Matching Schur Complement Approximations for Certain Saddle-Point Systems

Pearson, John W.; Wathen, Andrew J.

doi:10.1007/978-3-319-72456-0_44

Cited by 10 publications

(12 citation statements)

References 31 publications

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“…To estimate bounds for the eigenvalues of Q h , we highlight that this is similar to the case discussed in [31], but the conclusion reached does not directly apply to the matrices highlighted in this paper, due to the differing structures and algebraic properties of this matrix system. More specifically, in our case we have thatǏ h ̸ =Î h , and that L h + L ⊺ h is indefinite, which therefore will lead to eigenvalues spreading out of the desirable uniform interval [ 1 2 , 1] obtained in [31,Theorem 1]. For the upcoming analysis, we first recall the following lemma that provides a uniform lower bound for the eigenvalues of the preconditioned Schur complement for a very general matrix form.…”

Section: Eigenvalues Of the Preconditioned System: Lower Boundsmentioning

confidence: 90%

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Parameter-robust preconditioning for the optimal control of the wave equation

Pearson

2019

Numer Algor

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In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh-sizes in the spatial variables.

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Section: Eigenvalues Of the Preconditioned System: Lower Boundsmentioning

confidence: 90%

“…Inspired by the idea of matching Schur complement approximations, see e.g., [9,11,27,31], we propose a new Schur-complement type preconditioner B h in the following factorization form:…”

Section: A New Preconditioner With Eigenvalue Analysismentioning

confidence: 99%

Parameter-robust preconditioning for the optimal control of the wave equation

Pearson

2019

Numer Algor

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“…From (24), and due to the convenient structure of the matrices

A_{D}

and

C_{D}

in (19), we can devise an approximation

{\tilde{S}}_{int}

S_{int}

using the “matching strategy” discussed in References 5,45 as follows. We seek an approximation:

{\tilde{S}}_{int} = (Λ_{21} + \hat{M}) A^{- 1} {(Λ_{21} + \hat{M})}^{⊤} \approx S_{int}

such that

{\tilde{S}}_{int}

“captures” both terms of

S_{int}

, namely

C_{D}

and

B_{D} A_{D}^{- 1} B_{D}^{⊤}

(noting that

Λ_{21} A^{- 1} Λ_{21}^{⊤} = B_{D} A_{D}^{- 1} B_{D}^{⊤}

).…”

Section: Preconditioning Approachmentioning

confidence: 99%

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Leveque

Pearson

2021

Numerical Linear Algebra App

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In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all‐at‐once solution of optimal control problems with time‐dependent PDEs as constraints, including the heat equation and the non‐steady convection–diffusion equation. After applying an optimize‐then‐discretize approach, one is faced with continuous first‐order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first‐order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second‐order accurate) Crank–Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle‐point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation—these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely‐used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh‐sizes, regularization parameter, and diffusion coefficient.

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“…In subsection 5.1, we provide a full characterization of the spectrum of the preconditioned system which highlights both the dependence on β and on the random field extremal values. In subsection 5.2, we propose instead a β-robust preconditioner based on a more involved factorization of the Schur complement of (4.6), inspired by works on deterministic OCPs in [41,47,46,27,31,42] 5.1. A first Schur complement approximation.…”

Section: Algebraic Preconditionersmentioning

confidence: 99%

“…The first one, used to derive the preconditioner P LR is algebraic and has its roots in the seminal work of [38], where the authors proposed an optimal, but expensive, preconditioner for saddle point matrices which relies on the exact Schur complement. For deterministic OCP, several preconditioners based on approximations of the exact Schur complements have been study in the last decade [47,46,41,27,31,42]. Concerning OCPUU, the same strategy has been applied in [3] in combination with SGMs, but we remark once more that their problem is not an instance of robust OCP.…”

mentioning

confidence: 99%

Preconditioners for robust optimal control problems under uncertainty

Nobile¹,

Vanzan²

2021

Preprint

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The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient κ(x, ω) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables 1/ min x∈D κ(x, ω) and max x∈D κ(x, ω), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

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Matching Schur Complement Approximations for Certain Saddle-Point Systems

Cited by 10 publications

References 31 publications

Parameter-robust preconditioning for the optimal control of the wave equation

Parameter-robust preconditioning for the optimal control of the wave equation

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Preconditioners for robust optimal control problems under uncertainty

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