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7 records
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↗
DOE OSTI2024

Degradation of Poly- and Perfluoroalkyl Substances (PFAS) in Water via High Power, Energy-Efficient Electron Beam Accelerator

The goal of the 2-year workplan was to see if electron beam (EB) could be used to break down a sub-set of the larger chemical family of per and polyfluoroalkylated substances (PFAS) in an energy efficient and economical manner when compared to conventional water treatment technologies. Year one (Y1) work focused on sample EB treatment work in the Fermi National Accelerator Laboratory’s (FNALs) Accelerator Applications Demonstration and Development (A2D2) EB accelerator. While there are reportedly thousands of types of PFAS, for the point of most of the work herein, a small subset was examined, typically perfluorooctane sulfonate (PFOS) and perfluorooctanoate (PFOA). PFOA and PFOS are two of the most well studied PFAS and are studied for baseline evaluations and are considered most useful. The work from Y1 provided information about the optimal operating parameters and additives to use when treating PFOS and PFOA via EB. The data were then used to see where in a water treatment system an EB accelerator would be best suited to treat PFAS. A conventional water treatment technology, GAC, was then compared to e-beam treatment technology with respect to energy and costs for treatment. In year two (Y2), several conventional e-beam accelerator designs, and FNAL’s developmental compact SRF accelerator design, were evaluated for their suitability in PFAS treatment, from an energy efficiency and cost standpoint. Several EB parameters were evaluated and optimized for the removal of PFOA and PFOS from water at normal pressure and temperature, measured as total PFAS removal. Under the optimized test conditions both PFOA showed complete destruction to inorganic fluoride, and PFOS to inorganic fluoride and sulfate, with mass balance. The effect on PFAS removal relative to solution pH, total EB dose, EB dose rate, dissolved oxygen concentration (DO), temperature, and initial PFAS concentration were evaluated. In general, PFOA was easier to destroy than PFOS. Degradation products, typically observed under less-than-optimal EB conditions, provided insight to degradation mechanisms. Products were identified to rule out possible deleterious biproduct formation. The water radiolysis radical reaction kinetics with PFOS and PFOA were not dependent on the initial concentration over 5-orders of magnitude from 2 μg/L to 20 mg/L. This is thought to be because there was an overabundance of the reactive water radiolysis radicals relative to PFAS molecules and largely attributed to aqueous electrons. The reaction rates appeared to be diffusion limited. Testing at higher concentrations (100-200 mg/L) showed a decrease in removal efficiency, suggesting alternative kinetics, possibly second order rates, at higher concentrations. In all, we successfully defined a set of optimal EB parameters to treat PFOA and PFOS at concentrations of 20 mg/L in water with destruction efficiencies near 100%. We further tested the optimized EB parameters with other types of PFAS, including shorter and longer fluorocarbon chain homologs of PFOA and PFOS, and PFAS with alternative functional groups such as sulfonamides. Based on our results EB can be optimized as an effective destructive technology for removing PFAS from water. The conditions optimized for PFOA and PFOS were less effective with ultra-short fluorocarbon compounds like TFMS, PFES, PFPS and PFBS, and likely require re-optimization of parameters to them. In all, it was determined that from a cost and energy efficiency standpoint, EB would be best applied to waste streams with relatively high concentrations of PFOS and PFOA and is not as cost effective as GAC treatment for removing low concentrations of PFAS from water. Higher concentrations of PFAS can be found in the wastewater of conventional treatment processes such as RO and IE and therefore EB may be used to supplement such treatment technologies. Some real-world IE regeneration wash water and RO reject water containing higher concentrations of PFAS and obtained from pilot scale industrial wastewater treatment system at a fluorochemical manufacturing facility, showed that EB could remove PFAS from such types of wastewaters. The IE regenerant wash water appeared to be the most efficient of the two types of wastewaters tested. However, some further optimization of the EB parameters for the specific PFAS types present in those wastewaters may be required. Also, the effects of co-present TOC and mineral salts should be considered during such optimization efforts. From the experimental Y1 results it was seen that the aqueous electron drives degradation of the PFAS. In a hypothetical water treatment skid using EB for PFAS destruction the parameters of the system should be optimized to promote aqueous electron production. Before EB treatment, the PFAS should be preconcentrated when possible, the pH should be raised to pH 10 or higher to enhance aqueous electron production, and the water should be nitrogen purged to remove dissolved oxygen to minimize aqueous electron scavenging. An excel spreadsheet was created that calculates optimal conditions based on inlet PFAS concentration and desired outlet concentration, by optimizing the accelerator power, dose rate, water treatment rate, pH and dissolved oxygen levels to reach the desired endpoint. Given this information on accelerator operating conditions five different EB accelerator systems were compared. One EB system was a continuous-wave, linear superconducting accelerator being designed at Fermilab. Three other EB systems (IMPELA at 5% and 25% duty factor and the ILU-14) were normal conducting pulsed linear accelerators. The fifth system was an IBA Rhodotron which is a normal conducting, circular, continuous-wave accelerator. The accelerator efficiency (% of the incoming power that is used in water treatment) was the dominating factor in accelerator choice. The radio frequency (RF) power supply and the accelerator design (superconducting versus warm technology) drive the accelerator efficiency. The IBA Rhodotron was seen to be the most energy efficient commercially available technology with a wall-plug (total) power efficiency of 43% at 400 kW. The Fermilab design, with a prototype for a different application currently being fabricated, was the most energy efficient at 55% when driven by a Klystron RF power supply and as high as 77% when powered by a magnetron. As the Fermilab design was the most energy efficient by approximately 10-30%, further design work was done on the accelerator and beam delivery system specific to the destruction of PFAS in water. The Fermilab design is unique from industrial accelerators in that is superconducting. Superconducting technology allows for the acceleration of electrons without losses. The accelerator must be cooled to below the point where it is superconducting and is operated around 4 degrees Kelvin. The bulk of the design work for the accelerator is on making the accelerator as energy efficient as possible so that it does not require liquid helium and can be cooled with conduction cooling via cryocoolers. Final design work resulted in an EB accelerator that would operate at minimally 200 kW and 10 MeV. Prototype construction would cost $\$ $7.8 million dollars when driven by a Klystron power supply. A second version of the same accelerator would cost $\$ $5.5 million dollars when driven by a magnetron that is still under development. The commercially available 300 kW IBA Rhodotron cost was estimated at approximately $\$ $9 million. While it is hard to directly compare, an operational GAC system used by 3M for groundwater treatment capital cost (2022 dollars) was estimated to cost $\$ $3.3 million. While the capital expense of the EB accelerator systems was higher than GAC, the accelerator EB treatment would result in destruction of the PFAS and not just sequestration of PFAS to form a new waste stream that requires further treatment or disposal. The operating cost to destroy the PFAS via 400 kw EB system was less than $\$ $1000/kg of PFAS destroyed when treating at a 20 mg/L PFAS concentration, compared to GAC with operating costs that calculated at $\$ $27,530 per kg of PFAS sequestered when treating 100 μg/L PFOA and PFOS combined concentration.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Summary Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

