Multigrid Solver Research Articles

New Features in the pyCSEP Toolkit for Earthquake Forecast Development and Evaluation

Abstract The Collaboratory for the Study of Earthquake Predictability (CSEP) is a global community dedicated to advancing earthquake predictability research by rigorously testing probabilistic earthquake forecast models and prediction algorithms. At the heart of this mission is the recent introduction of pyCSEP, an open-source software tool designed to evaluate earthquake forecasts. pyCSEP integrates modules to access earthquake catalogs, visualize forecast models, and perform statistical tests. Contributions from the CSEP community have reinforced the role of pyCSEP in offering a comprehensive suite of tools to test earthquake forecast models. This article builds on Savran, Bayona, et al. (2022), in which pyCSEP was originally introduced, by describing new tests and recent updates that have significantly enhanced the functionality and user experience of pyCSEP. It showcases the integration of new features, including access to authoritative earthquake catalogs from Italy (Bolletino Seismico Italiano), New Zealand (GeoNet), and the world (Global Centroid Moment Tensor), the creation of multiresolution spatial forecast grids, the adoption of non-Poissonian testing methods, applying a global seismicity model to specific regions for benchmarking regional models and evaluating alarm-based models. We highlight the application of these recent advances in regional studies, specifically through the New Zealand case study, which showcases the ability of pyCSEP to evaluate detailed, region-specific seismic forecasts using statistical functions. The enhancements in pyCSEP also facilitate the standardization of how the CSEP forecast experiments are conducted, improving the reliability, and comparability of the earthquake forecasting models. As such, pyCSEP exemplifies collaborative research and innovation in earthquake predictability, supporting transparent scientific practices, and community-driven development approaches.

A Multigrid Solver for PDE-Constrained Optimization with Uncertain Inputs

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number N of samples used to discretized the probability space. We show that this reduced system can be solved with optimal O(N) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

GPU-Enabled Exascale Quantum Transport Modeling of Carbon Nanotube Devices

The work will describe our efforts to develop an exascale electrostatic-quantum transport framework, Quantum-eXstatic (https://github.com/AMReX-Microelectronics/eXstatic), currently supporting the modeling of carbon nanotube field-effect transistors (CNTFETs). The framework is capable of modeling realistic three-dimensional structures, efficiently, using CPU/GPU heterogeneous architectures of state-of-the-art supercomputers. It is built on top of the AMReX library, which is a GPU-enabled software library containing functionality for writing particle-mesh applications on structured grids. The Quantum-eXstatic framework comprises three major components: the electrostatic module, the quantum transport module, and the part that self-consistently couples the two modules. The electrostatic module using a finite-volume multigrid solver to compute the electrostatic potential induced by charges on the surface of carbon nanotubes, as well as by source, drain, and gate terminals, which can be modeled using an embedded boundary approach to allow for intricate shapes. The quantum transport module uses the Nonequilibrium Green's Function (NEGF) method to model induced charge. Currently, it supports coherent (ballistic) transport, mode-space approximation for subbands of nanotube, contacts modeled as semi-infinite leads, and Hamiltonian representation using the tight-binding approximation. Special features are being included for accurate modeling of long nanotubes, such as adaptive refinement of integration contours to resolve van Hove singularities in long channels. Particular attention is paid to overlap computations with communication to optimize GPU performance. The self-consistency between the two modules is achieved using Broyden's modified second algorithm, which is parallelized using CPUs and GPUs. Preliminary scaling studies show that the code exhibits ~77% weak-scaling efficiency on 512 NVIDIA A100 GPUs of Perlmutter supercomputer for modeling gate-all-around CNTFETs with a channel length of 13.5 microns at equilibrium conditions. This case involved computations of electrostatic potential in 14.5 billion computational cells and Green’s function computation for a Hamiltonian of size 131,072 x 131,072. For problems of this large size, the code takes about 5 min to reach self-consistency. Figure 1

A Discrete Resistance Network Based on a Multiresolution Grid for 3D Ground-Return Current Forward Modeling

A Discrete Resistance Network Based on a Multiresolution Grid for 3D Ground-Return Current Forward Modeling

LibEMMI_MGFD: A program of marine controlled-source electromagnetic modelling and inversion using frequency-domain multigrid solver

