Computational approaches for parametric imaging of dynamic PET data

Crisci, Serena; Piana, Michele; Ruggiero, Valeria; Scussolini, Mara

doi:10.48550/arxiv.1908.11139

Cited by 2 publications

(5 citation statements)

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“…• the computation of the square root of the matrix B k , to evaluate the right-hand side in (8a), to compute the step p(λ k ) from z(λ k ) by ( 9), and to update the trust region radius in (10), • the computation of an approximation to z(λ k ) through (8) requires the solution of the secular equation (cf (55) below) via Newton's method [6,27]. This requires the solution of a sequence of linear system of the form (8a), which are large scale systems.…”

Section: Exact Methods Inmentioning

confidence: 99%

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An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Bellavia¹,

Donatelli²,

Riccietti³

2020

Inverse Problems

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In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The ad-hoc choice of the trust region radius update and the structure of the step resulting from the use of the Lanczos approach, allows us to prove some regularizing properties of the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

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Section: Exact Methods Inmentioning

confidence: 99%

“…Many approaches have been proposed in the literature for this class of problems, such as nonstationary iterated Tikhonov regularization or regularized versions of Newton method or trust region methods [2,3,8,17,18,[35][36][37].…”

Section: Introductionmentioning

confidence: 99%

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Bellavia¹,

Donatelli²,

Riccietti³

2020

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…and p k = p(λ k ), the couple (λ k , p k ) is a KKT point for (6). The solution of ( 6) can be then found by solving (7), and through relation (9).…”

Section: Preliminariesmentioning

confidence: 99%

“…Many approaches have been proposed in the literature to deal with this problem, such as nonstationary iterated Tikhonov regularization or regularized versions of trust region methods [1,2,14,28,29,15,6,30].…”

Section: Introductionmentioning

confidence: 99%

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An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

Bellavia¹,

Donatelli²,

Riccietti³

2019

Preprint

View full text Add to dashboard Cite

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The trust region radius is chosen to ensure the resulting Tikhonov regularization parameter to satisfy a prescribed condition on the model, which is proved to ensure regularizing properties to the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

show abstract

Computational approaches for parametric imaging of dynamic PET data

Cited by 2 publications

References 0 publications

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

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