arXiv subjects

Scott P. MacLachlan

Publications and source records attributed to Scott P. MacLachlan.

16 recordsLinked to original sources

arXiv2026

Automated Galerkin time stepping in Irksome

As the study of temporal and spatial discretization schemes continues to advance, recent work has focused on the use of Galerkin-in-time discretization schemes that enable broader structure-preservation than is known for Runge-Kutta integrators. While the promise of such discretizations is immense, their realization has, until now, generally relied on bespoke implementations that have limited their wider use. In this work, we present automation in Irksome for both discontinuous Galerkin and continuous Petrov-Galerkin time stepping of semidiscrete variational problems. The implementation supports auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers, and it enables switching between Runge-Kutta and Galerkin-in-time formulations with minimal changes to user code. Numerical examples illustrate accuracy, solver performance, and structure preservation across representative PDE systems.

math.NA↗

arXiv2025

Automated Runge-Kutta-Nyström time stepping for finite element methods in Irksome

Irksome is a library based on the Unified Form Language (UFL) that automates the application of Runge-Kutta time-stepping methods for finite element spatial discretizations of partial differential equations (PDEs). This paper describes recent updates to Irksome that allow users to express semidiscrete forms of PDEs that contain second-order temporal derivatives, whence it generates stage-coupled variational problems to be solved at each time step for Runge-Kutta-Nyström methods. Firedrake then generates code for these variational problems and provides a rich interface to PETSc for solving them. Directly discretizing second-order time derivatives with Runge-Kutta-Nyström methods provides several advantages relative to discretizing a rewritten first-order system with a standard Runge-Kutta method. Besides working with an interface closer to the problem formulation in UFL, avoiding these auxiliary variables means that Runge-Kutta-Nyström methods lead to smaller algebraic systems and better run-time. Our numerical results indicate that, with effective preconditioning, fully implicit Runge-Kutta-Nyström methods can be made competitive with more traditional explicit methods for wave equations. They are also (essentially) required to discretize wave-type equations with higher-order spatial derivatives. We also provide numerical experiments for fully dynamic poroelasticity, a system of mixed temporal order, where our time-stepping and algebraic solvers perform effectively even as we approach the incompressible limit.

math.NA↗

arXiv2025

Extending Irksome: improvements in automated Runge--Kutta time stepping for finite element methods

Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to express semidiscrete forms of PDE, it generates UFL representations for the stage-coupled variational problems to be solved at each time step. The Firedrake package then generates efficient code for evaluating these variational problems and allows users a wide range of options to deploy efficient algebraic solvers in PETSc. In this paper, we describe several recent advances in Irksome. These include alternate formulations of the Runge--Kutta time-stepping methods and optimized support for diagonally implicit (DIRK) methods. Additionally, we present new and improved tools for building preconditioners for the resulting linear and linearized systems, demonstrating that these can lead to efficient approaches for solving fully implicit Runge-Kutta discretizations. The new features are demonstrated through a sequence of computational examples demonstrating the high-level interface and obtained solver performance.

math.NA↗

arXiv2024

Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

math.NA↗

arXiv2023

Finite-element discretization of the smectic density equation

The fourth-order PDE that models the density variation of smectic A liquid crystals presents unique challenges in its (numerical) analysis beyond more common fourth-order operators, such as the classical biharmonic. While the operator is positive definite, the equation has a "wrong-sign" shift, making it somewhat more akin to an indefinite Helmholtz operator, with lowest-energy modes consisting of plane waves. As a result, for large shifts, the natural continuity, coercivity, and inf-sup constants degrade considerably, impacting standard error estimates. In this paper, we analyze and compare three finite-element formulations for such PDEs, based on $H^2$-conforming elements, the $C^0$ interior penalty method, and a mixed finite-element formulation that explicitly introduces approximations to the gradient of the solution and a Lagrange multiplier. The conforming method is simple but is impractical to apply in three dimensions; the interior-penalty method works well in two and three dimensions but has lower-order convergence and (in preliminary experiments) seems difficult to precondition; the mixed method uses more degrees of freedom, but works well in both two and three dimensions, and is amenable to monolithic multigrid preconditioning. Our analysis reveals different behaviours of the error bounds with the shift parameter and mesh size for the different schemes. Numerical results verify the finite-element convergence for all discretizations, and illustrate the trade-offs between the three schemes.