We develop a software package libEMMI_MGFD for 3D frequency-domain marine controlled-source electromagnetic (CSEM) modelling and inversion. It is the first open-source C program tailored for geometrical multigrid (GMG) CSEM simulation. An volumetric anisotropic averaging scheme has been employed to compute effective medium for modelling over uniform and nonuniform grid. The computing coordinate is aligned with acquisition geometry by rotation with the azimuth and dip angles, facilitating the injection of the source and the extraction of data with arbitrary orientations. Efficient nonlinear optimization is achieved using quasi-Newton scheme assisted with bisection backtracking line search. In constructing the modularized Maxwell solver and evaluating the misfit and gradient for 3D CSEM inversion, the reverse communication technique is the key to the compaction of the software while maintaining the computational performance. A number of numeric tests demonstrate the efficiency of the modelling while preserving the solution accuracy. A 3D marine CSEM inversion example has been examined for resistivity imaging. Program summaryProgram Title: libEMMI_MGFDCPC Library link to program files:https://doi.org/10.17632/mpk7xrd38m.1Developer's repository link:https://github.com/yangpl/libEMMI_MGFDLicensing provisions: GNU General Public License v3.0Programming language: C, ShellSoftware dependencies: MPI [1]Nature of problem: 3D Controlled-source electromagnetic modelling and inversion.Solution method: Frequency-domain modelling on staggered grid; Geometrical multigrid (GMG) method as an iterative solver; Quasi-Newton method for nonlinear minimization; Bisection-based backtracking line search.

Thermo-fluidic and entropy generation analysis of MPCMS in straight, segmental, and rib-groove microchannel heat sinks: A numerical investigation

The efficient management of thermal systems is crucial for various technological applications, necessitating innovative cooling strategies to enhance performance and reduce entropy generation. Microchannels have emerged as a promising solution due to their high heat transfer rates and compact size. This study investigates the thermofluidic characteristics and entropy generation in straight, segmental, and rib-groove microchannels utilizing de-ionized water and microencapsulated phase change slurry (MPCMS) as working fluids. Employing numerical analysis, the investigation integrates a mixture model for the base fluid and micro-encapsulated phase change material (MPCM) using an equivalent heat capacity approach to incorporate the phase transition of the MPCM (composed of paraffin with a Ti shell). Operational parameters include an inlet fluid velocity ranging from 0.575 to 3.45 m/s, heat flux varying between 15 and 30 W/cm2, and MPCM volume fractions of 5% to 20%. The ANSYS Fluent Solver, a finite volume method using a multigrid solver, was used to control the simulation by species transport. Notably, segmental channels exhibit a notable decrease in wall temperature of 29.7% and 26.3% compared to straight and rib-groove channels, respectively, under constant heat flux and flow rate. However, the effectiveness of MPCM in enhancing thermal performance was greater in straight and rib-groove channels, surpassing segmental channels by 2.5% and 3.28%, respectively, under constant heat flux. On analyzing the irreversibility, segmental channels exhibit higher frictional entropy but lower thermal entropy generation due to the fin structure, resulting in an overall reduction in total entropy generation. Moreover, segmental microchannels, when utilizing MPCMS, demonstrate superior performance in reducing irreversibility by 60.66% and 49.6% compared to straight and rib-groove channels, respectively. Total entropy generation increases with higher velocity and heat flux, while it diminishes with higher MPCM concentration across all configurations.

Multiscale graph neural networks with adaptive mesh refinement for accelerating mesh-based simulations

Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems.

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

DC3DPAFEM: An efficient and accurate 3-D direct current resistivity anisotropic forward modeling software for complex geological settings

Nowadays, there is a growing trend that direct current (DC) field surveys are shifting towards challenging areas characterized by mountainous topography and electrical anisotropy. Given these complex geological settings, there is an urgent need for 3-D DC forward modeling software capable of effectively addressing large-scale problems and delivering accurate modeling results to interpret field data. However, most open-source software packages face certain limitations, such as the high numerical cost to handle complex surface topography, the lack of consideration for anisotropic conductivity, the absence of mesh refinement techniques to guarantee accuracy in forward modeling, and the lack of parallel computing techniques to solve large-scale problems. In this study, we develop an efficient and highly accurate 3-D DC anisotropic forward modeling software, namely DC3DPAFEM, using the adaptive finite element algorithm based on the unstructured tetrahedral mesh. Firstly, we construct a strong compatible boundary value problem (BVP) for 3-D anisotropic DC problems by adopting a specialized secondary potential approach to handle the surface topography efficiently. Then, we develop a goal-oriented adaptive mesh refinement (AMR) technique to ensure accurate forward modeling results, even with a coarse initial mesh. To ensure time and memory efficiency, we employ a robust conjugate gradient (CG) algorithm preconditioned by the algebraic multigrid (AMG) solver to solve the large-scale linear system of equations resulting from complex geological structures. We aim to investigate the performance of the AMG scheme in anisotropic DC cases. Furthermore, we incorporate the domain decomposition technique into the iterative solution scheme for further efficiency gains. This technique significantly improves computing efficiency for large-scale problems in parallel clusters. Finally, we conduct comprehensive performance tests for DC3DPAFEM using a two-layer anisotropic model and a 3-D complex model with undulating terrain. The results of both examples validate the accuracy of DC3DPAFEM, as they closely align with the analytical solutions and the solutions obtained from the existing 3-D DC forward modeling code. Compared to traditional direct solver MUMPS and ILU-preconditioned iterative solvers, DC3DPAFEM exhibits highly scalable performance for large-scale problems, offering significant advantages in terms of memory and time consumption. Overall, DC3DPAFEM demonstrates substantial advances in efficiency, accuracy, and practicality through a series of numerical examples. This open-source code provides an efficient and available tool for developing a 3-D DC inversion method that can deal with large-scale problems involving intricate topography and anisotropic media.

