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DOE OSTI2022

UPC++ v1.0 Programmer’s Guide (Revision 2022.3.0)

UPC++ is a C++ library that supports Partitioned Global Address Space (PGAS) programming. It is designed for writing efficient, scalable parallel programs on distributed-memory parallel computers. The key communication facilities in UPC++ are one-sided Remote Memory Access (RMA) and Remote Procedure Call (RPC). The UPC++ control model is single program, multiple-data (SPMD), with each separate constituent process having access to local memory as it would in C++. The PGAS memory model additionally provides one-sided RMA communication to a global address space, which is allocated in shared segments that are distributed over the processes. UPC++ also features Remote Procedure Call (RPC) communication, making it easy to move computation to operate on data that resides on remote processes. UPC++ was designed to support exascale high-performance computing, and the library interfaces and implementation are focused on maximizing scalability. In UPC++, all communication operations are syntactically explicit, which encourages programmers to consider the costs associated with communication and data movement. Moreover, all communication operations are asynchronous by default, encouraging programmers to seek opportunities for overlapping communication latencies with other useful work. UPC++ provides expressive and composable abstractions designed for efficiently managing aggressive use of asynchrony in programs. Together, these design principles are intended to enable programmers to write applications using UPC++ that perform well even on hundreds of thousands of cores.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

UPC++ v1.0 Programmer’s Guide, Revision 2023.3.0

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

UPC++ v1.0 Programmer’s Guide, Revision 2022.9.0

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Preparing an Incompressible-Flow Fluid Dynamics Code for Exascale-Class Wind Energy Simulations

The U.S. Department of Energy has identified exascale-class wind farm simulation as critical to wind energy scientific discovery. A primary objective of the ExaWind project is to build high-performance, predictive computational fluid dynamics (CFD) tools that satisfy these modeling needs. GPU accelerators will serve as the computational thoroughbreds of next-generation, exascale-class supercomputers. Here, we report on our efforts in preparing the ExaWind unstructured mesh solver, Nalu-Wind, for exascale-class machines. For computing at this scale, a simple port of the incompressible-flow algorithms to GPUs is insufficient. To achieve high performance, one needs novel algorithms that are application aware, memory efficient, and optimized for the latest-generation GPU devices. The result of our efforts are unstructured-mesh simulations of wind turbines that can effectively leverage thousands of GPUs. In particular, we demonstrate a first-of-its-kind, incompressible-flow simulation using Algebraic Multigrid solvers that strong scales to more than 4000 GPUs on the Summit supercomputer.

algebraic multigrid↗

DOE OSTI2024

MAGMA: Enabling exascale performance with accelerated BLAS and LAPACK for diverse GPU architectures

MAGMA (Matrix Algebra for GPU and Multicore Architectures) is a pivotal open-source library in the landscape of GPU-enabled dense and sparse linear algebra computations. With a repertoire of approximately 750 numerical routines across four precisions, MAGMA is deeply ingrained in the DOE software stack, playing a crucial role in high-performance computing. Notable projects such as ExaConstit, HiOP, MARBL, and STRUMPACK, among others, directly harness the capabilities of MAGMA. In addition, the MAGMA development team has been acknowledged multiple times for contributing to the vendors’ numerical software stacks. Looking back over the time of the Exascale Computing Project (ECP), we highlight how MAGMA has adapted to recent changes in modern HPC systems, especially the growing gap between CPU and GPU compute capabilities, as well as the introduction of low precision arithmetic in modern GPUs. We also describe MAGMA’s direct impact on several ECP projects. Maintaining portable performance across NVIDIA and AMD GPUs, and with current efforts toward supporting Intel GPUs, MAGMA ensures its adaptability and relevance in the ever-evolving landscape of GPU architectures.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A Co-design Framework for Online Data Analysis and Reduction

