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All sourcesNASA NTRSDOE OSTI
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 OSTI2026

A coupled hydrologic-agroeconomic modeling framework to evaluate adaptive irrigation strategies under groundwater withdrawal restrictions

Growing groundwater scarcity requires integrated tools to capture interactions among hydrology, agricultural production, markets, and land use. This study presents an iterative modeling framework that couples hydrologic, crop-yield, and economic models to capture two-way feedback among water availability, agricultural production, and market responses under groundwater constraints. The primary goal of this paper is to describe the methodological development of the coupled framework and demonstrate the significance of iterative model interaction. Applied to the western United States, we evaluated adaptive responses to restricting groundwater use beyond recharge levels, represented through changes in irrigation management and expansion or shrinkage of crop markets through land reallocation. Results demonstrate that the iterative coupling converges to stable equilibrium responses within 10 iterations. At equilibrium, deficit irrigation emerges as the dominant adaptation strategy in California, with irrigation levels stabilizing at approximately 70% of full irrigation demand, while Arizona and New Mexico experience stronger yield sensitivities. Early iterations produce commodity price increases of up to 10% for fruit and vegetable crops; however, these responses moderate as land allocation and production patterns adjust across regions. Deficit irrigation and spatial reallocation of irrigated land partially offset production losses, with variability observed across different states: California maintains yields primarily via deficit irrigation, whereas Arizona and New Mexico will rely mainly on reducing irrigated area to absorb the shock. By capturing feedback between biophysical and economic processes, this approach highlights how irrigation strategies and land-use decisions evolve under water stress and provides a transferable platform for evaluating water management policies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2015

Numerical analysis of modified Central Solenoid insert design

The United States ITER Project Office (USIPO) is responsible for fabrication of the Central Solenoid (CS) for ITER project. The ITER machine is currently under construction by seven parties in Cadarache, France. The CS Insert (CSI) project should provide a verification of the conductor performance in relevant conditions of temperature, field, currents and mechanical strain. The US IPO designed the CSI that will be tested at the Central Solenoid Model Coil (CSMC) Test Facility at JAEA, Naka. To validate the modified design we performed three-dimensional numerical simulations using coupled solver for simultaneous structural, thermal and electromagnetic analysis. Thermal and electromagnetic simulations supported structural calculations providing necessary loads and strains. According to current analysis design of the modified coil satisfies ITER magnet structural design criteria for the following conditions: (1) room temperature, no current, (2) temperature 4K, no current, (3) temperature 4K, current 60 kA direct charge, and (4) temperature 4K, current 60 kA reverse charge. Fatigue life assessment analysis is performed for the alternating conditions of: temperature 4K, no current, and temperature 4K, current 45 kA direct charge. Results of fatigue analysis show that parts of the coil assembly can be qualified for up to 1 million cycles. Distributions of the Current Sharing Temperature (TCS) in the superconductor were obtained from numerical results using parameterization of the critical surface in the form similar to that proposed for ITER. Lastly, special ADPL scripts were developed for ANSYS allowing one-dimensional representation of TCS along the cable, as well as three-dimensional fields of TCS in superconductor material. Published by Elsevier B.V.

42 ENGINEERING↗
DOE OSTI2026

Mechanical characterization of fine-grain dispersion-strengthened tungsten as a plasma facing material

Field-Assisted Sintering Technology (FAST) was used to produce fine-grained, dispersion-strengthened tungsten (W) materials. Investigated materials 4138, 4353, and 4355 composed of 3 wt% ZrC sintered at 1800 °C, 5 wt% ZrC sintered at 1800 °C, and 3 wt% ZrC sintered at 2000 °C, respectively. They were compared against ITER-grade W. A series of mechanical and thermal property testing and microstructure studies were conducted to study them as a potential plasma facing material (PFM) for fusion reactors. Hardness testing showed that manufacturing conditions substantially altered hardness. Material 4355 had an average HV10 value of 497.2 ± 16.8, slightly higher than ITER-grade at 378.5 ± 40.3. However, material 4353 was substantially higher with an HV10 value of 738.9 ± 31.7 over the investigated temperature range. Electron Backscatter Diffraction (EBSD) analysis showed that FAST produced substantially smaller grains than the hot-rolled ITER-grade W material, offering notable control over grain size. Materials 4353 and 4355 had grain sizes of 0.44 ± 0.20 µm and 3.67 ± 0.89 µm, respectively, whereas ITER-grade 27.14 ± 19.76 µm at room temperature. The fine grain structures showed no net coarsening after 1 hr. anneals up to 1800 °C, several hundred degrees above the 1100 – 1500 °C recrystallization range reported for conventional W. Inverse application of the Zener pinning relationship to the measured grain sizes indicates that these two FAST sintering conditions produce markedly different effective dispersoid populations, with effective particle diameters of approximately 90 nm at a peak sintering temperature of 1800 °C and approximately 460 nm at 2000 °C, respectively. This result demonstrates that the FAST thermal condition itself, and not the nominal ZrC content alone, governs the pinning effectiveness of the dispersion. Thermal diffusivity measurements support this finding independently. Materials of identical composition sintered at different temperatures differ by approximately 19% in measured thermal diffusivity with statistically indistinguishable density, while materials of different composition and sintering temperature converge to within approximately 2%. At a representative divertor heat flux of 10 MW/m², the lower thermal conductivity of the fine-grained materials corresponds to approximately 28 to 33 °C per millimeter of armor thickness relative to ITER-grade W, traded against a substantially larger margin to recrystallization-driven degradation. While high temperature tensile testing revealed likely contamination that motivates refinement of the manufacturing process, FAST-produced, fine-grained, dispersion-strengthened W offers process-controlled microstructural stability well above the operating temperatures of conventional W and supports its continued development as a PFM for economically viable commercial fusion power.

