Fourier Analysis of Periodic Stencils in Multigrid Methods

Summary For time‐dependent partial differential equations, parallel‐in‐time integration using the “parallel full approximation scheme in space and time” (PFASST) is a promising way to accelerate existing space‐parallel approaches beyond their scaling limits. Inspired by the classical Parareal method and multigrid ideas, PFASST allows to integrate multiple time steps simultaneously using a space–time hierarchy of spectral deferred correction sweeps. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. Very recently, however, PFASST for linear problems has been identified as a multigrid method. In this paper, we will use this multigrid formulation and, in particular, PFASST's iteration matrix to show that, in the nonstiff and stiff limit, PFASST indeed is a convergent iterative method. We will provide upper bounds for the spectral radius of the iteration matrix and investigate how PFASST performs for increasing numbers of parallel time steps. Finally, we will demonstrate that the results obtained here indeed relate to actual PFASST runs.

“…Proof The full iteration matrix T ( μ ) of PFASST can be written as

T (μ) = T (0) + O (μ) = E \otimes (H ({boldI}_{M N} - {boldT}_{C, Q}^{F} {boldT}_{F, Q}^{C} \otimes {boldT}_{C, A}^{F} {boldT}_{F, A}^{C})) + O (μ);

see the discussion leading to Theorem . As before, this yields

ρ (\lim_{L \to \infty} boldT false(μ false)) = ρ (\lim_{L \to \infty} boldT false(0 false)) = \sup_{x \in [- π, π]} ρ (\overset{boldT false(0 false)}{true} false(x false)),

following the work of Bolten et al Now, the symbol of the limit matrix T (0) is given by

\hat{T (0)} (x) = e^{- i x} H ({boldI}_{M N} - {boldT}_{C, Q}^{F} {boldT}_{F, Q}^{C} \otimes {boldT}_{C, A}^{F} {boldT}_{F, A}^{C}),

which makes the computation of the eigenvalues slightly more intricate. Using Theorem , we write

H ({boldI}_{M N})

…”

Section: Increasing the Number Of Time Stepsmentioning

confidence: 90%

Section: Increasing the Number Of Time Stepsmentioning

confidence: 98%

“…Following Equation , we write

\lim_{L \to \infty} {boldT}_{normalS} (μ) = \lim_{L \to \infty} {boldT}_{normalS} (0) + \lim_{L \to \infty} μ D

for some perturbation matrix

boldD \in R^{L M N \times L M N}

, which is bounded for μ small enough (i.e., for L large enough); see the discussion leading to . Now, this matrix D is bounded for all L so that, because μ →0 as L → ∞ , we have

\lim_{L \to \infty} {boldT}_{normalS} (μ) = \lim_{L \to \infty} {boldT}_{normalS} (0) .

The matrix T S (0) = E ⊗ H is a block Toeplitz matrix with periodic stencil and symbol

\hat{{boldT}_{normalS} (0)} (x) = e^{- i x} H

in the sense of the work of Bolten et al…”

Section: Increasing the Number Of Time Stepsmentioning

confidence: 99%

“…More precisely, there exists a transformation matrix

scriptF

, consisting of a permutation matrix for the Kronecker product and a Fourier matrix decomposing the spatial problem such that, for the smoother, we have

{boldT}_{normalS} = F diag ({boldB}_{1}, …, {boldB}_{N}) {scriptF}^{- 1}

for blocks

{boldB}_{n} = {boldI}_{L M} - {({boldI}_{L M} - μ λ_{n} {boldI}_{L} \otimes {boldQ}_{normalΔ})}^{- 1} ({boldI}_{L M} - μ λ_{n} {boldI}_{L} \otimes Q - E \otimes N) \in {double-struckR}^{L M \times L M} .

\begin{matrix} alignleft \\ align-1 \hat{{bold-italicB}_{n}} (x) & align-2 = {boldI}_{M} - {({boldI}_{M} - μ λ_{n} {boldQ}_{normalΔ})}^{- 1} ({boldI}_{M} - μ λ_{n} Q) + e^{- i x} N \\ align-1 & align-2 = {({boldI}_{M} - μ λ_{n} {boldQ}_{normalΔ})}^{- 1} μ λ_{n} (Q - {boldQ}_{normalΔ}) + e^{- i x} N . \end{matrix}

Then, for L → ∞ , we again use the work of Bolten et al and obtain

ρ (\lim_{L \to})

…”

Section: Increasing the Number Of Time Stepsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

Bolten

Moser

Speck

2018

Independence of placement for local Fourier analysis

2021

Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid‐transfer operator. In this work, we prove that the LFA representation for d‐dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid‐transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods.

Low‐order preconditioning of the Stokes equations

Voronin

MacLachlan

et al. 2021

A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor-Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the ℚ 1 iso ℚ 2 ∕ℚ 1 discretization of the Stokes operator as a preconditioner for the ℚ 2 ∕ℚ 1 discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the ℚ 2 ∕ℚ 1 system, our ultimate motivation is to apply algebraic multigrid within solvers for ℚ 2 ∕ℚ 1 systems via the ℚ 1 iso ℚ 2 ∕ℚ 1 discretization, which will be considered in a companion paper.