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All sources NASA NTRS DOE OSTI

DOE OSTI2021

Multifidelity computing for coupling full and reduced order models

Hybrid physics-machine learning models are increasingly being used in simulations of transport processes. Many complex multiphysics systems relevant to scientific and engineering applications include multiple spatiotemporal scales and comprise a multifidelity problem sharing an interface between various formulations or heterogeneous computational entities. To this end, we present a robust hybrid analysis and modeling approach combining a physics-based full order model (FOM) and a data-driven reduced order model (ROM) to form the building blocks of an integrated approach among mixed fidelity descriptions toward predictive digital twin technologies. At the interface, we introduce a long short-term memory network to bridge these high and low-fidelity models in various forms of interfacial error correction or prolongation. The proposed interface learning approaches are tested as a new way to address ROM-FOM coupling problems solving nonlinear advection-diffusion flow situations with a bifidelity setup that captures the essence of a broad class of transport processes.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2020

Interface learning of multiphysics and multiscale systems

Complex natural or engineered systems comprise multiple characteristic scales, multiple spatiotemporal domains, and even multiple physical closure laws. To address such challenges, we introduce an interface learning paradigm and put forth a data-driven closure approach based on memory embedding to provide physically correct boundary conditions at the interface. To enable the interface learning for hyperbolic systems by considering the domain of influence and wave structures into account, we put forth the concept of upwind learning toward a physics-informed domain decomposition. The promise of the proposed approach is shown for a set of canonical illustrative problems. Here, we highlight that high-performance computing environments can benefit from this methodology to reduce communication costs among processing units in emerging machine-learning-ready heterogeneous platforms toward exascale era.

42 ENGINEERING↗

DOE OSTI2021

A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis

In this work, a butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the O(N log 2 N) construction cost of the HOD-BF-compressed system matrix and O(N log 1.5 N) inversion cost for the preconditioner, where N is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a H matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.

42 ENGINEERING↗

DOE OSTI2024

P38 heterogeneous multi-tiled system with support for message queues (MoSAIC) v0.1

The proposed system is written in the hardware description language (HDL) verilog targeting an FPGA board. It is intended as a testbed to explore architecture tradeoffs in multi-tiled heterogeneous architectures. Although we target FPGAs, the system can be implemented as a monolithic SoC or a package comprised of many chiplets that are interconnected in the same package using a NoC. The proposed NoC is lightweight and follows an axi-lite interface. The endpoints of the NoC are a heterogeneous mix of "tiles" as endpoints that are general purpose processors, fixed function accelerators, and programmable accelerators. We assume that the network interfaces for the NoC endpoints are all addressable in a global name-space in that they represent an address range (for memory addresses) or a range of unique identifiers that are associated with each individual tile. This makes the functionality abstract from the standpoint of the NoC design details. Message queues offer a direct inter-processor interface between peer general purpose cores and diverse accelerators that comprise an SoC. Although they share the same NoC infrastructure for inter-tile communication within an SoC or SiP, the hardware message queues bypass the memory hierarchy and thus do not pollute the memory state or invoke the cache coherence mechanism.

Gonzalez, LouisaPatricia↗

DOE OSTI2025

CRCNS22 Learning Rules in the Hippocampus and their Mapping to Neuromorphic Systems (Final Technical Report)

