arXiv subjects

Niall Bootland

Publications and source records attributed to Niall Bootland.

15 recordsLinked to original sources

arXiv2024

Multipreconditioning with directional sweeping methods for high-frequency Helmholtz problems

We consider the use of multipreconditioning, which allows for multiple preconditioners to be applied in parallel, on high-frequency Helmholtz problems. Typical applications present challenging sparse linear systems which are complex non-Hermitian and, due to the pollution effect, either very large or else still large but under-resolved in terms of the physics. These factors make finding general purpose, efficient and scalable solvers difficult and no one approach has become the clear method of choice. In this work we take inspiration from domain decomposition strategies known as sweeping methods, which have gained notable interest for their ability to yield nearly-linear asymptotic complexity and which can also be favourable for high-frequency problems. While successful approaches exist, such as those based on higher-order interface conditions, perfectly matched layers (PMLs), or complex tracking of wave fronts, they can often be quite involved or tedious to implement. We investigate here the use of simple sweeping techniques applied in different directions which can then be incorporated in parallel into a multipreconditioned GMRES strategy. Preliminary numerical results on a two-dimensional benchmark problem will demonstrate the potential of this approach.

math.NA↗

arXiv2023

A robust and adaptive GenEO-type domain decomposition preconditioner for $\mathbf{H}(\mathbf{curl})$ problems in three-dimensional general topologies

In this paper we design, analyse and test domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$ problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. We also address the practical robustness of various solvers in the case of non-trivial topologies and/or high aspect ratio of the domain.

math.NA↗

arXiv2022

Numerical assessment of PML transmission conditions in a domain decomposition method for the Helmholtz equation

The convergence rate of domain decomposition methods (DDMs) strongly depends on the transmission condition at the interfaces between subdomains. Thus, an important aspect in improving the efficiency of such solvers is careful design of appropriate transmission conditions. In this work, we will develop an efficient solver for Helmholtz equations based on perfectly matched layers (PMLs) as transmission conditions at the interfaces within an optimised restricted additive Schwarz (ORAS) domain decomposition preconditioner, in both two and three dimensional domains. We perform a series of numerical simulations on a model problem and will assess the convergence rate and accuracy of our solutions compared to the situation where impedance boundary conditions are used.

math.NA↗

arXiv2022

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

math.NA↗

arXiv2021

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers which are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection-diffusion-reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalised eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts which are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the non-self-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus we obtain an iterative solver which is efficient in parallel and very effective for a wide range of convection--diffusion--reaction problems.

math.NA↗

arXiv2021

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell's equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm applied to the equations with absorption, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that can typically be encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.

math.NA↗

arXiv2021

Several ways to achieve robustness when solving wave propagation problems

Wave propagation problems are notoriously difficult to solve. Time-harmonic problems are especially challenging in mid and high frequency regimes. The main reason is the oscillatory nature of solutions, meaning that the number of degrees of freedom after discretisation increases drastically with the wave number, giving rise to large complex-valued problems to solve. Additional difficulties occur when the problem is defined in a highly heterogeneous medium, as is often the case in realistic physical applications. For time-discretised problems of Maxwell type, the main challenge remains the significant kernel in curl-conforming spaces, an issue that impacts on the design of robust preconditioners. This has already been addressed theoretically for a homogeneous medium but not yet in the presence of heterogeneities. In this review we provide a big-picture view of the main difficulties encountered when solving wave propagation problems, from the first step of their discretisation through to their parallel solution using two-level methods, by showing their limitations on a few realistic examples. We also propose a new preconditioner inspired by the idea of subspace decomposition, but based on spectral coarse spaces, for curl-conforming discretisations of Maxwell's equations in heterogeneous media.

math.NA↗

arXiv2021

A comparison of coarse spaces for Helmholtz problems in the high frequency regime

Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending on the problem and particular numerical setting.

math.NA↗

arXiv2021

Inexact subdomain solves using deflated GMRES for Helmholtz problems

We examine the use of a two-level deflation preconditioner combined with GMRES to locally solve the subdomain systems arising from applying domain decomposition methods to Helmholtz problems. Our results show that the direct solution method can be replaced with an iterative approach. This will be particularly important when solving large 3D high-frequency problems as subdomain problems can be too large for direct inversion or otherwise become inefficient. We additionally show that, even with a relatively low tolerance, inexact solution of the subdomain systems does not lead to a drastic increase in the number of outer iterations. As a result, it is promising that a combination of a two-level domain decomposition preconditioner with inexact subdomain solves could provide more economical and memory efficient numerical solutions to large-scale Helmholtz problems.

math.NA↗

arXiv2021

GenEO coarse spaces for heterogeneous indefinite elliptic problems

Motivated by recent work on coarse spaces for Helmholtz problems, we provide in this paper a comparative study on the use of spectral coarse spaces of GenEO type for heterogeneous indefinite elliptic problems within an additive overlapping Schwarz method. In particular, we focus here on two different but related formulations of local generalised eigenvalue problems and compare their performance numerically. Even though their behaviour seems to be very similar for several well-known heterogeneous test cases that are mildly indefinite, only one of the coarse spaces has so far been analysed theoretically, while the other one leads to a significantly more robust domain decomposition method when the indefiniteness is increased. We present a summary of upcoming results developing such a theory and describe how the numerical experiments illustrate it.

math.NA↗

arXiv2020

Two-level DDM preconditioners for positive Maxwell equations

In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations. Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement the near-kernel space made of the gradient of scalar functions. This extends the results in [2] to the variable coefficient case and non-convex domains at the expense of a larger coarse space.

math.NA↗

arXiv2020

Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews.

math.NA↗

arXiv2020

Multipreconditioning with application to two-phase incompressible Navier-Stokes flow

We consider the use of multipreconditioning to solve linear systems when more than one preconditioner is available but the optimal choice is not known. In particular, we consider a selective multipreconditioned GMRES algorithm where we incorporate a weighting that allows us to prefer one preconditioner over another. Our target application lies in the simulation of incompressible two-phase flow. Since it is not always known if a preconditioner will perform well within all regimes found in a simulation, we also consider robustness of the multipreconditioning to a poorly performing preconditioner. Overall, we obtain promising results with the approach.

math.NA↗

arXiv2019

On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

math.NA↗

arXiv2017

Preconditioners for Two-Phase Incompressible Navier-Stokes Flow

We consider iterative methods for solving the linearised Navier-Stokes equations arising from two-phase flow problems and the efficient preconditioning of such systems when using mixed finite element methods. Our target application is simulation within the Proteus toolkit; in particular, we will give results for a dynamic dam-break problem in 2D. We focus on a preconditioner motivated by approximate commutators which has proved effective, displaying mesh-independent convergence for the constant coefficient single-phase Navier-Stokes equations. This approach is known as the "pressure convection-diffusion" (PCD) preconditioner [H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, second ed., Oxford University Press, 2014]. However, the original technique fails to give comparable performance in its given form when applied to variable coefficient Navier-Stokes systems such as those arising in two-phase flow models. Here we develop a generalisation of this preconditioner appropriate for two-phase flow, requiring a new form for PCD. We omit considerations of boundary conditions to focus on the key features of two-phase flow. Before considering our target application, we present numerical results within the controlled setting of a simplified problem using a variety of different mixed elements. We compare these results with those for a straightforward extension to another commutator-based method known as the "least-squares commutator" (LSC) preconditioner, a technique also discussed in the aforementioned reference. We demonstrate that favourable properties of the original PCD and LSC preconditioners (without boundary adjustments) are retained with the new preconditioners in the two-phase situation.

math.NA↗