Efficient Visibility Approximation for Game AI using Neural Omnidirectional Distance Fields

Visibility information is critical in game AI applications, but the computational cost of raycasting-based methods poses a challenge for real-time systems. To address this challenge, we propose a novel method that represents a partitioned game scene as neural Omnidirectional Distance Fields (ODFs), allowing scalable and efficient visibility approximation between positions without raycasting. For each position of interest, we map its omnidirectional distance data from the spherical surface onto a UV plane. We then use multi-resolution grids and bilinearly interpolated features to encode directions. This allows us to use a compact multi-layer perceptron (MLP) to reconstruct the high-frequency directional distance data at these positions, ensuring fast inference speed. We demonstrate the effectiveness of our method through offline experiments and in-game evaluation. For in-game evaluation, we conduct a side-by-side comparison with raycasting-based visibility tests in three different scenes. Using a compact MLP (128 neurons and 2 layers), our method achieves an average cold start speedup of 9.35 times and warm start speedup of 4.8 times across these scenes. In addition, unlike the raycasting-based method, whose evaluation time is affected by the characteristics of the scenes, our method's evaluation time remains constant.

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

A parallel porosity model for large-scale modelling of river floods in urban areas

With the purpose of modelling at a large-scale river floods affecting urban areas, this work presents a numerical model solving the two-dimensional Shallow Water Equations with porosity. The distribution of buildings within urban fabrics is accounted for by means of a spatially-distributed field of the storage porosity parameter. The finite volume scheme conserves mass and it ensures the well-balancing property over complex topography. A structured multi-resolution grid is adopted for the domain discretization and the parallelization on Graphic Processing Units (GPUs) enables the reduction of the computational costs. Model testing is performed against experimental data and applications over real domains as well. The results provided by the proposed model agree with those obtained by resolving building footprints on a high-resolution mesh, meanwhile reducing the runtimes up to 6 times.

Direct Numerical Simulation of Mixed Convection Flow in Lid-Driven Cavities

The mixed convection of heat transfer and fluid flow in a - lid drivencubical cavity filled with air is investigated numerically in this study. The computational procedure is based on the finite volume method and a full multigrid acceleration solver. The top wall of the cavity is maintained at a constant high temperature - Th, and it can move with a constant velocity U0. The bottom wall is immobile and maintained at a cold temperature Tc. While, the remaining boundary parts of the cavity are motionless and kept thermally insulated. Several numerical simulations were conducted to investigate mixed convection heat transfer in a sliding cubical cavity for a range of Reynolds numbers from 1000 to 5000 and Richardson numbers from 0.001 to 10. The influence of mixed convection parameters, Reynolds number, Richardson number, and heat transfer rate on the flow behavior was analyzed through parametric studies. The results include flow and heat transfer characteristics, iso-surfaces, and streamlines for the entire range of Richardson numbers and Reynolds numbers investigated. The study shows that as Reynolds number is increased beyond a critical value, the flow becomes unstable and bifurcates.

An energy-efficient GMRES–multigrid solver for space-time finite element computation of dynamic poroelasticity

We present and analyze computationally Geometric MultiGrid (GMG) preconditioning techniques for Generalized Minimal RESidual (GMRES) iterations to space-time finite element methods (STFEMs) for a coupled hyperbolic–parabolic system modeling, for instance, flow in deformable porous media. By using a discontinuous temporal test basis, a time marching scheme is obtained. Higher order approximations that offer the potential to inherit most of the rich structure of solutions to the continuous problem on computationally feasible grids increase the block partitioning dimension of the algebraic systems, comprised of generalized saddle point blocks. Our V-cycle GMG preconditioner uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of 348 unknowns, the smoother is applied on patches of elements. This ensures damping of higher order error frequencies. By numerical experiments of increasing complexity, the efficiency of the solver for STFEMs of different polynomial order is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver’s energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms on the hardware.

