arXiv subjects

Matthew G. Knepley

Publications and source records attributed to Matthew G. Knepley.

At least 19 recordsLinked to original sources

arXiv2026

WING: A Simple Windowed Nonorthogonalized Initial Guess Procedure for Repeated Matrix Solves

Many numerical methods require solution of a sequence of linear systems with the same matrix and similar right-hand sides. Krylov subspace methods are a common tool for solving such linear systems, and a carefully chosen initial guess for the solution can reduce the total number of iterations, and thereby the total computational cost, required for convergence to a specified numerical tolerance. This paper introduces the WING algorithm, a modification of Fischer's second algorithm, which lowers the cost of forming an acceptably close initial guess by skipping orthogonalization and solving the possibly singular normal equations with a pseudoinverse. We demonstrate the efficacy of the new algorithm, particularly for solving linear systems with coarse relative tolerances, with numerical benchmarks based on fluid-structure interaction, mantle convection, and earthquake models.

math.NA↗

arXiv2026

Structure preservation using discrete gradients in the Vlasov-Poisson-Landau system

We present a novel structure-preserving framework for solving the Vlasov-Poisson-Landau system of equations using a particle in cell (PIC) discretization combined with discrete gradient time integrators. The Vlasov-Poisson-Landau system is an accurate model for studying hot plasma dynamics at a kinetic scale where small-angle Coulomb collisions dominate. Our scheme guarantees conservation of mass, momentum and energy as well as preservation of the monotonicity of entropy production in both the time-continuous and discrete systems. We employ the conservative integrator for both the Hamiltonian Vlasov-Poisson equations and the dissipative Landau equation using the PETSc library (www.mcs.anl.gov/petsc) to showcase structure-preserving properties.

physics.plasm-ph↗

arXiv2025

A projection method for particle resampling

Particle discretizations of partial differential equations are advantageous for high-dimensional kinetic models in phase space due to their better scalability than continuum approaches with respect to dimension. Complex processes collectively referred to as particle noise hamper long time simulations with particle methods. One approach to address this problem is particle mesh adaptivity or remapping, known as particle resampling. This paper introduces a resampling method that projects particles to and from a (finite element) function space. The method is simple; using standard sparse linear algebra and finite element techniques, it can adapt to almost any set of new particle locations and preserves all moments up to the order of polynomial represented exactly by the continuum function space. This work is motivated by the Vlasov-Maxwell-Landau model of magnetized plasmas with up to six dimensions, 3X in physical space and 3V in velocity space, and is developed in the context of a 1X + 1V Vlasov-Poisson model of Landau damping with logically regular particle and continuum phase space grids. Stable long time dynamics are demonstrated up to T = 500 and reproducibility artifacts and data with stable dynamics up to T = 1000 are publicly available.

physics.plasm-ph↗

arXiv2025

Transformations of Computational Meshes

Computational meshes, as a way to partition space, form the basis of much of PDE simulation technology, for instance for the finite element and finite volume discretization methods. In complex simulations, we are often driven to modify an input mesh, for example, to refine, coarsen, extrude, change cell types, or filter it. Mesh manipulation code can be voluminous, error-prone, spread over many special cases, and hard to understand and maintain by subsequent developers. We present a simple, table-driven paradigm for mesh transformation which can execute a large variety of transformations in a performant, parallel manner, along with experiments in the open source library PETSc which can be run by the reader.

cs.MS↗

arXiv2024

Efficient N-to-M Checkpointing Algorithm for Finite Element Simulations

In this work, we introduce a new algorithm for N-to-M checkpointing in finite element simulations. This new algorithm allows efficient saving/loading of functions representing physical quantities associated with the mesh representing the physical domain. Specifically, the algorithm allows for using different numbers of parallel processes for saving and loading, allowing for restarting and post-processing on the process count appropriate to the given phase of the simulation and other conditions. For demonstration, we implemented this algorithm in PETSc, the Portable, Extensible Toolkit for Scientific Computation, and added a convenient high-level interface into Firedrake, a system for solving partial differential equations using finite element methods. We evaluated our new implementation by saving and loading data involving 8.2 billion finite element degrees of freedom using 8,192 parallel processes on ARCHER2, the UK National Supercomputing Service.

cs.DC↗

arXiv2024

A performance portable, fully implicit Landau collision operator with batched linear solvers

