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At least 127 records · Page 7

All sources NASA NTRS DOE OSTI

DOE OSTI2018

Implementation of Krylov acceleration method for 3D S{sub N} transport code AETIUS with unstructured tetrahedral mesh - 25557

We have been developing an in-house deterministic neutron and gamma transport solver, AETIUS (An Easy modeling Transport code using Unstructured tetrahedral mesh, Shared memory parallel). To achieve fast results, BiCGStab (one of the Krylov acceleration methods) was implemented as a within-group iteration solver. In this paper, two test problems (simple double cube problem and helical tube in the outer cube problem) were used to verify the acceleration performance. The number of iterations and the calculation time were demonstrated and compared with the source iteration. (authors)

61 RADIATION PROTECTION AND DOSIMETRY↗

DOE OSTI2018

High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below M=10 -5 , which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the AUSM + -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Scalable PETSc Implementation of the Three-Phase Unbalanced AC Power Flow Solver

A revolutionary transition of energy infrastructure, spanning across supply and demand sectors, is happening all around the world. Integration of renewable energy sources into our ever-complex power distribution networks is central to the infrastructure planning of the future. One step towards achieving this goal is to solve unbalanced transmission and distribution systems. Traditional approaches and software tools are not suitable because they assume a completely balanced 3-phase system, hence a single-phase equivalent can be solved. Renewable energy such as wind and solar power are intermittent and can pose a challenge when integrating into an already unbalanced power grid. Furthermore, these power grids can be quite large at scale consisting of potentially thousands of buses, which can be computationally demanding. This poster presents preliminary work on designing a robust and scalable 3-phase unbalanced AC power flow solver. The proposed power flow solver will eventually be built on top of the PETSc library and utilizes state-of-the-art parallel data structures like the DMNetwork as well as iterative solvers and preconditioners.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2018

Numerically approximating centrality for graph ranking guarantees

Many real-world datasets can be represented as graphs. Using iterative solvers to approximate graph centrality measures allows us to obtain a ranking vector on the nodes of the graph, consisting of a number for each vertex in the graph identifying its relative importance. In this study the centrality measures we use are Katz Centrality and PageRank. Given an approximate solution, we use the residual to accurately estimate how much of the ranking matches the ranking given by the exact solution. Using probabilistic matrix norms, we obtain bounds on the accuracy of the approximation compared to the exact solution with respect to the highly ranked nodes and apply numerical analysis to the computation of centrality with iterative methods. This relates the numerical accuracy of the linear solver to the data analysis accuracy of finding the correct ranking. In particular, we answer the question of which pairwise rankings are reliable given an approximate solution to the linear system. Experiments on many real-world undirected and directed networks up to several million vertices and several hundred million edges validate our theory and show that we are able to accurately estimate large portions of the approximation. We also analyze the difference between global centrality scores and personalized scores (w.r.t. specific seed vertices). By analyzing convergence error, we develop confidence in the ranking schemes of data mining. We show we are able to accurately guarantee ranking of vertices with an approximation to centrality metrics faster than current methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Optimizing the hypre solver for manycore and GPU architectures

The solution of large-scale combustion problems with codes such as Uintah on modern computer architectures requires the use of multithreading and GPUs to achieve performance. Uintah uses a low-Mach number approximation that requires iteratively solving a large system of linear equations. The Hypre iterative solver has solved such systems in a scalable way for Uintah, but the use of OpenMP with Hypre leads to at least slowdown due to OpenMP overheads. The proposed solution uses the MPI Endpoints within Hypre, where each team of threads acts as a different MPI rank. This approach minimizes OpenMP synchronization overhead and performs as fast or (up to 1.44) faster than Hypre's MPI-only version, and allows the rest of Uintah to be optimized using OpenMP. The profiling of the GPU version of Hypre shows the bottleneck to be the launch overhead of thousands of micro-kernels. The GPU performance was improved by fusing these micro-kernels and was further optimized by using Cuda-aware MPI, resulting in an overall speedup of 1.16—1.44 compared to the baseline GPU implementation. The above optimization strategies were published in the International Conference on Computational Science 2020 [1]. This work extends the previously published research by carrying out the second phase of communication-centered optimizations in Hypre to improve its scalability on large-scale supercomputers. Additionally, this includes an efficient non-blocking inter-thread communication scheme, communication-reducing patch assignment, and expression of logical communication parallelism to a new version of the MPICH library that utilizes the underlying network parallelism [2]. The above optimizations avoid communication bottlenecks previously observed during strong scaling and improve performance by up to 2 on 256 nodes of Intel Knight's Landing processor.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2002