Dense viewpoint encoding of 3D light fields based on neural graphics primitives

The enormous rendering time is a challenge for generating the Projector Image Array (PIA) using parallax images. Based on Neural Radiance Fields (NeRF), a novel method for directly rendering sub-images of PIA is proposed to accelerate light field encoding. The method is a unique combination of 3D light field coding and recent instant Neural Graphics Primitives (NGP) technique, showing how to obtain high-quality PIA quickly. The NGP is used to accelerate the training and rendering of the NeRF, and to improve the quality of the rendered parallax images by utilizing a multi-resolution hash grid container. The light ray of the traditional light field coding method is viewed as the key object linking the display system and the implicit radiance field. By encoding the light field of the actual display system as input to the NGP network, the proposed method can directly obtain the sub-PIA displayed in a single projector. For a projector with a resolution of 720×1280, the sub-PIA rendering time is only 25ms. The PIA is then synthesized from all the sub-PIAs that have been rendered. Numerical simulation results show that this method runs faster than the traditional method with high-quality displays. Light field display experiments are conducted to demonstrate the method's feasibility.

Augmented Lagrangian Acceleration of Global-in-Time Pressure Schur Complement Solvers for Incompressible Oseen Equations

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier–Stokes equations.

Neural Physical Simulation with Multi-Resolution Hash Grid Encoding

We explore the generalization of the implicit representation in the physical simulation task. Traditional time-dependent partial differential equations (PDEs) solvers for physical simulation often adopt the grid or mesh for spatial discretization, which is memory-consuming for high resolution and lack of adaptivity. Many implicit representations like local extreme machine or Siren are proposed but they are still too compact to suffer from limited accuracy in handling local details and a long time of convergence. We contribute a neural simulation framework based on multi-resolution hash grid representation to introduce hierarchical consideration of global and local information, simultaneously. Furthermore, we propose two key strategies: 1) a numerical gradient method for computing high-order derivatives with boundary conditions; 2) a range analysis sample method for fast neural geometry boundary sampling with dynamic topologies. Our method shows much higher accuracy and strong flexibility for various simulation problems: e.g., large elastic deformations, complex fluid dynamics, and multi-scale phenomena which remain challenging for existing neural physical solvers.

A multiscale preconditioner for crack evolution in porous microstructures: Accelerating phase‐field methods

AbstractPhase‐field methods are attractive for simulating the mechanical failure of geometrically complex porous microstructures described by 2D/3D x‐ray CT images in subsurface (e.g., CO storage) and manufacturing (e.g., Li‐ion battery) applications. They capture the nucleation, growth, and branching of fractures without prior knowledge of the propagation path or having to remesh the domain. Their drawback lies in the high computational cost for the typical domain sizes encountered in practice. We present a multiscale preconditioner that significantly accelerates the convergence of Krylov solvers in computing solutions of linear(ized) systems arising from the sequential discretization of the momentum and crack‐evolution equations in phase‐field methods. The preconditioner is an algebraic reformulation of a recent pore‐level multiscale method (PLMM) by the authors and consists of a global preconditioner and a local smoother . Together, and attenuate low‐ and high‐frequency errors simultaneously. The proposed , used in the momentum equation only, is a simplification of a recent variant proposed by the authors that is much cheaper and easier to deploy in existing solvers. The smoother , used in both the momentum and crack‐evolution equations, is built such that it is compatible with and more robust and efficient than black‐box smoothers like ILU(). We test and systematically for static‐ and evolving‐crack problems on complex 2D/3D porous microstructures, and show that they outperform existing algebraic multigrid solvers. We also probe different strategies for updating as cracks evolve and show the associated cost can be minimized if is updated adaptively and infrequently. Both and are scalable on parallel machines and can be implemented non‐intrusively in existing codes.

MM-NeRF: Large-Scale Scene Representation with Multi-Resolution Hash Grid and Multi-View Priors Features

Reconstructing large-scale scenes using Neural Radiance Fields (NeRFs) is a research hotspot in 3D computer vision. Existing MLP (multi-layer perception)-based methods often suffer from issues of underfitting and a lack of fine details in rendering large-scale scenes. Popular solutions are to divide the scene into small areas for separate modeling or to increase the layer scale of the MLP network. However, the subsequent problem is that the training cost increases. Moreover, reconstructing large scenes, unlike object-scale reconstruction, involves a geometrically considerable increase in the quantity of view data if the prior information of the scene is not effectively utilized. In this paper, we propose an innovative method named MM-NeRF, which integrates efficient hybrid features into the NeRF framework to enhance the reconstruction of large-scale scenes. We propose employing a dual-branch feature capture structure, comprising a multi-resolution 3D hash grid feature branch and a multi-view 2D prior feature branch. The 3D hash grid feature models geometric details, while the 2D prior feature supplements local texture information. Our experimental results show that such integration is sufficient to render realistic novel views with fine details, forming a more accurate geometric representation. Compared with representative methods in the field, our method significantly improves the PSNR (Peak Signal-to-Noise Ratio) by approximately 5%. This remarkable progress underscores the outstanding contribution of our method in the field of large-scene radiance field reconstruction.

Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method

We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods are a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.