arXiv subjects

Luke Olson

Publications and source records attributed to Luke Olson.

13 recordsLinked to original sources

arXiv2026

Coarse Solvers for Exascale Solution of Poisson Problems

We present a two-level Schwarz method as an alternative to Algebraic Multigrid method(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poisson equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes equation. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

math.NA↗

arXiv2024

Exploiting mesh structure to improve multigrid performance for saddle point problems

In recent years, solvers for finite-element discretizations of linear or linearized saddle-point problems, like the Stokes and Oseen equations, have become well established. There are two main classes of preconditioners for such systems: those based on block-factorization approach and those based on monolithic multigrid. Both classes of preconditioners have several critical choices to be made in their composition, such as the selection of a suitable relaxation scheme for monolithic multigrid. From existing studies, some insight can be gained as to what options are preferable in low-performance computing settings, but there are very few fair comparisons of these approaches in the literature, particularly for modern architectures, such as GPUs. In this paper, we perform a comparison between a block-triangular preconditioner and a monolithic multigrid method with the three most common choices of relaxation scheme - Braess-Sarazin, Vanka, and Schur-Uzawa. We develop a performant Vanka relaxation algorithm for structured-grid discretizations, which takes advantage of memory efficiencies in this setting. We detail the behavior of the various CUDA kernels for the multigrid relaxation schemes and evaluate their individual arithmetic intensity, performance, and runtime. Running a preconditioned FGMRES solver for the Stokes equations with these preconditioners allows us to compare their efficiency in a practical setting. We show monolithic multigrid can outperform block-triangular preconditioning, and that using Vanka or Braess-Sarazin relaxation is most efficient. Even though multigrid with Vanka relaxation exhibits reduced performance on the CPU (up to $100\%$ slower than Braess-Sarazin), it is able to outperform Braess-Sarazin by more than $20\%$ on the GPU, making it a competitive algorithm, especially given the high amount of algorithmic tuning needed for effective Braess-Sarazin relaxation.

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arXiv2023

Learning from Integral Losses in Physics Informed Neural Networks

This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for training. As a result, accurate evaluation may be impractical, and we show that naive approximations at replacing these integrals with unbiased estimates lead to biased loss functions and solutions. To overcome this bias, we investigate three types of potential solutions: the deterministic sampling approaches, the double-sampling trick, and the delayed target method. We consider three classes of PDEs for benchmarking; one defining Poisson problems with singular charges and weak solutions of up to 10 dimensions, another involving weak solutions on electro-magnetic fields and a Maxwell equation, and a third one defining a Smoluchowski coagulation problem. Our numerical results confirm the existence of the aforementioned bias in practice and also show that our proposed delayed target approach can lead to accurate solutions with comparable quality to ones estimated with a large sample size integral. Our implementation is open-source and available at https://github.com/ehsansaleh/btspinn.

cs.LG↗

arXiv2023

Generalizing Lloyd's algorithm for graph clustering

Clustering is a commonplace problem in many areas of data science, with applications in biology and bioinformatics, understanding chemical structure, image segmentation, building recommender systems, and many more fields. While there are many different clustering variants (based on given distance or graph structure, probability distributions, or data density), we consider here the problem of clustering nodes in a graph, motivated by the problem of aggregating discrete degrees of freedom in multigrid and domain decomposition methods for solving sparse linear systems. Specifically, we consider the challenge of forming balanced clusters in the graph of a sparse matrix for use in algebraic multigrid, although the algorithm has general applicability. Based on an extension of the Bellman-Ford algorithm, we generalize Lloyd's algorithm for partitioning subsets of Rn to balance the number of nodes in each cluster; this is accompanied by a rebalancing algorithm that reduces the overall energy in the system. The algorithm provides control over the number of clusters and leads to "well centered" partitions of the graph. Theoretical results are provided to establish linear complexity and numerical results in the context of algebraic multigrid highlight the benefits of improved clustering.

math.NA↗

arXiv2023

MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods

Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters, prescribing overlap, subdomain boundary conditions, and other properties of the DDM. While some work has been done on optimizing these parameters, it has mostly focused on the one-level setting or special cases such as structured-grid discretizations with regular subdomain construction. In this paper, we propose multigrid graph neural networks (MG-GNN), a novel GNN architecture for learning optimized parameters in two-level DDMs\@. We train MG-GNN using a new unsupervised loss function, enabling effective training on small problems that yields robust performance on unstructured grids that are orders of magnitude larger than those in the training set. We show that MG-GNN outperforms popular hierarchical graph network architectures for this optimization and that our proposed loss function is critical to achieving this improved performance.

cs.LG↗

arXiv2022

Generalizing Reduction-Based Algebraic Multigrid

Algebraic Multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.

