arXiv subjects

James Brannick

Publications and source records attributed to James Brannick.

At least 19 recordsLinked to original sources

arXiv2026

Nodal Coarsening and Sparse Ideal Interpolation for H(curl) Problems in Algebraic Multigrid

We propose a sparse interpolation construction and a practical coarsening algorithm for the algebraic multigrid (AMG) method, tailored towards H(curl). Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. We establish the weak approximation property and the commuting relation under certain assumptions. Combined with matching block smoothers, the proposed interpolation yields an effective algebraic multilevel method. Numerical experiments show robustness under strong coefficient jumps, where the proposed methods substantially outperform standard geometric multigrid.

math.NA↗

arXiv2025

Multiscale Graph Reduction for Heterogeneous and Anisotropic Discrete Diffusion Processes

We present multiscale graph-based reduction algorithms for upscaling heterogeneous and anisotropic diffusion problems. The proposed coarsening approaches begin by constructing a partitioning of the computational domain into a set of balanced local subdomains, resulting in a standard type of domain decomposition. Given this initial decomposition, general coarsening techniques based on spectral clustering are applied within each subgraph in order to accurately identify the key microscopic features of a given system. The spectral clustering algorithm is based on local generalized eigen-decompositions applied to the signed graph Laplacian. The resulting coarse-fine splittings are combined with two variants of energy-minimizing strategies for constructing coarse bases for diffusion problems. The first is an unconstrained minimization formulation in which local harmonic extensions are applied column-wise to construct multi-vector preserving interpolation in each region, whereas the second approach is a variant of the constrained energy minimization formulations derived in the context of non-local multi-continua upscaling techniques. We apply the resulting upscaling algorithms to a variety of tests coming from the graph Laplacian, including diffusion in the perforated domain, channelized media, highly anisotropic settings, and discrete pore network models to demonstrate the potential and robustness of the proposed coarsening approaches. We show numerically and theoretically that the proposed approaches lead to accurate coarse-scale models.

math.NA↗

arXiv2025

Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems

Consider an algebraic two-level method applied to the $n$-dimensional linear system $A \mathbf{x} = \mathbf{b}$ using fine-space preconditioner (i.e., ``relaxation'' or ``smoother'') $M$, with $M \approx A$, restriction and interpolation $R$ and $P$, and algebraic coarse-space operator ${A_c := R^*AP}$. Then, what are the the best possible transfer operators $R$ and $P$ of a given dimension $n_c < n$? Brannick et al. (2018) showed that when $A$ and $M$ are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the $n_c$ smallest generalized eigenvectors of the matrix pencil $(A, M)$. Recently, in Ali et al. (2025) we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of $(A, M)$ and defining corresponding nonsymmetric transfer operators $\{R_\#,P_\#\}$. Tight convergence bounds for $\{R_\#,P_\#\}$ are derived in spectral radius, as well as a proof of pseudo-optimality. Note, $\{R_\#,P_\#\}$ are typically complex valued, which is not practical for real-valued problems. Here we build on Ali et al. (2025), first characterizing all inner products in which the coarse-space correction defined by $\{R_\#,P_\#\}$ is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators $\{R_\#,P_\#\}$ are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from Brannick et al. (2018). Finally, we show how to construct optimal, real-valued transfer operators in the case of that $A$ and $M$ are real valued, but are not HPD. Numerical examples arising from discretized advection and wave-equation problems are used to verify and illustrate the theory.

math.NA↗

arXiv2025

Nodal AMG Coarsening and Interpolation for PDE Systems

We present an approach to constructing a practical coarsening algorithm and interpolation operator for the algebraic multigrid (AMG) method, tailored towards systems of partial differential equations (PDEs) with large near-kernels, such as H(curl) and H(div). Our method builds on compatible relaxation (CR) and the ideal interpolation model within the generalized AMG (GAMG) framework but introduces several modifications to define an AMG method for PDE systems. We construct an interpolation operator through a coarsening process that first coarsens a nodal dual problem and then builds the coarse and fine variables using a matching algorithm. Our interpolation follows the ideal formulation; however, we enhance the sparsity of ideal interpolation by decoupling the fine and coarse variables completely. When the coarse variables align with the geometric refinement, our method reproduces re-discretization on unstructured meshes. Together with an automatic smoother construction scheme that identifies the local near kernels, our approach forms a complete two-grid method. Finally, we also show numerical results that demonstrate the effectiveness of this interpolation scheme by applying it to targeted problems and the Stokes system.