Higher Order, Hybrid BEM/FEM Methods Applied to Antenna Modeling

In this presentation, the authors address topics relevant to higher order modeling using hybrid BEM/FEM formulations. The first of these is the limitation on convergence rates imposed by geometric modeling errors in the analysis of scattering by a dielectric sphere. The second topic is the application of an Incomplete LU Threshold (ILUT) preconditioner to solve the linear system resulting from the BEM/FEM formulation. The final tOpic is the application of the higher order BEM/FEM formulation to antenna modeling problems. The authors have previously presented work on the benefits of higher order modeling. To achieve these benefits, special attention is required in the integration of singular and near-singular terms arising in the surface integral equation. Several methods for handling these terms have been presented. It is also well known that achieving ~he high rates of convergence afforded by higher order bases may als'o require the employment of higher order geometry models. A number of publications have described the use of quadratic elements to model curved surfaces. The authors have shown in an EFIE formulation, applied to scattering by a PEC .sphere, that quadratic order elements may be insufficient to prevent the domination of modeling errors. In fact, on a PEC sphere with radius r = 0.58 Lambda(sub 0), a quartic order geometry representation was required to obtain a convergence benefi.t from quadratic bases when compared to the convergence rate achieved with linear bases. Initial trials indicate that, for a dielectric sphere of the same radius, - requirements on the geometry model are not as severe as for the PEC sphere. The authors will present convergence results for higher order bases as a function of the geometry model order in the hybrid BEM/FEM formulation applied to dielectric spheres. It is well known that the system matrix resulting from the hybrid BEM/FEM formulation is ill -conditioned. For many real applications, a good preconditioner is required to obtain usable convergence from an iterative solver. The authors have examined the use of an Incomplete LU Threshold (ILUT) preconditioner . to solver linear systems stemming from higher order BEM/FEM formulations in 2D scattering problems. Although the resulting preconditioner provided aD excellent approximation to the system inverse, its size in terms of non-zero entries represented only a modest improvement when compared with the fill-in associated with a sparse direct solver. Furthermore, the fill-in of the preconditioner could not be substantially reduced without the occurrence of instabilities. In addition to the results for these 2D problems, the authors will present iterative solution data from the application of the ILUT preconditioner to 3D problems.

Fink, P. W.↗

DOE OSTI2022

Comparison of Some RANS Solvers

We will take a look at solving the Reynolds-averaged Navier-Stokes (RANS) equations that are encountered in the context of wind farm performance simulations and optimizations. We will compare some of the more popular ways to solve these equations with a focus on using iterative solvers for the linear solve. We will compare their performance and reliability to a direct solve as we scale the problem both by adding more and parallel resources and by increasing the size of the domain, both in two and three dimensions. There are many strategies that can be applied to solving the RANS equations, some are very efficient, while others are very insensitive to, for example, the Reynolds number. The first contender we will consider is the Pressure-Convection-Diffusion (PCD) preconditioner. Early results suggest that PCD is indeed a very efficient solver, in particular in two dimensions, as long as the Reynold's number remains small. Next we will try to reorder our degrees of freedom such that we can use GMRES with ILU for our linear solve. Another popular choice we will consider for solving the RANS equations is SIMPLE (and its derivatives). For all of our implementations we make use of either FEniCS or Firedrake, basing our work on both existing implementations of some of these solvers while also writing new extensions for others.