math.NA↗

arXiv2022

Parallel Energy-Minimization Prolongation for Algebraic Multigrid

Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches.

math.NA↗

arXiv2022

Learning Interface Conditions in Domain Decomposition Solvers

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured- and unstructured-grid problems.

cs.LG↗

arXiv2022

Performance Analysis and Optimal Node-Aware Communication for Enlarged Conjugate Gradient Methods

Krylov methods are a key way of solving large sparse linear systems of equations, but suffer from poor strong scalabilty on distributed memory machines. This is due to high synchronization costs from large numbers of collective communication calls alongside a low computational workload. Enlarged Krylov methods address this issue by decreasing the total iterations to convergence, an artifact of splitting the initial residual and resulting in operations on block vectors. In this paper, we present a performance study of an Enlarged Krylov Method, Enlarged Conjugate Gradients (ECG), noting the impact of block vectors on parallel performance at scale. Most notably, we observe the increased overhead of point-to-point communication as a result of denser messages in the sparse matrix-block vector multiplication kernel. Additionally, we present models to analyze expected performance of ECG, as well as, motivate design decisions. Most importantly, we introduce a new point-to-point communication approach based on node-aware communication techniques that increases efficiency of the method at scale.

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arXiv2021

Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learning

Large sparse linear systems of equations are ubiquitous in science and engineering, such as those arising from discretizations of partial differential equations. Algebraic multigrid (AMG) methods are one of the most common methods of solving such linear systems, with an extensive body of underlying mathematical theory. A system of linear equations defines a graph on the set of unknowns and each level of a multigrid solver requires the selection of an appropriate coarse graph along with restriction and interpolation operators that map to and from the coarse representation. The efficiency of the multigrid solver depends critically on this selection and many selection methods have been developed over the years. Recently, it has been demonstrated that it is possible to directly learn the AMG interpolation and restriction operators, given a coarse graph selection. In this paper, we consider the complementary problem of learning to coarsen graphs for a multigrid solver, a necessary step in developing fully learnable AMG methods. We propose a method using a reinforcement learning (RL) agent based on graph neural networks (GNNs), which can learn to perform graph coarsening on small planar training graphs and then be applied to unstructured large planar graphs, assuming bounded node degree. We demonstrate that this method can produce better coarse graphs than existing algorithms, even as the graph size increases and other properties of the graph are varied. We also propose an efficient inference procedure for performing graph coarsening that results in linear time complexity in graph size.

cs.LG↗

arXiv2019

Reducing Communication in Algebraic Multigrid with Multi-step Node Aware Communication

Algebraic multigrid (AMG) is often viewed as a scalable $\mathcal{O}(n)$ solver for sparse linear systems. Yet, parallel AMG lacks scalability due to increasingly large costs associated with communication, both in the initial construction of a multigrid hierarchy as well as the iterative solve phase. This work introduces a parallel implementation of AMG to reduce the cost of communication, yielding an increase in scalability. Standard inter-process communication consists of sending data regardless of the send and receive process locations. Performance tests show notable differences in the cost of intra- and inter-node communication, motivating a restructuring of communication. In this case, the communication schedule takes advantage of the less costly intra-node communication, reducing both the number and size of inter-node messages. Node-centric communication extends to the range of components in both the setup and solve phase of AMG, yielding an increase in the weak and strong scalability of the entire method.

cs.DC↗

arXiv2018

Learning with Analytical Models

To understand and predict the performance of scientific applications, several analytical and machine learning approaches have been proposed, each having its advantages and disadvantages. In this paper, we propose and validate a hybrid approach for performance modeling and prediction, which combines analytical and machine learning models. The proposed hybrid model aims to minimize prediction cost while providing reasonable prediction accuracy. Our validation results show that the hybrid model is able to learn and correct the analytical models to better match the actual performance. Furthermore, the proposed hybrid model improves the prediction accuracy in comparison to pure machine learning techniques while using small training datasets, thus making it suitable for hardware and workload changes.

cs.PF↗

arXiv2018

FFT, FMM, and Multigrid on the Road to Exascale: performance challenges and opportunities

FFT, FMM, and multigrid methods are widely used fast and highly scalable solvers for elliptic PDEs. However, emerging large-scale computing systems are introducing challenges in comparison to current petascale computers. Recent efforts (Dongarra et al. 2011) have identified several constraints in the design of exascale software that includes massive concurrency, resilience management, exploiting the high performance of heterogeneous systems, energy efficiency, and utilizing the deeper and more complex memory hierarchy expected at exascale. In this paper, we perform a model-based comparison of the FFT, FMM, and multigrid methods in the context of these projected constraints. In addition, we use performance models to offer predictions about the expected performance on upcoming exascale system configurations based on current technology trends.

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