math.NA↗

arXiv2024

Generalized Optimal AMG Convergence Theory for Nonsymmetric and Indefinite Problems

Algebraic multigrid (AMG) is known to be an effective solver for many sparse symmetric positive definite (SPD) linear systems. For SPD systems, the convergence theory of AMG is well-understood in terms of the $A$-norm, but in a nonsymmetric setting, such an energy norm is non-existent. For this reason, convergence of AMG for nonsymmetric systems of equations remains an open area of research. A particular aspect missing from theory of nonsymmetric and indefinite AMG is the incorporation of general relaxation schemes. In the SPD setting, the classical form of optimal AMG interpolation provides a useful insight in determining the best possible two-grid convergence rate of a method based on an arbitrary symmetrized relaxation scheme. In this work, we discuss a generalization of the optimal AMG convergence theory targeting nonsymmetric problems, using a certain matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem relating the system matrix and relaxation operator. We show that using this generalization of the optimal convergence theory, one can obtain a measure of the spectral radius of the two grid error transfer operator that is mathematically equivalent to the derivation in the SPD setting for optimal interpolation, which instead uses norms. In addition, this generalization of the optimal AMG convergence theory can be further extended for symmetric indefinite problems, such as those arising from saddle point systems so that one can obtain a precise convergence rate of the resulting two-grid method based on optimal interpolation. We provide supporting numerical examples of the convergence theory for nonsymmetric advection-diffusion problems, two-dimensional Dirac equation motivated by $\gamma_5$-symmetry, and the mixed Darcy flow problem corresponding to a saddle point system.

math.NA↗

arXiv2023

Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems

This paper focuses on developing a reduction-based algebraic multigrid method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based algebraic multigrid (AMG) approach, $\ell$AIR (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion dominated problems in two or three dimensions. Motivated by the success of $\ell$AIR in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy minimization AMG methods with the local approximation of ideal operators used in $\ell$AIR. The resulting constrained $\ell$AIR (C$\ell$AIR) algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that has been previously difficult for reduction-based methods.

math.NA↗

arXiv2019

The Role of Energy Minimization in Algebraic Multigrid Interpolation

Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development of improved interpolation operators. Such development is often based on AMG convergence theory. However, convergence theory in AMG tends to have a disconnect with AMG in practice due to the practical constraints of (i) maintaining matrix sparsity in transfer and coarse-grid operators, and (ii) retaining linear complexity in the setup and solve phase. This paper presents a review of fundamental results in AMG convergence theory, followed by a discussion on how these results can be used to motivate interpolation operators in practice. A general weighted energy minimization functional is then proposed to form interpolation operators, and a novel `diagonal' preconditioner for Sylvester- or Lyapunov-type equations developed simultaneously. Although results based on the weighted energy minimization typically underperform compared to a fully constrained energy minimization, numerical results provide new insight into the role of energy minimization and constraint vectors in AMG interpolation.

math.NA↗

arXiv2018

Analysis of the energy dissipation laws in multi-component phase field models

In this paper, two approaches for modeling three-component fluid flows using diffusive interface method are discussed. Thermodynamic consistency of the proposed models is preserved when using an energetic variational framework to derive the coupled systems of partial differential equations that comprise the resulting models. The issue of algebraic and dynamic consistency is investigated. In addition, the two approaches that are presented are compared analytically and numerically.

math.AP↗

arXiv2017

Optimal interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap algebraic multigrid setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the BAMG method with sparse interpolation matrix (and spanning multiple levels) outperforms the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

math.NA↗

arXiv2015

A Bootstrap Multigrid Eigensolver

This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid (BAMG) setup algorithm that includes an AMG eigensolver, it is illustrated how the algorithm can be simplified for the case of a discretized partial differential equation (PDE), thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for: (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace-Beltrami eigenvalue problem.