math.NA↗

arXiv2023

Multigrid preconditioning for regularized least-squares problems

In this paper, we are concerned with efficiently solving the sequences of regularized linear least squares problems associated with employing Tikhonov-type regularization with regularization operators designed to enforce edge recovery. An optimal regularization parameter, which balances the fidelity to the data with the edge-enforcing constraint term, is typically not known a priori. This adds to the total number of regularized linear least squares problems that must be solved before the final image can be recovered. Therefore, in this paper, we determine effective multigrid preconditioners for these sequences of systems. We focus our approach on the sequences that arise as a result of the edge-preserving method introduced in [6], where we can exploit an interpretation of the regularization term as a diffusion operator; however, our methods are also applicable in other edge-preserving settings, such as iteratively reweighted least squares problems. Particular attention is paid to the selection of components of the multigrid preconditioner in order to achieve robustness for different ranges of the regularization parameter value. In addition, we present a parameter culling approach that, when used with the L-curve heuristic, reduces the total number of solves required. We demonstrate our preconditioning and parameter culling routines on examples in computed tomography and image deblurring.

math.NA↗

arXiv2023

Coarse-Grid Selection Using Simulated Annealing

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying problem and its discretization, algebraic multigrid aims to be less intrusive, requiring less knowledge about the origin of the linear system. A key step in algebraic multigrid is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In work by MacLachlan and Saad, a constrained combinatorial optimization problem is used to define the ``best'' coarse grid within the setting of a two-level reduction-based algebraic multigrid method and is shown to be NP-complete. Here, we develop a new coarsening algorithm based on simulated annealing to approximate solutions to this problem, which yields improved results over the greedy algorithm developed previously. We present numerical results for test problems on both structured and unstructured meshes, demonstrating the ability to exploit knowledge about the underlying grid structure if it is available.

math.NA↗

arXiv2022

A new mixed finite-element method for $H^2$ elliptic problems

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

math.NA↗

arXiv2021

A Boundary-Layer Preconditioner for Singularly Perturbed Convection Diffusion

Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the study of the numerical approximation of solutions to differential equations. Consequently, much is known about how to accurately and stably discretize such equations on \textit{a priori} adapted meshes, in order to properly resolve the layer structure present in their continuum solutions. However, despite being a key step in the numerical simulation process, much less is known about the efficient and accurate solution of the linear systems of equations corresponding to these discretizations. In this paper, we discuss problems associated with the application of direct solvers to these discretizations, and we propose a preconditioning strategy that is tuned to the matrix structure induced by using layer-adapted meshes for convection-diffusion equations, proving a strong condition-number bound on the preconditioned system in one spatial dimension, and a weaker bound in two spatial dimensions. Numerical results confirm the efficiency of the resulting preconditioners in one and two dimensions, with time-to-solution of less than one second for representative problems on $1024\times 1024$ meshes and up to $40\times$ speedup over standard sparse direct solvers.

math.NA↗

arXiv2021

Optimizing multigrid reduction-in-time (MGRIT) and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent PDEs in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically non-scalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge-Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping. Our insight of tracking characteristics on coarse grids provides a key idea for solving the long-standing problem of efficient parallel-in-time integration for hyperbolic PDEs.

math.NA↗

arXiv2020

A local Fourier analysis of additive Vanka relaxation for the Stokes equations

Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid-transfer operators is crucial to the design of optimally efficient algorithms. It is well--known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite-element discretizations of the Stokes equations. The analysis offers insight into the choice of "patches" for the Vanka relaxation, revealing that smaller patches offer more effective convergence per floating point operation. Parameters that minimize the two-grid convergence factor are proposed and numerical experiments are presented to validate the LFA predictions.

math.NA↗

arXiv2019

On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improved coarse-grid time-stepping operators. For several high-order discretizations of the advection equation, we demonstrate that there exist non-standard coarse-grid time stepping operators that yield significant improvements over the standard choice of rediscretization.

math.NA↗

arXiv2019

Convergence analysis for parallel-in-time solution of hyperbolic systems

Parallel-in-time algorithms have been successfully employed for reducing time-to-solution of a variety of partial differential equations, especially for diffusive (parabolic-type) equations. A major failing of parallel-in-time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel-in-time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (1) space-time local Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time, and (3) a two-level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi-algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.

math.NA↗

arXiv2019

Local Fourier analysis for mixed finite-element methods for the Stokes equations

In this paper, we develop a local Fourier analysis of multigrid methods based on block-structured relaxation schemes for stable and stabilized mixed finite-element discretizations of the Stokes equations, to analyze their convergence behavior. Three relaxation schemes are considered: distributive, Braess-Sarazin, and Uzawa relaxation. From this analysis, parameters that minimize the local Fourier analysis smoothing factor are proposed for the stabilized methods with distributive and Braess-Sarazin relaxation. Considering the failure of the local Fourier analysis smoothing factor in predicting the true two-grid convergence factor for the stable discretization, we numerically optimize the two-grid convergence predicted by local Fourier analysis in this case. We also compare the efficiency of the presented algorithms with variants using inexact solvers. Finally, some numerical experiments are presented to validate the two-grid and multigrid convergence factors.

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↗

arXiv2017

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.

cond-mat.soft↗