Parker, Gabe [ORNL] (ORCID:0000000190394100)↗
DOE OSTI2019

Retention properties in displacement damaged ultra-fine grain tungsten exposed to divertor plasma

One of the main advantages of using tungsten (W) as a plasma facing material (PFM) is its low uptake and retention of tritium. However, in high purity (ITER grade) W, hydrogenic retention increases significantly with neutron-induced displacement damage in the W lattice. This experiment examines an alternative W grade PFM, ultra-fine grain (UFG) W, to compare its retention properties with ITER grade W after 12 MeV Si ion displacement damage up to 0.6 dpa (displacements per atom.) Following exposure to plasma in the DIII-D divertor, D retention was then assessed with Nuclear Reaction Analysis (NRA) depth profiling up to 3.5 µm and thermal desorption spectrometry (TDS). Undamaged specimens were also included in our test matrix for comparison. For all samples, D release peaks were observed during TDS at approximately 200 °C and 750 °C. For the ITER-grade W specimens, the intensity of the 750 °C release peak was more pronounced for specimens that had been pre-damaged. Conversely, UFG samples that had been damaged by 12 MeV Si showed enhancement of the lower temperature release peak (200 °C). NRA profiles also reveal a higher D concentration for UFG W samples up to the peak in the damage profile at a depth of 2 m. Overall, we observed that the total trapped inventory in UFG W was 20% higher than ITER grade W in the undamaged case and 10% higher in the damaged case. A comparison of NRA and TDS data indicates that a larger fraction of the total retained D is trapped near the surface (86–100%) in UFG W pre-damaged to 0.6 dpa compared with ITER grade W (39–61%). Further examination of the UFG material with microscopy is recommended for a definitive determination of the types of defects responsible for D trapping. Our results highlight some potential trade-offs associated UFG W regarding its performance from a tritium retention standpoint. That said, our TDS results indicate that this enhanced inventory can be released by baking at relatively low temperatures (<500 °C), providing an avenue for minimizing tritium retention in this material that would be practical for implementation in a tokamak.

Barton, J↗
DOE OSTI2019

Agent-Based Dynamic Traffic Assignment with Information Mixing

This study develops an approach for network assignment convergence using individualized agent-based routing and agent-specific link cost information as well as historical travel times. Each traveler is routed individually from their starting and ending network link at a specific departure time. The approach is gap-based in two ways. First, as in previous approaches, the re-assignment decision during the convergence process is based on the gap between the routed and experienced travel time from the previous iteration. Secondly, the historical time-dependent and prevailing traffic conditions are averaged into a single expected value for each agent using a weight calculated by a modified two-parameter Weibull survival function. This weight is individualized based on the relative gap of the traveler from the previous iteration, as well as the iteration number; a novel aspect of this work. The methodology is tested on a medium-scale network of Bloomington, IL. The algorithm converges after only two iterations, which is promising as computational time of a single iteration can be high for large-scale networks.

agent-based modeling↗
DOE OSTI2017

Projected Commutator DIIS Method for Accelerating Hybrid Functional Electronic Structure Calculations