Large scale biologically-realistic computational models are key to investigating the interplay between structure and function in nervous systems, thus paving the way to new clinical methods and neuro-inspired computing solutions. This project focuses on the hippocampus, in particular the CA3-CA1 regions, due to their role in associative learning and memory, pattern separation and completion, and spatial navigation. Investigations into the neuronal organization and learning rule(s) of this circuit can shed light into how declarative memories are formed, stored, recalled and forgotten and inform computational, experimental and clinical neuroscience work. Our project aims at developing a novel data-driven methodology supported by a broad heterogeneous base of neuroscience experimental knowledge and inspired from advances in computer science and engineering. Specifically, this work will benchmark existing and new learning rules within a full-scale spiking neural network simulation of the CA3-CA1 region. The model will be based on an open-source repository, called the Hippocampome, which contains neuronal morphologies, firing patterns, synapse probabilities, and most other required parameters for all known neuron types in the rodent hippocampal formation. The model will be first trained in a supervised fashion for associative memory tasks using backpropagation through time traditionally used in computer science, enhanced with a new technique called the surrogate gradient method. This optimization method will be used to obtain a global loss minimization, but it is not biologically inspired as it assumes the use of data not locally available to the synapses. However, we propose its use as a benchmarking tool, to compare the training performance of local biologically plausible and hardware-mappable learning rules at scale. New rules or combinations will be proposed and tested as needed, based on the obtained results. Progress in this area will also drive the development of novel hardware-mappable algorithms for continual lifelong learning and categorization of new events from few presented examples. This project goes beyond the existing state-of-the-art by looking at large scale realistic neuronal circuits as networks trainable via global optimization methods such as surrogate gradient descent. The objective function of the brain that supports learning is largely unknown, but it is likely that it operates through local learning rules. Studying network trajectories around local minima as proposed in this work represents a useful strategy for understanding whether a network is training by using a specific (set of) learning rule(s). Starting from a completely untrained network is a challenging test since it is difficult to determine how the learning rule affects the trajectory of the network. This interdisciplinary project will help understand what rule governs learning in these regions or if multiple learning rules are involved. The work will develop a robust methodology to measure if the network is converging to the target solution, oscillating around it, or diverging away.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Studying performance portability of LAMMPS across diverse GPU-based platforms

The molecular dynamics simulation software, LAMMPS, utilizes the Kokkos acceleration library to port computation to a diverse set of architectures including those based on GPU accelerators. In addition to Kokkos, LAMMPS contains a vast code base that leverages the CUDA application programming interface using library functions such as cuFFT, CUDA's fast-fourier transform (FFT) library, and, more recently, also support for AMD's Heterogeneous Interface for Portability (HIP) that is rapidly growing. While preparing LAMMPS tests for the AMD GPU-based test system precursors to Frontier, we investigated several strategies for accelerating LAMMPS on AMD GPUs, using the AMD Instinct MI100 and MI250X. In this work, we integrated the HIP FFT library, hipFFT, into the particle-particle particle-mesh (PPPM) long-range solver, which allowed the porting of PPPM calculations to the GPUs. Kokkos behavior on the MI100 and MI250X was also investigated through the package kokkos command of LAMMPS, targeting communication, memory usage, and particle grid decomposition. The Tersoff, Reax, Lennard-Jones (LJ), EAM, Granular, and PPPM potentials were investigated in this effort, and results from these experiments are provided. In conclusion, the selected potentials were run on Spock (AMD Instinct MI100), Crusher (AMD Instinct MI250X), AFW HPC11 (NVIDIA A100) and Summit (NVIDIA V100), for comparison. Operational roofline models were constructed and analyzed for the Tersoff, Reax, and Lennard–Jones potentials on Crusher and Summit.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Studying Performance Portability of LAMMPS across Diverse GPU-based Platforms

The molecular dynamics simulation software, LAMMPS, utilizes the Kokkos acceleration library to port computation to a diverse set of architectures including those based on GPU accelerators. In addition to Kokkos, LAMMPS contains a vast code base that leverages the CUDA application programming interface using library functions such as cuFFT, CUDA’s fast-fourier transform (FFT) library, and, more recently, also support for AMD’s Heterogeneous Interface for Portability (HIP) that is rapidly growing. While preparing LAMMPS tests for the AMD GPU-based test system precursors to Frontier, we investigated several strategies for accelerating LAMMPS on AMD GPUs, using the AMD Instinct MI100 and MI250X. In this work, we integrated the HIP FFT library, hipFFT, into the particle-particle particle-mesh (PPPM) long-range solver, which allowed the porting of PPPM calculations to the GPUs. Kokkos behavior on the MI100 and MI250X was also investigated through the package kokkos command of LAMMPS, targeting com- munication, memory usage, and particle grid decomposition. The Tersoff, Reax, Lennard-Jones (LJ), EAM, Granular, and PPPM potentials were investigated in this effort, and results from these experiments are provided. The selected potentials were run on Spock (AMD Instinct MI100), Crusher (AMD Instinct MI250X), AFW HPC11 (NVIDIA A100) and Summit (NVIDIA V100), for comparison. Operational roofline models were constructed and analyzed for the Tersoff, Reax, and Lennard-Jones potentials on Crusher and Summit.