Science applications preparing for the exascale era are increasingly exploring in situ computations comprising of simulation-analysis-reduction pipelines coupled in-memory. Efficient composition and execution of such complex pipelines for a target platform is a codesign process that evaluates the impact and tradeoffs of various application- and system-specific parameters. In this article, we describe a toolset for automating performance studies of composed HPC applications that perform online data reduction and analysis. We describe Cheetah, a new framework for composing parametric studies on coupled applications, and Savanna, a runtime engine for orchestrating and executing campaigns of codesign experiments. Furthermore, this toolset facilitates understanding the impact of various factors such as process placement, synchronicity of algorithms, and storage versus compute requirements for online analysis of large data. Ultimately, we aim to create a catalog of performance results that can help scientists understand tradeoffs when designing next-generation simulations that make use of online processing techniques. We illustrate the design of Cheetah and Savanna, and present application examples that use this framework to conduct codesign studies on small clusters as well as leadership class supercomputers.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Efficient exascale discretizations: High-order finite element methods

Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially assembled high-order finite element methods, since these methods can increase the accuracy and/or lower the computational time due to reduced data motion. In this paper we provide an overview of the research and development activities in the Center for Efficient Exascale Discretizations (CEED), a co-design center in the Exascale Computing Project that is focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware. CEED is a research partnership involving more than 30 computational scientists from two US national labs and five universities, including members of the Nek5000, MFEM, MAGMA and PETSc projects. We discuss the CEED co-design activities based on targeted benchmarks, miniapps and discretization libraries and our work on performance optimizations for large-scale GPU architectures. We also provide a broad overview of research and development activities in areas such as unstructured adaptive mesh refinement algorithms, matrix-free linear solvers, high-order data visualization, and list examples of collaborations with several ECP and external applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

E3SM: Improved Climate Prediction with Exascale Capability

The Energy Exascale Earth System Model (E3SM) project is an ongoing, state-of-the-science earth system modeling, simulation, and prediction effort that optimizes Department of Energy (DOE) computing resources to meet the science needs of the nation and the agency’s mission objectives. Climate simulation has become a proven tool for identifying and quantifying the impacts of climate change, but even greater accuracy is required at all levels to improve forecast precision. Understanding the impact of climate change on global and regional water cycles is one of the highest priorities and most difficult challenges in climate change prediction. As part of a subproject of DOE’s Exascale Computing Project, a multidisciplinary team including geophysical and computational scientists developed a multiscale modeling framework (MMF) to refine cloud representation in E3SM climate simulation on GPU accelerated supercomputers, making higher resolution, more computationally efficient predictions possible.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Holistic Measurement Driven Resilience: Combining Operational Fault and Failure Measurements and Fault Injection for Quantifying Fault Detection, Propagation and Impact. Final report

For HPC systems to date, application resilience to faults and failures has been accomplished by the brute- force method of checkpoint/restart, which allows an application to make forward progress in the face of system and application faults, errors, and failures independent of root cause or end result. It has remained the primary resilience mechanism because we lack a way to identify faults and anticipate consequences early enough to take meaningful mitigating action. However, checkpoint/restart implementations put a tremendous burden on system resources and on the applications themselves and is becoming less feasible at scale. Because we have not yet operated at scales at which checkpoint/restart fails to provide forward progress, despite increasing costs, vendors have had little motivation to provide the instrumentation necessary for early identification of faults and failures. However, as we move from petascale to exascale, component mean time to failure (MTTF) will render the existing techniques ineffectual and/or too expensive. Furthermore, fault recovery mechanisms such as failover and/or error correction introduce performance inconsistency. Instrumentation allowing early indication of problems and tools to enable use of such information by systems, operating systems, and applications offer an alternative, more scalable and less costly solution. In the HMDR project, we built on our experience and expertise developed and accumulated over years of research on design, monitoring, measurement, and assessment of resilient computing systems. Analysis of field data on the current and past generations of extreme-scale systems revealed several challenges that, if not addressed in increasingly larger and more complex systems, may hinder the effectiveness of future exascale computing systems. Specifically, i) file systems and interconnects in current-generation large-scale systems already operate at the margins of resiliency, including consistent performance, and may not scale to larger deployments; ii) automated, software-based failover mechanisms are frequently inadequate and can introduce wider failures, such that failures during recovery may lead to system/application failures, including system-wide outages; and iii) silent data corruption represents a critical fault mode and will require efficient detection mechanisms if next-generation applications are to take full advantage of exascale hardware. To address the above challenges, we assembled a team of world-renowned experts in resilient extreme- scale computing from the University of Illinois (Electrical and Computer Engineering, Computer Science, and NCSA), SNL, LANL, NERSC, and Cray. Our team includes representatives from centers that house many of the largest HPC resources in the world, both today and over the coming years. The team has a unique track record of research in i) system and application failure characterization based on the analysis of field data, ii) data-driven design of fault/error detection mechanisms, and iii) experimental characterization of system/application resiliency. The team includes system owners/operators who provide continuous data collection and access and ensure installation of appropriate analysis tools.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Preparing MPICH for exascale