The commutator direct inversion of the iterative subspace (commutator DIIS or C-DIIS) method developed by Pulay is an efficient and the most widely used scheme in quantum chemistry to accelerate the convergence of self-consistent field (SCF) iterations in Hartree–Fock theory and Kohn–Sham density functional theory. The C-DIIS method requires the explicit storage of the density matrix, the Fock matrix, and the commutator matrix. Hence, the method can only be used for systems with a relatively small basis set, such as the Gaussian basis set. We develop a new method that enables the C-DIIS method to be efficiently employed in electronic structure calculations with a large basis set such as planewaves for the first time. The key ingredient is the projection of both the density matrix and the commutator matrix to an auxiliary matrix called the gauge-fixing matrix. The resulting projected commutator-DIIS method (PC-DIIS) only operates on matrices of the same dimension as that consists of Kohn–Sham orbitals. The cost of the method is comparable to that of standard charge mixing schemes used in large basis set calculations. The PC-DIIS method is gauge-invariant, which guarantees that its performance is invariant with respect to any unitary transformation of the Kohn–Sham orbitals. We demonstrate that the PC-DIIS method can be viewed as an extension of an iterative eigensolver for nonlinear problems. We use the PC-DIIS method for accelerating Kohn–Sham density functional theory calculations with hybrid exchange–correlation functionals, and demonstrate its superior performance compared to the commonly used nested two-level SCF iteration procedure. Furthermore, we demonstrate that in the context of ab initio molecular dynamics (MD) simulation with hybrid functionals one can extrapolate the gauge-fixing matrix to achieve the goal of extrapolating the entire density matrix implicitly along the MD trajectory. Here, numerical results indicate that the new method significantly reduces the number of SCF iterations per MD step, compared to the commonly used strategy of extrapolating the electron density.

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

Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations

We describe here a novel iterative strategy for Kohn–Sham density functional theory calculations aimed at large systems (>1,000 electrons), applicable to metals and insulators alike. In lieu of explicit diagonalization of the Kohn–Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ a two-level Chebyshev polynomial filter based complementary subspace strategy to (1) compute a set of vectors that span the occupied subspace of the Hamiltonian; (2) reduce subspace diagonalization to just partially occupied states; and (3) obtain those states in an efficient, scalable manner via an inner Chebyshev filter iteration. By reducing the necessary computation to just partially occupied states and obtaining these through an inner Chebyshev iteration, our approach reduces the cost of large metallic calculations significantly, while eliminating subspace diagonalization for insulating systems altogether. We describe the implementation of the method within the framework of the discontinuous Galerkin (DG) electronic structure method and show that this results in a computational scheme that can effectively tackle bulk and nano systems containing tens of thousands of electrons, with chemical accuracy, within a few minutes or less of wall clock time per SCF iteration on large-scale computing platforms. We anticipate that our method will be instrumental in pushing the envelope of large-scale ab initio molecular dynamics. As a demonstration of this, we simulate a bulk silicon system containing 8,000 atoms at finite temperature, and obtain an average SCF step wall time of 51 s on 34,560 processors; thus allowing us to carry out 1.0 ps of ab initio molecular dynamics in approximately 28 h (of wall time).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Full wave modeling of radio-frequency beams in tokamaks in the electron cyclotron frequency range

Simulation of full wave, without paraxial approximation, high-resolution solution of wave equations in frequency domain in the electron cyclotron resonance (ECR) frequency range for realistic Tokamak plasma parameters became possible by using recently formulated hybrid iterative algorithm [Svidzinski et al., Phys. Plasmas 25, 082509 (2018)] for numerically solving discretized wave equations. This approach combines time evolution and iterative relaxation techniques into iteration cycles. This algorithm is implemented in 2D code FullWave, solving wave equations in Tokamaks in cold and hot plasma models, and it has been tested in 3D full wave iterative RF beams simulation tool, which is presently being developed to model 3D ECRH RF beams in fusion devices using dynamic grid adaptation. The results of 2D full wave modeling, assuming specified toroidal mode number, of ECRH RF beams in DIII-D plasma, performed in the cold and hot plasma models for outboard and top launch scenarios using FullWave are presented. Nonlocal hot plasma response model, based on accurate numerical solution of linearized Vlasov equation, is used to model beam propagation and absorption in the 2nd electron cyclotron harmonic region. Demonstration of capability of the hybrid iterative algorithm to model ECRH RF beams in 3D is made by simulating a substantial part of realistic beam in DIII-D, launched from outboard side of the machine. All relevant physics of RF beam propagation, most of which is not captured in paraxial approximation, such as beam's divergence, interference between the X and O modes in the beam, X-O mode conversion, beam splitting into the X and O mode beams, transformation of beam's cross section, and absorption at the 2nd electron cyclotron harmonic, is captured in the simulations. A numerical technique to find an optimal beam polarization at the launcher to launch a nearly pure X or O mode beam in plasma is developed and tested.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are introduced, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the conclusions in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

