Iterative Solution of Linear Variational Problems in Hilbert Spaces: Some Conjugate Gradients Success Stories

Glowinski, Roland; Lapin, Sergey

doi:10.1007/978-3-642-18560-1_15

Cited by 3 publications

(7 citation statements)

References 10 publications

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“…Existence of such an index is justified using Lemma 2 for r = j 2 and ζ = ζ . Now, by (38) and using the same argument as in the proof of Lemma 2, where we have shown that from (22) implies (31), we can infer that 2 3 a +1 ≤ α k+j+2 ≤ δ sup for every j 2 ≤ j ≤ j 3 − 2.…”

Section: General Casementioning

confidence: 78%

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Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces

Azmi¹,

Kunisch²

2018

Preprint

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Due to simplicity, computational cheapness, and efficiency, the Barzilai and Borwein (BB) gradient method has received a significant amount of attention in different fields of optimization. In the first part of this paper, based on spectral analysis, R-linear global convergence for the BB-method is proven for strictly convex quadratic problems posed in infinite-dimensional Hilbert spaces. Then this result is strengthened to R-linear local convergence for a class of twice continuously Frećhet-differentiable functions. In the second part, aiming at problems governed by partial differential equations (PDE), the mesh-independent principle is investigated for the BB-method. The applicability of these results is demonstrated for three different types of PDE-constrained optimization problems. Numerical experiments illustrate the theoretical results.

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Section: General Casementioning

confidence: 78%

“…2. If α k+j+1 − λ < 0, then by (31) and using the fact that λ ≤ b +1 ≤ a +1 + η for λ ∈ I +1 , we obtain…”

Section: General Casementioning

confidence: 99%

Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces

Azmi¹,

Kunisch²

2018

Preprint

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“…There are several methods to solve Problems 2.1. In this article, we will make use of three methods: Landweber's (see, e.g., [15,Chapter 6]), Nesterov's (see [47,46]), and the conjugate gradient (CG) methods (see [29,27,13,34,26,20,48,16,4,22,42]). As we will see, the knowledge of the adjoint operator is essential for all of these iterative methods.…”

Section: Iterative Methods In Hilbert Spacesmentioning

confidence: 99%

Analysis of Iterative Methods in Photoacoustic Tomography with Variable Sound Speed

Haltmeier¹,

Nguyen²

2017

SIAM J. Imaging Sci.

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In this article, we revisit iterative methods for solving the inverse problem of photoacoustic tomography in free space. Recently, there have been interesting developments on explicit formulations of the adjoint operator, demonstrating that iterative methods is an attractive choice for photoacoustic image reconstruction. In this work, we propose several modifications of current formulations of the adjoint operator which help speed up the convergence and yield improved error estimates. We establish a stability analysis and show that, with our choices of the adjoint operator, the iterative methods can achieve a linear rate of convergence, in the L 2 -norm as well as in the H 1 -norm. In addition, we analyze the normal operator from the microlocal analysis point of view. This gives insight into the convergence speed of the iterative methods and choosing proper weights for the mapping spaces. Finally, we present numerical results using various iterative reconstruction methods for full as well as limited view data. Our results demonstrate that Nesterov's fast gradient and the CG methods converge faster than Landweber's and iterative time reversal methods in the visible as well as the invisible case.

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“…These methods were originally introduced in [20] and [30], respectively, for the case X = R n . Generalizations to infinite dimensional Hilbert spaces have been considered in [18,5,10,28,8,2,14,27] and the references therein, see also the references in [15]. In most of the above publications, a problem Bx = c was considered, where B was assumed to map X into itself.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, this complies with the present setting when we set B := RA, where R ∈ L(X * , X) denotes the Riesz isomorphism. This point of view was taken, at least implicitly, in [2,14,23] and in [3,Section 3]. In the present paper, we prefer the setting (1.1), and we keep the Riesz map explicit.…”

Section: Introductionmentioning

confidence: 99%

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Herzog¹,

Sachs²

2015

SIAM J. Numer. Anal.

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Iterative Solution of Linear Variational Problems in Hilbert Spaces: Some Conjugate Gradients Success Stories

Cited by 3 publications

References 10 publications

Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces

Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces

Analysis of Iterative Methods in Photoacoustic Tomography with Variable Sound Speed

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

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