Hagerty, Nick↗

DOE OSTI2022

Integrating Deep Learning and Hydrodynamic Modeling to Improve the Great Lakes Forecast

The Laurentian Great Lakes, one of the world’s largest surface freshwater systems, pose a modeling challenge in seasonal forecast and climate projection. While physics-based hydrodynamic modeling is a fundamental approach, improving the forecast accuracy remains critical. In recent years, machine learning (ML) has quickly emerged in geoscience applications, but its application to the Great Lakes hydrodynamic prediction is still in its early stages. This work is the first one to explore a deep learning approach to predicting spatiotemporal distributions of the lake surface temperature (LST) in the Great Lakes. Our study shows that the Long Short-Term Memory (LSTM) neural network, trained with the limited data from hypothetical monitoring networks, can provide consistent and robust performance. The LSTM prediction captured the LST spatiotemporal variabilities across the five Great Lakes well, suggesting an effective and efficient way for monitoring network design in assisting the ML-based forecast. Furthermore, we employed an explainable artificial intelligence (XAI) technique named SHapley Additive exPlanations (SHAP) to uncover how the features impact the LSTM prediction. Our XAI analysis shows air temperature is the most influential feature for predicting LST in the trained LSTM. The relatively large bias in the LSTM prediction during the spring and fall was associated with substantial heterogeneity of air temperature during the two seasons. In contrast, the physics-based hydrodynamic model performed better in spring and fall yet exhibited relatively large biases during the summer stratification period. Finally, we developed a statistical integration of the hydrodynamic modeling and deep learning results based on the Best Linear Unbiased Estimator (BLUE). The integration further enhanced prediction accuracy, suggesting its potential for next-generation Great Lakes forecast systems.

Xue, Pengfei (ORCID:000000025702421X)↗

DOE OSTI2021

JTAG-based PLC memory acquisition framework for industrial control systems

In industrial control systems (ICS), programmable logic controllers (PLC) are the embedded devices that directly control and monitor critical industrial infrastructure processes such as nuclear plants and power grid stations. Cyberattacks often target PLCs to sabotage a physical process. A memory forensic analysis of a suspect PLC can answer questions about an attack, including compromised firmware and manipulation of PLC control logic code and I/O devices. Given physical access to a PLC, collecting forensic information from the PLC memory at the hardware-level is risky and challenging. It may cause the PLC to crash or hang since PLCs have proprietary, legacy hardware with heterogeneous architecture. This paper addresses this research problem and proposes a novel JTAG (Joint Test Action Group)-based framework, Kyros, for reliable PLC memory acquisition. Kyros systematically creates a JTAG profile of a PLC through hardware assessment, JTAG pins identification, memory map creation, and optimizing acquisition parameters. It also facilitates the community of interest (such as ICS owners, operators, and vendors) to develop the JTAG profiles of PLCs. Further, we present a case study of Kyros implementation over Allen-Bradley 1756-A10/B to help understand the framework's application on a real-world PLC used in industry settings. The sample PLC memory dumps are shared with the research community to facilitate further research.