CFD↗

DOE OSTI2015

Petascale solvers for anisotropic PDEs in atmospheric modelling on GPU clusters

Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To solve systems with trillions ( O ( 10 12 ) ) unknowns the code has to make efficient use of several million individual processor cores on large GPU clusters. We describe the multi-GPU implementation of two algorithmically optimal iterative solvers for anisotropic PDEs which are encountered in (semi-) implicit time stepping procedures in atmospheric modelling. In this application the condition number is large but independent of the grid resolution and both methods are asymptotically optimal, albeit with different absolute performance. In particular, an important constant in the discretisation is the CFL number; only the multigrid solver is robust to changes in this constant. We parallelise the solvers and adapt them to the specific features of GPU architectures, paying particular attention to efficient global memory access. We achieve a performance of up to 0.78 PFLOPs when solving an equation with 0.55 · 10 12 unknowns on 16384 GPUs; this corresponds to about 3% of the theoretical peak performance of the machine and we use more than 40% of the peak memory bandwidth with a Conjugate Gradient (CG) solver. Although the other solver, a geometric multigrid algorithm, has a slightly worse performance in terms of FLOPs per second, overall it is faster as it needs less iterations to converge; the multigrid algorithm can solve a linear PDE with half a trillion unknowns in about one second.

Computer Science↗

NASA NTRS1991

Iterative solution techniques in boundary element analysis

Iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis (BEA) method are discussed. Continuum structural response analysis problems are considered, employing single- and multizone boundary element models with and without zone condensation. The impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices, resulting in multizone BEA, is studied. Both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported. Included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods.

Kane, J. H.↗

NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗

DOE OSTI2019

Taylor states in stellarators: A fast high-order boundary integral solver

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

Parallel Preconditioning for CFD Problems on the CM-5

Up to today, preconditioning methods on massively parallel systems have faced a major difficulty. The most successful preconditioning methods in terms of accelerating the convergence of the iterative solver such as incomplete LU factorizations are notoriously difficult to implement on parallel machines for two reasons: (1) the actual computation of the preconditioner is not very floating-point intensive, but requires a large amount of unstructured communication, and (2) the application of the preconditioning matrix in the iteration phase (i.e. triangular solves) are difficult to parallelize because of the recursive nature of the computation. Here we present a new approach to preconditioning for very large, sparse, unsymmetric, linear systems, which avoids both difficulties. We explicitly compute an approximate inverse to our original matrix. This new preconditioning matrix can be applied most efficiently for iterative methods on massively parallel machines, since the preconditioning phase involves only a matrix-vector multiplication, with possibly a dense matrix. Furthermore the actual computation of the preconditioning matrix has natural parallelism. For a problem of size n, the preconditioning matrix can be computed by solving n independent small least squares problems. The algorithm and its implementation on the Connection Machine CM-5 are discussed in detail and supported by extensive timings obtained from real problem data.

Simon, Horst D.↗

DOE OSTI2016

CAMELLIA

Camellia offers a framework for development of solvers for the discontinuous Petrov-Galerkin methodology of Demkowicz and Gopalakrishnan as well as other finite element methodologies. Spatially 1D-3D meshes are supported, and space-time elements are supported for all spatial topologies. Both h and p adaptivity are supported. A general interface for direct and iterative solvers is provided; a geometric multigrid preconditioner is also provided. The core computational kernels take advantage of MPI to execute in parallel.