math.NA↗

arXiv2015

Microphysics of Neutron Star Outer Envelopes in the Periodized, Magnetic Thomas-Fermi Model

Static and dynamic properties of low density outer envelopes of neutron stars are calculated within the nonlinear magnetic Thomas-Fermi model, assuming degenerate electrons. A novel domain decomposition enables proper description of lattice symmetry and may be seen as a prototype for the general class of problems involving nonlinear charge screening of periodic, quasi-low-dimensionality structures, e.g. liquid crystals. We describe a scalable implementation of the method using Hypre. Phase velocity of long wavelength transverse phonons is found to be a factor of 5-7 larger than in the corresponding Coulomb crystal model, which could have implications for low temperature phonon-mediated thermal conductivity. Other findings include $c'<0$ elastic instabilities for both bcc and fcc lattices, reminiscent of the situation in some light actinides, and suggestive of a symmetry-lowering transition to a tetragonal or orthorhombic lattice.

astro-ph.SR↗

arXiv2014

Multigrid Preconditioning for the Overlap Operator in Lattice QCD

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing a motivation why our preconditioner works well in computational practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of up to one order of magnitude.

hep-lat↗

arXiv2014

Bootstrap Algebraic Multigrid: status report, open problems, and outlook

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.

math.NA↗

arXiv2014

An Adaptively Constructed Algebraic Multigrid Preconditioner for Irreducible Markov Chains

The computation of stationary distributions of Markov chains is an important task in the simulation of stochastic models. The linear systems arising in such applications involve non-symmetric M-matrices, making algebraic multigrid methods a natural choice for solving these systems. In this paper we investigate extensions and improvements of the bootstrap algebraic multigrid framework for solving these systems. This is achieved by reworking the bootstrap setup process to use singular vectors instead of eigenvectors in constructing interpolation and restriction. We formulate a result concerning the convergence speed of GMRES for singular systems and experimentally justify why rapid convergence of the proposed method can be expected. We demonstrate its fast convergence and the favorable scaling behavior for various test problems.

math.NA↗

arXiv2013

Local Fourier Analysis of Multigrid Methods with Polynomial Smoothers and Aggressive coarsening

We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite difference/element discretizations of the Laplace equation. Using local Fourier analysis we determine automatically the optimal values for the parameters involved in defining the polynomial smoothers and achieve fast convergence of cycles with aggressive coarsening. We also present numerical tests supporting the theoretical results and the heuristic ideas. The methods we introduce are highly parallelizable and efficient multigrid algorithms on structured and semi-structured grids in two and three spatial dimensions.

math.NA↗

arXiv2013

Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system

We develop an algebraic multigrid method for solving the non-Hermitian Wilson discretization of the 2-dimensional Dirac equation. The proposed approach uses a bootstrap setup algorithm based on a multigrid eigensolver. It computes test vectors which define the least squares interpolation operators by working mainly on coarse grids, leading to an efficient and integrated self learning process for defining algebraic multigrid interpolation. The algorithm is motivated by the \gamma-symmetry of the Dirac equation, which carries over to the Wilson discretization. This discrete \gamma-symmetry is used to reduce a general Petrov Galerkin bootstrap setup algorithm to a Galerkin method for the Hermitian and indefinite formulation of the Wilson matrix. Kaczmarz relaxation is used as the multigrid smoothing scheme in both the setup and solve phases of the resulting Galerkin algorithm. The overall method is applied to the odd-even reduced Wilson matrix, which also fulfills the discrete \gamma-symmetry. Extensive numerical results are presented to motivate the design and demonstrate the effectiveness of the proposed approach.

math.NA↗

arXiv2013

Aggregation-based aggressive coarsening with polynomial smoothing

This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient iteration in the solve phase. We show that by combining these techniques it is possible to design a simple and scalable algorithm. Results of the algorithm applied to graph Laplacian systems arising from the standard linear finite element discretization of the scalar Poisson problem are reported.

math.NA↗

arXiv2013

Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a $\ell^1$-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.

math.NA↗