Rais, Muhammad Haris↗

DOE OSTI2021

Early warm-season mesoscale convective systems dominate soil moisture–precipitation feedback for summer rainfall in central United States

Land–atmosphere interactions play an important role in summer rainfall in the central United States, where mesoscale convective systems (MCSs) contribute to 30 to 70% of warm-season precipitation. Previous studies of soil moisture–precipitation feedbacks focused on the total precipitation, confounding the distinct roles of rainfall from different convective storm types. Here, we investigate the soil moisture–precipitation feedbacks associated with MCS and non-MCS rainfall and their surface hydrological footprints using a unique combination of these rainfall events in observations and land surface simulations with numerical tracers to quantify soil moisture sourced from MCS and non-MCS rainfall. Furthermore, we find that early warm-season (April to June) MCS rainfall, which is characterized by higher intensity and larger area per storm, produces coherent mesoscale spatial heterogeneity in soil moisture that is important for initiating summer (July) afternoon rainfall dominated by non-MCS events. On the other hand, soil moisture sourced from both early warm-season MCS and non-MCS rainfall contributes to lower-level atmospheric moistening favorable for upscale growth of MCSs at night. However, soil moisture sourced from MCS rainfall contributes to July MCS rainfall with a longer lead time because with higher intensity, MCS rainfall percolates into deeper soil that has a longer memory. Therefore, early warm-season MCS rainfall dominates soil moisture–precipitation feedback. This motivates future studies to examine the contribution of early warm-season MCS rainfall and associated soil moisture anomalies to predictability of summer rainfall in the major agricultural region of the central United States and other continental regions frequented by MCSs.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

End-to-end online performance data capture and analysis for scientific workflows

With the increased prevalence of employing workflows for scientific computing and a push towards exascale computing, it has become paramount that we are able to analyze characteristics of scientific applications to better understand their impact on the underlying infrastructure and vice-versa. Such analysis can help drive the design, development, and optimization of these next generation systems and solutions. Here, we present the architecture, integrated with existing well-established and newly developed tools, to collect online performance statistics of workflow executions from various, heterogeneous sources and publish them in a distributed database (Elasticsearch). Using this architecture, we are able to correlate online workflow performance data, with data from the underlying infrastructure, and present them in a useful and intuitive way via an online dashboard. We have validated our approach by executing two classes of real-world workflows, both under normal and anomalous conditions. The first is an I/O-intensive genome analysis workflow; the second, a CPU- and memory-intensive material science workflow. Based on the data collected in Elasticsearch, we are able to demonstrate that we can correctly identify anomalies that we injected. The resulting end-to-end data collection of workflow performance data is an important resource of training data for automated machine learning analysis.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of 𝑂⁡(𝑁 2/3 polylog 𝑁 ⋅log (1/𝜖)), outperforming the best classical methods (with run times of 𝑂⁡(𝑁⁢log 𝑁 ⋅log (1/𝜖))) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.

58 GEOSCIENCES↗

DOE OSTI2026

Thin-film lithium niobate on diamond as a platform for efficient spin-phonon coupling

Negatively charged silicon vacancy (SiV − ) centers in diamond are leading candidates for solid-state quantum memory that can be controlled using electromagnetic or acoustic waves. The latter are particularly promising due to the strong strain response of SiV − , enabling large spin-phonon interaction strengths. Indeed, coherent spin control via surface acoustic waves (SAWs) has been demonstrated and is essential for developing on-chip phononic quantum networks. However, the absence of piezoelectricity in diamond requires an interface with a piezoelectric material for efficient transduction and delivery of acoustic waves to spins in diamonds. Here, we demonstrate a heterogeneously integrated phononic platform that combines thin-film lithium niobate with diamond to enable acoustic control of single SiV − spins. Additionally, leveraging large SAW-induced strain at the location of SiV − , we achieve coherent acoustic control of an electron spin with more than twofold improvement in Rabi frequency compared to state-of-the-art devices based on aluminum nitride on diamond. Furthermore, this work represents a crucial step towards realizing phonon-based quantum information processing systems.