3-D↗
DOE OSTI2015

Overview of the JET results

Since the installation of an ITER-like wall, the JET programme has focused on the consolidation of ITER design choices and the preparation for ITER operation, with a specific emphasis given to the bulk tungsten melt experiment, which has been crucial for the final decision on the material choice for the day-one tungsten divertor in ITER. Integrated scenarios have been progressed with the re-establishment of long-pulse, high-confinement H-modes by optimizing the magnetic configuration and the use of ICRH to avoid tungsten impurity accumulation. Stationary discharges with detached divertor conditions and small edge localized modes have been demonstrated by nitrogen seeding. The differences in confinement and pedestal behaviour before and after the ITER-like wall installation have been better characterized towards the development of high fusion yield scenarios in DT. Post-mortem analyses of the plasma-facing components have confirmed the previously reported low fuel retention obtained by gas balance and shown that the pattern of deposition within the divertor has changed significantly with respect to the JET carbon wall campaigns due to the absence of thermally activated chemical erosion of beryllium in contrast to carbon. Lastly, transport to remote areas is almost absent and two orders of magnitude less material is found in the divertor.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Enhanced understanding of non-axisymmetric intrinsic and controlled field impacts in tokamaks

Here, an extensive study of intrinsic and controlled non-axisymmetric field (δB) impacts in KSTAR has enhanced the understanding about non-axisymmetric field physics and its implications, in particular, on resonant magnetic perturbation (RMP) physics and power threshold (P th ) for L–H transition. The n=1 intrinsic non-axisymmetric field in KSTAR was measured to remain as low as δB/B 0 ~ 4×10 –5 even at high-beta plasmas (β N ~ 2), which corresponds to approximately 20% below the targeted ITER tolerance level. As for the RMP edge-localized-modes (ELM) control, robust n=1 RMP ELM-crash-suppression has been not only sustained for more than ~90 τ E , but also confirmed to be compatible with rotating RMP. An optimal window of radial position of lower X-point (i.e. R x = $1.44\pm 0.02\,$ m) proved to be quite critical to reach full n=1 RMP-driven ELM-crash-suppression, while a constraint of the safety factor could be relaxed (q 95 = 5 $\pm $ 0.25). A more encouraging finding was that even when R x cannot be positioned in the optimal window, another systematic scan in the vicinity of the previously optimal R x allows for a new optimal window with relatively small variations of plasma parameters. Also, we have addressed the importance of optimal phasing (i.e. toroidal phase difference between adjacent rows) for n=1 RMP-driven ELM control, consistent with an ideal plasma response modeling which could predict phasing-dependent ELM suppression windows. In support of ITER RMP study, intentionally misaligned RMPs have been found to be quite effective during ELM-mitigation stage in lowering the peaks of divertor heat flux, as well as in broadening the 'wet' areas. Besides, a systematic survey of P th dependence on non-axisymmetric field has revealed the potential limit of the merit of low intrinsic non-axisymmetry. Considering that the ITER RMP coils are composed of 3-rows, just like in KSTAR, further 3D physics study in KSTAR is expected to help us minimize the uncertainties of the ITER RMP coils, as well as establish an optimal 3D configuration for ITER and future reactors.

3D physics↗
DOE OSTI2022

DIII-D research advancing the physics basis for optimizing the tokamak approach to fusion energy

DIII-D physics research addresses critical challenges for the operation of ITER and the next generation of fusion energy devices. This is done through a focus on innovations to provide solutions for high performance long pulse operation, coupled with fundamental plasma physics understanding and model validation, to drive scenario development by integrating high performance core and boundary plasmas. Substantial increases in off-axis current drive efficiency from an innovative top launch system for EC power, and in pressure broadening for Alfven eigenmode control from a co-/counter-I p steerable off-axis neutral beam, all improve the prospects for optimization of future long pulse/steady state high performance tokamak operation. Fundamental studies into the modes that drive the evolution of the pedestal pressure profile and electron vs ion heat flux validate predictive models of pedestal recovery after ELMs. Understanding the physics mechanisms of ELM control and density pumpout by 3D magnetic perturbation fields leads to confident predictions for ITER and future devices. Validated modeling of high-Z shattered pellet injection for disruption mitigation, runaway electron dissipation, and techniques for disruption prediction and avoidance including machine learning, give confidence in handling disruptivity for future devices. For the non-nuclear phase of ITER, two actuators are identified to lower the L–H threshold power in hydrogen plasmas. With this physics understanding and suite of capabilities, a high poloidal beta optimized-core scenario with an internal transport barrier that projects nearly to Q = 10 in ITER at ~8 MA was coupled to a detached divertor, and a near super H-mode optimized-pedestal scenario with co-I p beam injection was coupled to a radiative divertor. The hybrid core scenario was achieved directly, without the need for anomalous current diffusion, using off-axis current drive actuators. Also, a controller to assess proximity to stability limits and regulate β N in the ITER baseline scenario, based on plasma response to probing 3D fields, was demonstrated. Finally, innovative tokamak operation using a negative triangularity shape showed many attractive features for future pilot plant operation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Global adjoint tomography: first-generation model