The advent of exascale supercomputers heralds a new era of scientific discovery, yet it introduces significant architectural challenges that must be overcome for MPI applications to fully exploit its potential. Among these challenges is the adoption of heterogeneous architectures, particularly the integration of GPUs to accelerate computation. Additionally, the complexity of multithreaded programming models has also become a critical factor in achieving performance at scale. The efficient utilization of hardware acceleration for communication, provided by modern NICs, is also essential for achieving low latency and high throughput communication in such complex systems. In response to these challenges, the MPICH library, a high-performance and widely used Message Passing Interface (MPI) implementation, has undergone significant enhancements. Here, this paper presents four major contributions that prepare MPICH for the exascale transition. First, we describe a lightweight communication stack that leverages the advanced features of modern NICs to maximize hardware acceleration. Second, our work showcases a highly scalable multithreaded communication model that addresses the complexities of concurrent environments. Third, we introduce GPU-aware communication capabilities that optimize data movement in GPU-integrated systems. Finally, we present a new datatype engine aimed at accelerating the use of MPI derived datatypes on GPUs. These improvements in the MPICH library not only address the immediate needs of exascale computing architectures but also set a foundation for exploiting future innovations in high-performance computing. By embracing these new designs and approaches, MPICH-derived libraries from HPE Cray and Intel were able to achieve real exascale performance on OLCF Frontier and ALCF Aurora respectively.

Guo, Yanfei [Argonne National Laboratory (ANL), Ar↗

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 OSTI2024

HydraGNN_Predictive_GFM_2024 - Ensemble of predictive graph foundation models for ground state atomistic materials modeling