Optically detected magnetic resonance↗

DOE OSTI2023

Session Introduction: Graph Representations and Algorithms in Biomedicine

Connectivity is a fundamental property of biological systems: on the cellular level, proteins interact with each other to form protein-protein interaction networks (PPIs); on the organism level, neurons are arranged in a network; and on a community-level, species can have complex relationships with one another that drive the development and balance of an ecosystem. Graphs, representations of systems consisting of entities as vertices and their connections as edges, are a useful structure to characterize many such systems. Such models can be used to understand biological systems that naturally have a network structure, including PPIs, biological neurons, and ecosystems. In today’s information age, graph representations and algorithms (often in combination with machine learning techniques) are used to organize massive amounts of related data, much of which may be heterogeneous or unstructured, and identify patterns that represent novel biological insights. PSB’s 2023 session “Graph Representations and algorithms in Biomedicine,” encompasses modern developments in graph theory and its applications to various fields of biomedicine. This session includes a wide range of research - knowledge graphs built from text-mined health data, heterogeneous networks using multi-omic databases, and graphs refined to represent uncertainty or improve memory usage.

Chrisman, Brianna S.↗

DOE OSTI2026

FitCache: A Transparent Drop-In Framework for Multi-Tier Caching to Accelerate Distributed Deep Learning Workloads

Training in Deep learning (DL) remains highly compute- and data-intensive, with I/O becoming a critical bottleneck as models and datasets scale. Recent studies report that data loading can dominate training time, especially on large-scale HPC systems with shared parallel file systems (PFS). Existing caching approaches either rely on single-tier designs or require intrusive modifications to training pipelines, limiting their portability and effectiveness. In this work, we present FitCache, a transparent drop-in framework for multi-tier caching to accelerate distributed DL training by coordinating fast local memory (e.g., DRAM, Persistent Memory (PMem)) and NVMe as hierarchical caches atop PFS. Our design adapts to hardware diversity, i.e., if NVMe is missing, memory transparently acts as a caching tier, ensuring stable performance. FitCache transparently intercepts I/O requests and issues concurrent fetches across all tiers, returning data from the fastest responder without centralized metadata or static redirection paths. FitCache adapts to dynamic workloads and heterogeneous clusters while maintaining POSIX compatibility. Experiments on Frontier (2048 GPUs) and smaller research clusters show that FitCache reduces training time by up to 40% and per-batch I/O latency by up to 71.6% compared to Lustre Orion PFS, offering a drop-in solution for scalable DL training.

Hu, Guangxing [ORNL] (ORCID:0009000283203614)↗

NASA NTRS2017

On the Development and Application of High Data Rate Architecture (HiDRA) in Future Space Networks

Historically, space missions have been severely constrained by their ability to downlink the data they have collected. These constraints are a result of relatively low link rates on the spacecraft as well as limitations on the time during which data can be sent. As part of a coherent strategy to address existing limitations and get more data to the ground more quickly, the Space Communications and Navigation (SCaN) program has been developing an architecture for a future solar system Internet. The High Data Rate Architecture (HiDRA) project is designed to fit into such a future SCaN network. HiDRA's goal is to describe a general packet-based networking capability which can be used to provide assets with efficient networking capabilities while simultaneously reducing the capital costs and operational costs of developing and flying future space systems.Along these lines, this paper begins by reviewing various characteristics of modern satellite design as well as relevant characteristics of emerging technologies (such as free-space optical links capable of working at 100+ Gbps). Next, the paper describes HiDRA's design, and how the system is able to both integrate and support the operation of not only today's high-rate systems, but also the high-rate systems likely to be found in the future. This section also explores both existing and future networking technologies, such as Delay Tolerant Networking (DTN) protocol (RFC4838 citeRFC:1, RFC5050citeRFC:2), and explains how HiDRA supports them. Additionally, this section explores how HiDRA is used for scheduling data movement through both proactive and reactive link management. After this, the paper moves on to explore a reference implementation of HiDRA. This implementation is currently being realized based on a Field Programmable Gate Array (FPGA) memory and interface controller that is itself controlled by a local computer running DTN software. Next, this paper explores HiDRA's natural evolution, which includes an integration path for software-defined networking (SDN) switches. This section also describes considerations for both near-Earth and deep-space instantiations of HiDRA, describing how differences in latencies between the environments will necessarily influence how the system is configured and the networks operate. Finally, this paper describes future work. This section includes a description of a potential ISS implementation which will allow rapid advancement through the technology readiness levels (TRL). This section also explores work being done to support HiDRA's successful implementation and operation in a heterogeneous network: such a network could include communications equipment spanning many vintages and capabilities, and one significant aspect of HiDRA's future development involves balancing compatibility with capability.