Ghai, Aditi↗
DOE OSTI2022

Novel Solver Algorithms for Nearly Singular Linear Systems Arising in Combustion Modelling

Direct Numerical Simulations of realistic combustion devices are extremely challenging due to the wide separation of scales in the simulation, for example an internal combustion (IC) engine chamber, and the flame thickness of a high-pressure flame. The PeleLMeX solver uses adaptive mesh refinement (AMR) to evolve multi-species reacting flows in the low Mach number limit at the Exascale and relies on an embedded boundary (EB) approach to represent complex geometries. In that framework, the EB geometries often give rise to very small cut-cells along the boundary, which translate into extreme ill-conditioning of the pressure-projection, with eigenvalues that span 15-16 orders of magnitude. In this talk, we focus on the case of a typical IC piston bowl geometry for which we present on a novel approach towards solving these nearly singular linear systems with ILU-based, C-AMG smoothers on massively parallel architectures. In particular, we use scaling and equilibration algorithms to handle the non-normality of the upper triangular factors. This enables us to approximate the highly sequential triangular solve algorithm, embedded in the AMG smoothing-solve phase, with Jacobi iterations. This approximation can be written as a convergent Neumann series whose terms are composed of highly parallel sparse matrix vector multiplications. The result is an algorithm that substantially decreases setup and solve time, compared to state-of-the-art, for these challenging linear systems.

combustion modelling↗
DOE OSTI2021

Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems

Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.

Preconditioner, Nonlinear, Porous media, Multiphas↗
DOE OSTI2023

Scaled ILU Smoothers for Navier-Stokes Pressure Projection

Incomplete LU (ILU) smoothers are effective in the algebraic multigrid (AMG) V-cycle for reducing high-frequency components of the error. However, the requisite direct triangular solves are comparatively slow on GPUs. Previous work has demonstrated the advantages of Jacobi iteration as an alternative to direct solution of these systems. Depending on the threshold and fill-level parameters chosen, the factors can be highly nonnormal and Jacobi is unlikely to converge in a low number of iterations. We demonstrate that row scaling can reduce the departure from normality, allowing us to replace the inherently sequential solve with a rapidly converging Richardson iteration. There are several advantages beyond the lower compute time. Scaling is performed locally for a diagonal block of the global matrix because it is applied directly to the factor. Further, an ILUT Schur complement smoother maintains a constant GMRES iteration count as the number of MPI ranks increases, and thus parallel strong-scaling is improved. Our algorithms have been incorporated into hypre, and we demonstrate improved time to solution for linear systems arising in the Nalu-Wind and PeleLM pressure solvers. For large problem sizes, GMRES+AMG executes at least five times faster when using iterative triangular solves compared with direct solves on massively parallel GPUs.

algebraic multigrid↗
DOE OSTI2020

Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

97 MATHEMATICS AND COMPUTING↗