We provide the ensemble of fifteen pre-trained graph foundation models (GFMs) for atomistic materials modeling applications. Each one of the fifteen GFMs has been trained on five open-source datasets that (once aggregated) amount to over 154 million atomistic structures, which cover over two-thirds of the natural elements of the periodic table and that comprises a broad set of organic and inorganic compounds. This vast set of atomistic structures comprises ground state configurations that are dynamically stable (i.e., equilibrated structures with atomic forces approximately close to zero values) as well as dynamically unstable structures (i.e., non-equilibrium structures with non-negligible non-zero values of atomic forces). The ensemble of datasets aggregated does NOT include excited states. The datasets have been curated to remove atomistic structures with spectral norm of the force tensor above 100 eV/angstrom. Moreover, a linear term of the energy was computed for each dataset using a linear regression model that uses the chemical concentration of each natural element as regressor. The linear term predicted by the linear regression model has been subtracted from each original energy value to perform a re-alignment of the energy values across different electronic structures approximation theories performed to generate the diverse multi-source, multi-fidelity datasets. The folder "ADIOS_files" contains the set of pre-processed datasets in Adaptable I/O System (ADIOS) format (https://www.exascaleproject.org/research-project/adios/) that have been used for the development and training of GFMs in this work. The "ADIOS_files" directory contains 6 sub-directories named as follows: - ANI1x-v3.bp - MPTrj-v3.bp - OC2020-20M-v3.bp - OC2020-v3.bp - OC2022-v3.bp - qm7x-v3.bp Each sub-directory contains the pre-processed datasets converted in Adaptable I/O System (ADIOS) format (https://www.exascaleproject.org/research-project/adios/) that have been used to the development, training, and performance testing of the ensemble go predictive graph foundation models. Each GFM was developed using HydraGNN (https://github.com/ORNL/HydraGNN) as underlying graph neural network (GNN) architecture. The multi-task learning (MTL) capability of HydraGNN was used to simultaneously train the GFMs on labeled values for direct predictions of energy (a total system property of an atomistic structure that measures the chemical stability) and atomic forces (an atomic level property of an atomistic structure that measures the dynamical stability). The hyper parameters of the GFM have been tuned using scalable hyperparameter optimization (HPO) algorithms implemented in the software DeepHyper (https://github.com/deephyper/deephyper). The pre-training of each HPO trial was performed using distributed data parallelism (DDP) to scale the training across 128 compute nodes of the exascale OLCF supercomputer Frontier. Each HPO trial was trained only for 10 epochs and an early stopping was performed to avoid wasting significant computational resources on GNN architectures that were clearly underperforming. For each HPO trial, the 'omnistat' tool developed by (AMD Research - Advanced Micro Device) was used to measure the total energy consumption in kWh. The ensemble of GFMs was obtained by selecting the fifteen best performing HPO trials. Four models have been selected for their clear advantage in accuracy, and these are the GFMs with IDs 229, 156, 147, 260. Additional eleven models have been selected based on judicious balance between accuracy and energy consumption needed for training, and these are the GFMs with IDs 165, 78, 137, 1, 175, 171, 181, 67, 179, 167, 351. Each selected GFM of the ensemble was continued to cumulate a total of at most 30 epochs. In some cases, the total number of epochs actually performed was les than 30 due to two combined factors: (1) the size of the GFM (i.e., the number of model parameters to train) and (2) the total wall-clock time for which the computational resources could be allocated on OLCF-Frontier. The "Ensemble_of_models" directory contains 15 sub-directories named as follows: - gfm_0.229 - gfm_0.156 - gfm_0.147 - gfm_0.260 - gfm_0.165 - gfm_0.78 - gfm_0.137 - gfm_0.1 - gfm_0.175 - gfm_0.171 - gfm_0.181 - gfm_0.67 - gfm_0.179 - gfm_0.167 - gfm_0.351 Each one of these sub-directories refers to one of the fifteen HPO trials that have been selected to continue the pre-training with at most 30 epochs. With each sub-directory associated with a specific HPO trial, the following files can be found: - config.json: file for argument parsing to develop and train an HydraGNN architecture - gfm_0.ID_epoch_N.pk: file with model parameters for HPO ID trial after N epochs of training The ensemble of fifteen GFM architectures was used for (1) ensemble averaging to stabilize the predictions of energy and atomic forces after pre-training for post-processing analysis and (2) ensemble uncertainty quantification (UQ). The code used to develop, pre-train, and load the pre-trained models for post-processing analysis is available on the ORNL-GitHub at the following link: https://github.com/ORNL/HydraGNN/tree/Predictive_GFM_2024