Modern accelerators use hierarchical parallel programming models that enable massive multithreading within a processing element (PE), with multiple PEs per device driven by traditional processes. Batching is a technique for exposing PE-level parallelism in algorithms that have traditionally run on MPI processes or multiple threads within a single process. Opportunities for batching arise in, for example, kinetic discretizations of magnetized plasmas where collisions are advanced in velocity space at each spatial point independently. This paper builds on previous work on a high-performance, fully nonlinear, Landau collision operator by batching the linear solver, as well as batching the spatial point problems and adding new support for multiple grids for multiscale, multi-species problems. An anisotropic relaxation verification test that agrees well with previous published results and analytical models is presented. The performance results from NVIDIA A100 and AMD MI250X nodes are presented with hardware utilization analysis for each architecture. The entire implicit Landau operator time advance is implemented in Kokkos for performance portability, running entirely on the device and is available in the PETSc numerical library.

physics.plasm-ph↗

arXiv2023

On the order of accuracy for finite difference approximations of partial differential equations using stencil composition

Stencil composition uses the idea of function composition, wherein two stencils with arbitrary orders of derivative are composed to obtain a stencil with a derivative order equal to sum of the orders of the composing stencils. In this paper, we show how stencil composition can be applied to form finite difference stencils in order to numerically solve partial differential equations (PDEs). We present various properties of stencil composition and investigate the relationship between the order of accuracy of the composed stencil and that of the composing stencils. We also present comparisons between the stability restrictions of composed higher-order PDEs to their compact versions and numerical experiments wherein we verify the order of accuracy by convergence tests. To demonstrate an application to PDEs, a boundary value problem involving the two-dimensional biharmonic equation is numerically solved using stencil composition and the order of accuracy is verified by performing a convergence test. The method is then applied to the Cahn-Hilliard phase-field model. In addition to sample results in 2D and 3D for this benchmark problem, the scalability, spectral properties, and sparsity is explored.

math.NA↗

arXiv2023

A Numerical Study of Landau Damping with PETSc-PIC

We present a study of the standard plasma physics test, Landau damping, using the Particle-In-Cell (PIC) algorithm. The Landau damping phenomenon consists of the damping of small oscillations in plasmas without collisions. In the PIC method, a hybrid discretization is constructed with a grid of finitely supported basis functions to represent the electric, magnetic and/or gravitational fields, and a distribution of delta functions to represent the particle field. Approximations to the dispersion relation are found to be inadequate in accurately calculating values for the electric field frequency and damping rate when parameters of the physical system, such as the plasma frequency or thermal velocity, are varied. We present a full derivation and numerical solution for the dispersion relation, and verify the PETSC-PIC numerical solutions to the Vlasov-Poisson for a large range of wave numbers and charge densities.

physics.plasm-ph↗

arXiv2023

Tetrahedralization of a Hexahedral Mesh

Two important classes of three-dimensional elements in computational meshes are hexahedra and tetrahedra. While several efficient methods exist that convert a hexahedral element to a tetrahedral elements, the existing algorithm for tetrahedralization of a hexahedral complex is the marching tetrahedron algorithm which limits pre-selection of face divisions. We generalize a procedure for tetrahedralizing triangular prisms to tetrahedralizing cubes, and combine it with certain heuristics to design an algorithm that can triangulate any hexahedra.

cs.CG↗

arXiv2022

Parallel Metric-Based Mesh Adaptation in PETSc using ParMmg

This research note documents the integration of the MPI-parallel metric-based mesh adaptation toolkit ParMmg into the solver library PETSc. This coupling brings robust, scalable anisotropic mesh adaptation to a wide community of PETSc users, as well as users of downstream packages. We demonstrate the new functionality via the solution of Poisson problems in three dimensions, with both uniform and spatially-varying right-hand sides.

cs.MS↗

arXiv2022

Conservative Projection Between Finite Element and Particle Bases

Particle-in-Cell (PIC) methods employ particle representations of unknown fields, but also employ continuum fields for other parts of the problem. Thus projection between particle and continuum bases is required. Moreover, we often need to enforce conservation constraints on this projection. We derive a mechanism for enforcement based on weak equality, and implement it in the PETSc libraries. Scalability is demonstrated to more than 1B particles.

physics.plasm-ph↗

arXiv2021

PCPATCH: software for the topological construction of multigrid relaxation methods

Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gauß-Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this paper we present a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods. Space decompositions are specified by collecting topological entities in a mesh (such as all vertices or faces) and applying a construction rule (such as taking all degrees of freedom in the cells around each entity). The software is implemented in PETSc and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. In turn, this allows for the very rapid development of fast solvers for difficult problems.

cs.MS↗

arXiv2021

Understanding performance variability in standard and pipelined parallel Krylov solvers

In this work, we collect data from runs of Krylov subspace methods and pipelined Krylov algorithms in an effort to understand and model the impact of machine noise and other sources of variability on performance. We find large variability of Krylov iterations between compute nodes for standard methods that is reduced in pipelined algorithms, directly supporting conjecture, as well as large variation between statistical distributions of runtimes across iterations. Based on these results, we improve upon a previously introduced nondeterministic performance model by allowing iterations to fluctuate over time. We present our data from runs of various Krylov algorithms across multiple platforms as well as our updated non-stationary model that provides good agreement with observations. We also suggest how it can be used as a predictive tool.

cs.MS↗

arXiv2020

Fully Parallel Mesh I/O using PETSc DMPlex with an Application to Waveform Modeling

Large-scale PDE simulations using high-order finite-element methods on unstructured meshes are an indispensable tool in science and engineering. The widely used open-source PETSc library offers an efficient representation of generic unstructured meshes within its DMPlex module. This paper details our recent implementation of parallel mesh reading and topological interpolation (computation of edges and faces from a cell-vertex mesh) into DMPlex. We apply these developments to seismic wave propagation scenarios on Mars as an example application. The principal motivation is to overcome single-node memory limits and reach mesh sizes which were impossible before. Moreover, we demonstrate that scalability of I/O and topological interpolation goes beyond 12'000 cores, and memory-imposed limits on mesh size vanish.

cs.MS↗

arXiv2018

A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

We present a new method for approximating solutions to the incompressible miscible displacement problem in porous media. At the discrete level, the coupled nonlinear system has been split into two linear systems that are solved sequentially. The method is based on a hybridizable discontinuous Galerkin method for the Darcy flow, which produces a mass--conservative flux approximation, and a hybridizable discontinuous Galerkin method for the transport equation. The resulting method is high order accurate. Due to the implicit treatment of the system of partial differential equations, we observe computationally that no slope limiters are needed. Numerical experiments are provided that show that the method converges optimally and is robust for highly heterogeneous porous media in 2D and 3D.

cs.CE↗

arXiv2018

Comparative study of finite element methods using the Time-Accuracy-Size (TAS) spectrum analysis

We present a performance analysis appropriate for comparing algorithms using different numerical discretizations. By taking into account the total time-to-solution, numerical accuracy with respect to an error norm, and the computation rate, a cost-benefit analysis can be performed to determine which algorithm and discretization are particularly suited for an application. This work extends the performance spectrum model in Chang et. al. 2017 for interpretation of hardware and algorithmic tradeoffs in numerical PDE simulation. As a proof-of-concept, popular finite element software packages are used to illustrate this analysis for Poisson's equation.

cs.MS↗

arXiv2018

A hybridizable discontinuous Galerkin method for two-phase flow in heterogeneous porous media

We present a new method for simulating incompressible immiscible two-phase flow in porous media. The semi-implicit method decouples the wetting phase pressure and saturation equations. The equations are discretized using a hybridizable discontinuous Galerkin (HDG) method. The proposed method is of high order, conserves global/local mass balance, and the number of globally coupled degrees of freedom is significantly reduced compared to standard interior penalty discontinuous Galerkin methods. Several numerical examples illustrate the accuracy and robustness of the method. These examples include verification of convergence rates by manufactured solutions, common 1D benchmarks and realistic discontinuous permeability fields.

cs.CE↗

arXiv2018

Programming Languages for Scientific Computing

Scientific computation is a discipline that combines numerical analysis, physical understanding, algorithm development, and structured programming. Several yottacycles per year on the world's largest computers are spent simulating problems as diverse as weather prediction, the properties of material composites, the behavior of biomolecules in solution, and the quantum nature of chemical compounds. This article is intended to review specfic languages features and their use in computational science. We will review the strengths and weaknesses of different programming styles, with examples taken from widely used scientific codes.

cs.PL↗