An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

Baumann, Manuel; Rengifo, R.A. Astudillo; Qiu, Yue; Ang, Yew-Hock; Gijzen, Martin B. van; Plessix, R. E.

doi:10.1007/s10596-017-9667-7

Cited by 11 publications

(34 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The multilevel sequentially semiseparable (MSSS) preconditioning technique was first studied for PDE‐constrained optimization problems in Reference 27 and some benchmark problems of computational fluid dynamics problems, 28 and later extended to computational geoscience problems 68 . The global MSSS preconditioner computes an approximate factorization of the global (generalized) saddle‐point matrix up to a prescribed accuracy in linear computational complexity using MSSS matrix computations.…”

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

“…This is shown by Figure 3. For higher order FEM discretization, the MSSS structure can also be obtained, we refer to Reference 68 for more details.…”

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

“…In this article, we focus on 2D problems where the Schur complement is 1D and can be approximated by one‐level MSSS matrix computations. For the application of MSSS preconditioners for 3D problems, we refer to References 27,68. Moreover, we have the following theorem that gives the bound of the MSSS preconditioned spectrum for 2D problems.…”

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

See 2 more Smart Citations

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Qiu

Gijzen

Wingerden

et al. 2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

show abstract

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

“…This is shown by Figure 3. For higher order FEM discretization, the MSSS structure can also be obtained, we refer to Reference 68 for more details.…”

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

Section: Multilevel Sequentially Semiseparable Preconditionersmentioning

confidence: 99%

See 1 more Smart Citation

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Qiu

Gijzen

Wingerden

et al. 2020

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…Alternatively, to keep the number of reconstruction parameters to a minimum and to minimize possible stability issues, preconditioners can be used to reduce the number of iterations required for solving the least squares problem . The incomplete Cholesky factorization and hierarchically structured matrices are examples of preconditioners that reduce the number of iterations drastically in many applications . The drawback of these type of preconditioners is that the full system matrix needs to be built before the reconstruction starts, which for larger problem sizes can only be done on a very powerful computer due to memory limitations.…”

Section: Introductionmentioning

confidence: 99%

Accelerating compressed sensing in parallel imaging reconstructions using an efficient circulant preconditioner for cartesian trajectories

Koolstra

Gemert

Börnert

et al. 2018

Magnetic Resonance in Med

View full text Add to dashboard Cite

PurposeDesign of a preconditioner for fast and efficient parallel imaging (PI) and compressed sensing (CS) reconstructions for Cartesian trajectories.TheoryPI and CS reconstructions become time consuming when the problem size or the number of coils is large, due to the large linear system of equations that has to be solved in ℓ1 and ℓ2‐norm based reconstruction algorithms. Such linear systems can be solved efficiently using effective preconditioning techniques.MethodsIn this article we construct such a preconditioner by approximating the system matrix of the linear system, which comprises the data fidelity and includes total variation and wavelet regularization, by a matrix that is block circulant with circulant blocks. Due to this structure, the preconditioner can be constructed quickly and its inverse can be evaluated fast using only two fast Fourier transformations. We test the performance of the preconditioner for the conjugate gradient method as the linear solver, integrated into the well‐established Split Bregman algorithm.ResultsThe designed circulant preconditioner reduces the number of iterations required in the conjugate gradient method by almost a factor of 5. The speed up results in a total acceleration factor of approximately 2.5 for the entire reconstruction algorithm when implemented in MATLAB, while the initialization time of the preconditioner is negligible.ConclusionThe proposed preconditioner reduces the reconstruction time for PI and CS in a Split Bregman implementation without compromising reconstruction stability and can easily handle large systems since it is Fourier‐based, allowing for efficient computations.

show abstract

“…that have arisen as a natural algebraic model for discretized partial differential equations, possibly including stochastic terms or parameter dependent coefficient matrices [4,8,28,30], for PDEconstrained optimization problems [39], data assimilation [13], and many other applied contexts, including building blocks of other numerical procedures [23]; see also [35] for further references. The general matrix equation (1.2) covers two well known cases, the (generalized) Sylvester equation (for = 2), and the Lyapunov equation…”

mentioning

confidence: 99%

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

Palitta

Simoncini

2020

Vietnam J. Math.

View full text Add to dashboard Cite

Galerkin and Petrov-Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry over to the (Petrov-)Galerkin methods applied for the solution of linear matrix equations.Some novel considerations about the use of Galerkin and Petrov-Galerkin schemes in the numerical treatment of general linear matrix equations are expounded and the use of constrained minimization techniques in the Petrov-Galerkin framework is proposed.

show abstract

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

Cited by 11 publications

References 47 publications

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Accelerating compressed sensing in parallel imaging reconstructions using an efficient circulant preconditioner for cartesian trajectories

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

Contact Info

Product

Resources

About