36 MATERIALS SCIENCE↗

DOE OSTI2024

Towards exascale for wind energy simulations

We examine large-eddy-simulation modeling approaches and computational performance of two open-source computational fluid dynamics codes for the simulation of atmospheric boundary layer flows that are of direct relevance to wind energy production. The first code, NekRS, is a high-order, unstructured-grid, spectral element code. The second code, AMR-Wind, is a second-order, block-structured, finite-volume code with adaptive mesh refinement capabilities. The objective of this study is to co-develop these codes in order to improve model fidelity and performance for each. These features will be critical for running ABL-based applications such as wind farm analysis on advanced computing architectures. To this end, we investigate the performance of NekRS and AMR-Wind on the Oak Ridge Leadership Facility supercomputers Summit, using 4 to 800 nodes (24 to 4,800 NVIDIA V100 GPUs), and Crusher, the testbed for the Frontier exascale system, using 18 to 384 Graphics Compute Dies on AMD MI250X GPUs. We compare strong- and weak-scaling capabilities, linear solver performance, and time to solution. We also identify leading inhibitors to parallel scaling.

17 WIND ENERGY↗

DOE OSTI2022

Pele: An Exascale-Ready Suite of Combustion Codes

High fidelity simulations of realistic combustion devices are extremely demanding computationally because of the requirements to capture complex fuel chemical decomposition, its intricate interactions with turbulent, often multiphase, flows, and the wide separation of space and time scales between the thin flame and the device boundaries. Software required to carry out such computations tends to be extremely complex, particularly when designed to exploit hardware accelerators, and can be difficult to port and maintain. We present Pele, a performance portable suite of tools for the simulation of combustion systems, including codes to evolve reactive multiphase configurations in the low Mach number and compressible flow regimes, along with a set of inter-compatible post processing and in situ analysis tools. The Pele suite of tools is built on top of the AMReX framework for block-structured adaptive mesh refinement, which provides efficient data structures and algorithms that enable the development of a wide variety of efficient mesh and particle based PDE integration schemes. A hierarchical MPI+X parallelism scheme supports CPU-only and accelerated architectures, where X can be OpenMP, CUDA, and HIP based approaches for intra-node computational work distribution. The algorithms and data structures underlying the Pele simulation and analysis tools are highly scalable and performant across a wide variety of high-performance computing platforms, including DOEs newest exascale-class machines, Frontier and Aurora. The simulation and analysis tools are fully documented and freely distributed as open source via GitHub. We present key algorithmic and software challenges, solution strategies, performance and resulting set of capabilities.

AMReX↗

DOE OSTI2021

Measurement and analysis of GPU-accelerated applications with HPCToolkit

To address the challenge of performance analysis on the US DOE’s forthcoming exascale supercomputers, Rice University has been extending its HPCToolkit performance tools to support measurement and analysis of GPU-accelerated applications. To help developers understand the performance of accelerated applications as a whole, HPCToolkit’s measurement and analysis tools attribute metrics to calling contexts that span both CPUs and GPUs. To measure GPU-accelerated applications efficiently, HPCToolkit employs a novel wait-free data structure to coordinate monitoring and attribution of GPU performance. To help developers understand the performance of complex GPU code generated from high-level programming models, HPCToolkit constructs sophisticated approximations of call path profiles for GPU computations. To support fine-grained analysis and tuning, HPCToolkit uses PC sampling and instrumentation to measure and attribute GPU performance metrics to source lines, loops, and inlined code. To supplement fine-grained measurements, HPCToolkit can measure GPU kernel executions using hardware performance counters. To provide a view of how an execution evolves over time, HPCToolkit can collect, analyze, and visualize call path traces within and across nodes. Finally, on NVIDIA GPUs, HPCToolkit can derive and attribute a collection of useful performance metrics based on measurements using GPU PC samples. Here, we illustrate HPCToolkit’s new capabilities for analyzing GPU-accelerated applications with several codes developed as part of the Exascale Computing Project.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Tensor Network Quantum Virtual Machine for Simulating Quantum Circuits at Exascale