Roberts, Nathan↗

NASA NTRS1992

Iterative methods for large scale static analysis of structures on a scalable multiprocessor supercomputer

A parallel Preconditioned Conjugate Gradient (PCG) iterative solver has been developed and implemented on the iPSC-860 scalable hypercube. This new implementation makes use of the Parallel Automated Runtime Toolkit at ICASE (PARTI) primitives to efficiently program irregular communications patterns that exist in general sparse matrices and in particular in the finite element sparse stiffness matrices. The iterative PCG has been used to solve the finite element equations that result from discretizing large scale aerospace structures. In particular, the static response of the High Speed Civil Transport (HSCT) finite element model is solved on the iPSC-860.

Sobh, Nahil Atef↗

DOE OSTI2018

Characterization of the Impact of Soft Errors on Iterative Methods

Soft errors caused by transient bit flips have the potential to significantly impact an application’s behavior. This has motivated the design of an array of techniques to detect, isolate, and correct soft errors using microarchitectural, architectural, compilation-based, or application-level techniques to minimize their impact on the executing application. The first step towards the design of good error detection/correction techniques involves an understanding of an application’s vulnerability to soft errors. In this paper, we present the first comprehensive characterization of the impact of soft errors on the convergence characteristics of six iterative methods using application-level fault injection. In particular, we consider the use of iterative methods to incrementally solve a linear systems of equations, which constitutes the core kernel in many scientific applications. We analyze the impact of soft errors in terms of the type of error (single- vs multi-bit), the distribution and location of bits affected, the data structure and the statement impacted, and variation with time. In addition to understanding the vulnerability of iterative solvers to soft errors, this characterization can aid the design of fault injection campaigns that ensure systematic coverage.

fault injection, iterative methods, Silent data co↗

DOE OSTI2021

ANTS

The ANTS code (Alternate Non-Linear Two-phase Solver) is based on a novel non-linear solution algorithm for the solution of the two-phase, subchannel fluid equations. It achieves its performance through decoupling of the two-phase momentum equations (axial and transverse) from the axial phasic mass and energy equations which allows for a nested non-linear iteration scheme. This enables a plane-by-plane solution which the inner iteration focuses on a reduced non-linear equation set for the primitives in phasic mass flow rate, enthalpy and void for each node edge. Single node edges are coupled as part of the outer iteration via surface mass fluxes which appear as source terms in the inner iteration scheme. The outer iteration readily accommodates two-phase flow phenomena closure relationships for subchannel mixing and void drift. A primary feature is the use of a non-staggered mesh computational mesh and steady-state iterative solver in contrast to all existing subchannel codes.

Kropaczek, David J↗

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↗

NASA NTRS2011

Technique for Solving Electrically Small to Large Structures for Broadband Applications

Fast iterative algorithms are often used for solving Method of Moments (MoM) systems, having a large number of unknowns, to determine current distribution and other parameters. The most commonly used fast methods include the fast multipole method (FMM), the precorrected fast Fourier transform (PFFT), and low-rank QR compression methods. These methods reduce the O(N) memory and time requirements to O(N log N) by compressing the dense MoM system so as to exploit the physics of Green s Function interactions. FFT-based techniques for solving such problems are efficient for spacefilling and uniform structures, but their performance substantially degrades for non-uniformly distributed structures due to the inherent need to employ a uniform global grid. FMM or QR techniques are better suited than FFT techniques; however, neither the FMM nor the QR technique can be used at all frequencies. This method has been developed to efficiently solve for a desired parameter of a system or device that can include both electrically large FMM elements, and electrically small QR elements. The system or device is set up as an oct-tree structure that can include regions of both the FMM type and the QR type. The system is enclosed with a cube at a 0- th level, splitting the cube at the 0-th level into eight child cubes. This forms cubes at a 1st level, recursively repeating the splitting process for cubes at successive levels until a desired number of levels is created. For each cube that is thus formed, neighbor lists and interaction lists are maintained. An iterative solver is then used to determine a first matrix vector product for any electrically large elements as well as a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large and small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within the predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter. The solution for the desired parameter is then presented to a user in a tangible form; for example, on a display.

Jandhyala, Vikram↗