An efficient method to set up a Lanczos based preconditioner for discrete ill-posed problems

Erfani, Shervan; Tavakoli, Ali; Salkuyeh, Davod Khojasteh

doi:10.1016/j.apm.2013.03.064

Cited by 5 publications

(3 citation statements)

References 9 publications

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“…We can gain more insight by plotting the coefficients |u T i b| /σ i , the projections |u T i b|, and σ i together. This is known as a discrete Picard plot [39], and is shown in Figure (13). In order to…”

Section: Computed Sensitivitymentioning

confidence: 99%

“…To mitigate the effect of ill-conditioning on the sensitivity d J/ds for a given system, we can either use (39) with a suitable choice of l or regularise the MSS system. Regularisation has a similar filtering effect, i.e.…”

Section: Regularization Of the Preconditioned Systemmentioning

confidence: 99%

“…Figure(12) shows d ū /dc for the KS equation, obtained when different values of l are used to terminate the summation(39). On the same plot, we superimpose the corresponding singular values (right vertical axis).…”

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confidence: 99%

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A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

Shawki

Papadakis

2019

Journal of Computational Physics

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We propose a preconditioner that can accelerate the rate of convergence of the Multiple Shooting Shadowing (MSS) method [1]. This recently proposed method can be used to compute derivatives of time-averaged objectives (also known as sensitivities) to system parameter(s) for chaotic systems. We propose a block diagonal preconditioner, which is based on a partial singular value decomposition of the MSS constraint matrix. The preconditioner can be computed using matrix-vector products only (i.e. it is matrix-free) and is fully parallelised in the time domain. Two chaotic systems are considered, the Lorenz system and the 1D Kuramoto Sivashinsky equation. Combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range. This combination results in a significant reduction in the number of iterations, and renders the convergence rate almost independent of the number of degrees of freedom of the system, and the length of the trajectory that is used to compute the time-averaged objective. This can potentially allow the method to be used for large chaotic systems (such as turbulent flows) and optimal control applications. The singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities. In fact, neglecting the singular modes affected by noise, we recover the correct values of sensitivity that match closely with those obtained with finite differences for the Kuramoto Sivashinsky equation in the light turbulent regime.

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Section: Computed Sensitivitymentioning

confidence: 99%

Section: Regularization Of the Preconditioned Systemmentioning

confidence: 99%