The numerical simulation of quantum circuits is an indispensable tool for development, verification, and validation of hybrid quantum-classical algorithms intended for near-term quantum co-processors. The emergence of exascale high-performance computing (HPC) platforms presents new opportunities for pushing the boundaries of quantum circuit simulation. Here, we present a modernized version of the Tensor Network Quantum Virtual Machine (TNQVM) that serves as the quantum circuit simulation backend in the eXtreme-scale ACCelerator (XACC) framework. The new version is based on the scalable tensor network processing library ExaTN (Exascale Tensor Networks). It provides multiple configurable quantum circuit simulators that perform either an exact quantum circuit simulation via the full tensor network contraction or an approximate simulation via a suitably chosen tensor factorization scheme. Upon necessity, stochastic noise modeling from real quantum processors is incorporated into the simulations by modeling quantum channels with Kraus tensors. By combining the portable XACC quantum programming frontend and the scalable ExaTN numerical processing backend, we introduce an end-to-end virtual quantum development environment that can scale from laptops to future exascale platforms. We report initial benchmarks of our framework, which include a demonstration of the distributed execution, incorporation of quantum decoherence models, and simulation of the random quantum circuits used for the certification of quantum supremacy on Google’s Sycamore superconducting architecture.

Nguyen, Thien↗

DOE OSTI2022

Parthenon—a performance portable block-structured adaptive mesh refinement framework

On the path to exascale the landscape of computer device architectures and corresponding programming models has become much more diverse. While various low-level performance portable programming models are available, support at the application level lacks behind. To address this issue, we present the performance portable block-structured adaptive mesh refinement (AMR) framework Parthenon, derived from the well-tested and widely used Athena++ astrophysical magnetohydrodynamics code, but generalized to serve as the foundation for a variety of downstream multi-physics codes. Parthenon adopts the Kokkos programming model, and provides various levels of abstractions from multidimensional variables, to packages defining and separating components, to launching of parallel compute kernels. Parthenon allocates all data in device memory to reduce data movement, supports the logical packing of variables and mesh blocks to reduce kernel launch overhead, and employs one-sided, asynchronous MPI calls to reduce communication overhead in multi-node simulations. Using a hydrodynamics miniapp, we demonstrate weak and strong scaling on various architectures including AMD and NVIDIA GPUs, Intel and AMD x86 CPUs, IBM Power9 CPUs, as well as Fujitsu A64FX CPUs. At the largest scale on Frontier (the first TOP500 exascale machine), the miniapp reaches a total of 1.7 × 10 13 zone-cycles/s on 9216 nodes (73,728 logical GPUs) at [Formula: see text] weak scaling parallel efficiency (starting from a single node). In combination with being an open, collaborative project, this makes Parthenon an ideal framework to target exascale simulations in which the downstream developers can focus on their specific application rather than on the complexity of handling massively-parallel, device-accelerated AMR.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

R-Adaptivity to Enable Compression of Elementary Computations in Extreme-Scale Finite Element Simulators

Modern computing systems are capable of exascale calculations, which are revolutionizing the development and application of high-fidelity numerical models in computational science and engineering. While these systems continue to grow in processing power, the available system memory has not increased commensurately, and electrical power consumption continues to grow. A predominant approach to limit the memory usage in large-scale applications is to exploit the abundant processing power and continually recompute many low-level simulation quantities, rather than storing them. However, this approach can adversely impact the throughput of the simulation and diminish the benefits of modern computing architectures. We present three novel contributions to reduce the memory burden while maintaining, and sometimes improving, performance in simulations based on finite element discretizations. The first contribution develops dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh. While these schemes are shown to reduce memory requirements by more than 99% on meshes with large numbers of identical mesh cells, there are applications where this structure does not exist. The second contribution leverages a recently developed augmented Lagrangian optimization algorithm to enable r-adaptivity for meshes with the goal of enhancing the redundancies in the mesh. The third contribution extends these methods to patch-based linear solvers and preconditioners by compressing local matrices. Numerical results demonstrate the effectiveness of the proposed methods to detect, enhance and exploit mesh structure on a suite of examples inspired by large-scale applications.

97 MATHEMATICS AND COMPUTING↗