We present the first-generation global tomographic model constructed based on adjoint tomography, an iterative full-waveform inversion technique. Synthetic seismograms were calculated using GPU-accelerated spectral-element simulations of global seismic wave propagation, accommodating effects due to 3-D anelastic crust & mantle structure, topography & bathymetry, the ocean load, ellipticity, rotation, and self-gravitation. Fréchet derivatives were calculated in 3-D anelastic models based on an adjoint-state method. The simulations were performed on the Cray XK7 named ‘Titan’, a computer with 18 688 GPU accelerators housed at Oak Ridge National Laboratory. The transversely isotropic global model is the result of 15 tomographic iterations, which systematically reduced differences between observed and simulated three-component seismograms. Our starting model combined 3-D mantle model S362ANI with 3-D crustal model Crust2.0. We simultaneously inverted for structure in the crust and mantle, thereby eliminating the need for widely used ‘crustal corrections’. We used data from 253 earthquakes in the magnitude range 5.8 ≤ M w ≤ 7.0. We started inversions by combining ~30 s body-wave data with ~60 s surface-wave data. The shortest period of the surface waves was gradually decreased, and in the last three iterations we combined ~17 s body waves with ~45 s surface waves. We started using 180 min long seismograms after the 12th iteration and assimilated minor- and major-arc body and surface waves. The 15th iteration model features enhancements of well-known slabs, an enhanced image of the Samoa/Tahiti plume, as well as various other plumes and hotspots, such as Caroline, Galapagos, Yellowstone and Erebus. Furthermore, we see clear improvements in slab resolution along the Hellenic and Japan Arcs, as well as subduction along the East of Scotia Plate, which does not exist in the starting model. Point-spread function tests demonstrate that we are approaching the resolution of continental-scale studies in some areas, for example, underneath Yellowstone. Here, this is a consequence of our multiscale smoothing strategy in which we define our smoothing operator as a function of the approximate Hessian kernel, thereby smoothing gradients less wherever we have good ray coverage, such as underneath North America.

58 GEOSCIENCES↗
DOE OSTI2017

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Avoiding Communication in Primal and Dual Block Coordinate Descent Methods

Primal and dual block coordinate descent methods are iterative methods for solving regularized and unregularized optimization problems. Distributed-memory parallel implementations of these methods have become popular in analyzing large machine learning datasets. However, existing implementations communicate at every iteration, which, on modern data center and supercomputing architectures, often dominates the cost of floating-point computation. Recent results on communication-avoiding Krylov subspace methods suggest that large speedups are possible by re-organizing iterative algorithms to avoid communication. We show how applying similar algorithmic transformations can lead to primal and dual block coordinate descent methods that only communicate every $s$ iterations---where $s$ is a tuning parameter---instead of every iteration for the regularized least-squares problem. We show that the communication-avoiding variants reduce the number of synchronizations by a factor of $s$ on distributed-memory parallel machines without altering the convergence rate and attain strong scaling speedups of up to $6.1\times$ over the “standard algorithm" on a Cray XC30 supercomputer.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. Key to the efficiency of the method is a technique for fast computation of the product of the blob-blob mobility matrix and a vector. For unbounded suspensions, we rely on existing analytical expressions for the Rotne-Prager-Yamakawa tensor combined with a fast multipole method (FMM) to obtain linear scaling in the number of particles. For suspensions sedimented against a single no-slip boundary, we use a direct summation on a graphical processing unit (GPU), which gives quadratic asymptotic scaling with the number of particles. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently developed rigid-body immersed boundary method by B. Kallemov, A. P. S. Bhalla, B. E. Griffith, and A. Donev (Commun. Appl. Math. Comput. Sci. 11 (2016), no. 1, 79-141) to suspensions of freely moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid-body equations converges in a bounded number of iterations regardless of the system size. In our approach, each iteration only requires a few cycles of a geometric multigrid solver for the Poisson equation, and an application of the block-diagonal preconditioner, leading to linear scaling with the number of particles. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