on unstructured meshes. Results demonstrate that scientific data compression can significantly reduce storage requirements and improve data management for large-scale CFD simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems

Herein this paper focuses on developing a reduction-based algebraic multigrid (AMG) method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based AMG approach, $\ell \text{AIR}$ (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection-dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion-dominated problems in two or three dimensions. Motivated by the success of $\ell \text{AIR}$ in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion-dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy-minimization AMG methods with the local approximation of ideal operators used in $\ell \text{AIR}$. The resulting constrained $\ell \text{AIR}$ algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that was previously difficult for reduction-based methods.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

New multigrid approach for three-dimensional unstructured, adaptive grids

A new multigrid method with adaptive unstructured grids is presented. The three-dimensional Euler equations are solved on tetrahedral grids that are adaptively refined or coarsened locally. The multigrid method is employed to propagate the fine grid corrections more rapidly by redistributing the changes-in-time of the solution from the fine grid to the coarser grids to accelerate convergence. A new approach is employed that uses the parent cells of the fine grid cells in an adapted mesh to generate successively coaser levels of multigrid. This obviates the need for the generation of a sequence of independent, nonoverlapping grids as well as the relatively complicated operations that need to be performed to interpolate the solution and the residuals between the independent grids. The solver is an explicit, vertex-based, finite volume scheme that employs edge-based data structures and operations. Spatial discretization is of central-differencing type combined with a special upwind-like smoothing operators. Application cases include adaptive solutions obtained with multigrid acceleration for supersonic and subsonic flow over a bump in a channel, as well as transonic flow around the ONERA M6 wing. Two levels of multigrid resulted in reduction in the number of iterations by a factor of 5.

Parthasarathy, Vijayan↗
NASA NTRS1995

Convergence acceleration of the Proteus computer code with multigrid methods

This report presents the results of a study to implement convergence acceleration techniques based on the multigrid concept in the two-dimensional and three-dimensional versions of the Proteus computer code. The first section presents a review of the relevant literature on the implementation of the multigrid methods in computer codes for compressible flow analysis. The next two sections present detailed stability analysis of numerical schemes for solving the Euler and Navier-Stokes equations, based on conventional von Neumann analysis and the bi-grid analysis, respectively. The next section presents details of the computational method used in the Proteus computer code. Finally, the multigrid implementation and applications to several two-dimensional and three-dimensional test problems are presented. The results of the present study show that the multigrid method always leads to a reduction in the number of iterations (or time steps) required for convergence. However, there is an overhead associated with the use of multigrid acceleration. The overhead is higher in 2-D problems than in 3-D problems, thus overall multigrid savings in CPU time are in general better in the latter. Savings of about 40-50 percent are typical in 3-D problems, but they are about 20-30 percent in large 2-D problems. The present multigrid method is applicable to steady-state problems and is therefore ineffective in problems with inherently unstable solutions.

Demuren, A. O.↗
NASA NTRS1993

Thin-layer and full Navier-Stokes calculations for turbulent supersonic flow over a cone at an angle of attack

The proper use of a computational fluid dynamics code requires a good understanding of the particular code being applied. In this report the application of CFL3D, a thin-layer Navier-Stokes code, is compared with the results obtained from PARC3D, a full Navier-Stokes code. In order to gain an understanding of the use of this code, a simple problem was chosen in which several key features of the code could be exercised. The problem chosen is a cone in supersonic flow at an angle of attack. The issues of grid resolution, grid blocking, and multigridding with CFL3D are explored. The use of multigridding resulted in a significant reduction in the computational time required to solve the problem. Solutions obtained are compared with the results using the full Navier-Stokes equations solver PARC3D. The results obtained with the CFL3D code compared well with the PARC3D solutions.

Smith, Crawford F.↗
NASA NTRS1984

Multigrid techniques for the numerical solution of the diffusion equation

An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multigrid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.

Phillips, R. E.↗
NASA NTRS1993

Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗
NASA NTRS1993

A multigrid nonoscillatory method for computing high speed flows

A multigrid method using different smoothers has been developed to solve the Euler equations discretized by a nonoscillatory scheme up to fourth order accuracy. The best smoothing property is provided by a five-stage Runge-Kutta technique with optimized coefficients, yet the most efficient smoother is a backward Euler technique in factored and diagonalized form. The singlegrid solution for a hypersonic, viscous conic flow is in excellent agreement with the solution obtained by the third order MUSCL and Roe's method. Mach 8 inviscid flow computations for a complete entry probe have shown that the accuracy is at least as good as the symmetric TVD scheme of Yee and Harten. The implicit multigrid method is four times more efficient than the explicit multigrid technique and 3.5 times faster than the single-grid implicit technique. For a Mach 8.7 inviscid flow over a blunt delta wing at 30 deg incidence, the CPU reduction factor from the three-level multigrid computation is 2.2 on a grid of 37 x 41 x 73 nodes.

Li, C. P.↗
NASA NTRS1996

Multigrid Approach to Incompressible Viscous Cavity Flows

Two-dimensional incompressible viscous driven-cavity flows are computed for Reynolds numbers on the range 100-20,000 using a loosely coupled, implicit, second-order centrally-different scheme. Mesh sequencing and three-level V-cycle multigrid error smoothing are incorporated into the symmetric Gauss-Seidel time-integration algorithm. Parametrics on the numerical parameters are performed, achieving reductions in solution times by more than 60 percent with the full multigrid approach. Details of the circulation patterns are investigated in cavities of 2-to-1, 1-to-1, and 1-to-2 depth to width ratios.

Wood, William A.↗