DTN↗

DOE OSTI2021

A Deep Learning Modeling Framework to Capture Mixing Patterns in Reactive-Transport Systems

Prediction and control of chemical mixing are vital for many scientific areas such as subsurface reactive transport, climate modeling, combustion, epidemiology, and pharmacology. Due to the complex nature of mixing in heterogeneous and anisotropic media, the mathematical models related to this phenomenon are not analytically tractable. Numerical simulations often provide a viable route to predict chemical mixing accurately. However, contemporary modeling approaches for mixing cannot utilize available spatial-temporal data to improve the accuracy of the future prediction and can be compute-intensive, especially when the spatial domain is large and for long-term temporal predictions. To address this knowledge gap, in this work we will present in this paper a deep learning (DL) modeling framework applied to predict the progress of chemical mixing under fast bimolecular reactions. This framework uses convolutional neural networks (CNN) for capturing spatial patterns and long short-term memory (LSTM) networks for forecasting temporal variations in mixing. By careful design of the framework—placement of non-negative constraint on the weights of the CNN and the selection of activation function, the framework ensures non-negativity of the chemical species at all spatial points and for all times. Our DL-based framework is fast, accurate, and requires minimal data for training. The time needed to obtain a forecast using the model is a fraction (≈ O(-6)) of the time needed to obtain the result using a high-fidelity simulation. To achieve an error of 10% (measured using the infinity norm) for capturing local-scale mixing features such as interfacial mixing, only 24% to 32% of the sequence data for model training is required. To achieve the same level of accuracy for capturing global-scale mixing features, the sequence data required for model training is 64% to 70% of the total spatial-temporal data. Hence, the proposed approach—a fast and accurate way to forecast long-time spatial-temporal mixing patterns in heterogeneous and anisotropic media—will be a valuable tool for modeling reactive-transport in a wide range of applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

A Comparison of Linear Solvers for Resolving Flow in Three-Dimensional Discrete Fracture Networks

We compare various methods for resolving steady flow within three-dimensional discrete fracture networks, including direct methods, Krylov subspace methods with and without preconditioning, and multi-grid methods. We compared the performance of the methods based on compute times and scaling of the solution as a function of the number of grid nodes and log-variance of the hydraulic aperture. The methods are applied to three test cases: (a) variable density of networks with a truncated power-law distribution of fracture lengths, (b) a fixed network composed of monodisperse fracture sizes but varied permeability/aperture heterogeneity, (c) and a network based on field site in Nevada, US. We chose these cases to allow us to study the impact of the mesh size and flow properties, as well as to demonstrate our conclusions on a large-scale, realistic problem (more than 40 million mesh nodes). A direct solution using Cholesky factorization outperformed other methods for every example but was closely followed in performance by some algebraic multigrid (AMG) preconditioned Krylov subspace methods. Among the Krylov methods, conjugate gradients (CG) with an AMG preconditioner performs the best. Generally, Cholesky factorization is recommended, but CG with an AMG preconditioner may be suitable for very large problems beyond 40 million nodes where the entire linear system cannot reside in memory.

58 GEOSCIENCES↗