On the Similarities Between the Quasi-Newton Inverse Least Squares Method and GMRes

Haelterman, Rob; Degroote, Joris; Heule, Dirk Van; Vierendeels, Jan

doi:10.1137/090750354

Cited by 26 publications

(18 citation statements)

References 10 publications

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“…is the solution obtained at the k-th step of the full GMRES [71]. Indeed, as proved in [44,Eq. (3.3)], the iterates of the full GMRES for solving the system Ax = (I − M )x = d can be written as a Schur complement.…”

Section: The Reduced Rank Extrapolationmentioning

confidence: 95%

Shanks Sequence Transformations and Anderson Acceleration

Brezinski¹,

Redivo–Zaglia²,

Saad³

2018

SIAM Rev.

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This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that the Minimal Polynomial Extrapolation (MPE), the Modified Minimal Polynomial Extrapolation (MMPE), the Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard general techniques designed for accelerating arbitrary sequences and/or solving nonlinear equations, all fall into this framework. Their properties and their connections with quasi-Newton and Broyden methods are studied. The paper then exploits this framework to compare these methods. In the linear case, it is known that AA and GMRES are 'essentially' equivalent in a certain sense while GMRES and RRE are mathematically equivalent. This paper discusses the connection between AA, the RRE, the MPE, and other methods in the nonlinear case.

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Section: The Reduced Rank Extrapolationmentioning

confidence: 95%

Shanks Sequence Transformations and Anderson Acceleration

Brezinski¹,

Redivo–Zaglia²,

Saad³

2018

SIAM Rev.

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“…The interface quasi-Newton method with an approximation of the inverse of the Jacobian from a least-squares model (IQN-ILS) was introduced by Degroote et al [13] to couple the interface variables of the flow solver and the structural solver in a partitioned fluid-structure interaction simulation. This method was also analyzed for linear problems by Haelterman et al [14] in a more general framework, where the method was called QN-ILS. In this paper we also call the method QN-ILS as the position is computed for all material points and not only for those lying on the interface.…”

Section: Quasi-newton Strategy With An Approximation Of the Jacobian'mentioning

confidence: 99%

“…This means that the rank of the matrices W i would always be equal to one, what would prohibit the convergence of the method since then the rank of the approximation of the inverse of the Jacobian will also become at maximum one. However, for a linear system it has been proven that the use of equations (2.21) and (2.22) returns an exact Jacobian -and thus a converged result -in at most n+1 iterations, with n the number of degrees of freedom of the problem [14]. Note, however, that in practical applications the convergence rate is much higher.…”

Section: Quasi-newton Strategy With An Approximation Of the Jacobian'mentioning

confidence: 99%

Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry andin vivostress incorporation

Bols

Degroote

Trachet³

et al. 2013

ESAIM: M2AN

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Abstract.In vivo visualization of cardiovascular structures is possible using medical images. However, one has to realize that the resulting 3D geometries correspond to in vivo conditions. This entails an internal stress state to be present in the in vivo measured geometry of e.g. a blood vessel due to the presence of the blood pressure. In order to correct for this in vivo stress, this paper presents an inverse method to restore the original zero-pressure geometry of a structure, and to recover the in vivo stress field of the final, loaded structure. The proposed backward displacement method is able to solve the inverse problem iteratively using fixed point iterations, but can be significantly accelerated by a quasiNewton technique in which a least-squares model is used to approximate the inverse of the Jacobian. The here proposed backward displacement method allows for a straightforward implementation of the algorithm in combination with existing structural solvers, even if the structural solver is a black box, as only an update of the coordinates of the mesh needs to be performed.Mathematics Subject Classification. 65D18, 74L15, 49Q10, 65N21, 90C53.

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“…The only requirement for the relaxation parameter ω is that it avoids excessive divergence in the second coupling iteration, which could cause errors in the solvers, for example due to a corrupted mesh. Previous research has shown that the IQN-ILS method is robust with respect to this parameter and that it does not influence the long-term convergence [51]. A typical value is ω = 10 −2 .…”

Section: Coupling Iterationsmentioning

confidence: 99%

Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem

Degroote

Hojjat

Stavropoulou

et al. 2012

Struct Multidisc Optim

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Unsteady fluid-structure interaction (FSI) simulations are generally time-consuming. Gradient-based methods are preferred to minimise the computational cost of parameter identification studies (and more in general optimisation) with a high number of parameters. However, calculating the cost function's gradient using finite differences becomes prohibitively expensive for a high number of parameters. Therefore, the adjoint equations of the unsteady FSI problem are solved to obtain this gradient at a cost almost independent of the number of parameters.Here, both the forward and the adjoint problems are solved in a partitioned way, which means that the flow equations and the structural equations are solved separately. The application of interest is the identification of the arterial wall's stiffness by comparing the motion of the arterial wall with a reference, possibly obtained from non-invasive imaging. Due to the strong interaction between the fluid and the structure, quasi-Newton coupling iterations are applied to stabilise the partitioned solution of both the forward and the adjoint problem.

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On the Similarities Between the Quasi-Newton Inverse Least Squares Method and GMRes

Cited by 26 publications

References 10 publications

Shanks Sequence Transformations and Anderson Acceleration

Shanks Sequence Transformations and Anderson Acceleration

Inverse modelling of image-based patient-specific blood vessels: zero-pressure geometry andin vivostress incorporation

Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem

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