A characterization of the behavior of the Anderson acceleration on linear problems

Potra, Florian A.; Engler, Hans

doi:10.1016/j.laa.2012.09.008

Cited by 54 publications

(53 citation statements)

References 6 publications

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“…Anderson acceleration is related to multisecant methods (extensions of quasi-Newton methods involving multiple secant conditions); in fact, Eyert [16] proves that it is equivalent to the so-called "bad" Broyden's method [11,28], and a similar analysis is done by Fang and Saad [17] and Rohwedder and Schneider [43]. For linear systems, if m k = k for each k then Anderson acceleration is essentially equivalent to the generalized minimal residual (GMRES) method [44], as shown by Potra and Engler [36], Rohwedder and Schneider [43], and Walker and Ni [50]. For nonlinear problems Rohwedder and Schneider [43] show that Anderson acceleration is locally linearly convergent under certain conditions.…”

Section: Introductionmentioning

confidence: 84%

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Higham

Strabić

2015

Numer Algor

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In a wide range of applications it is required to compute the nearest correlation matrix in the Frobenius norm to a given symmetric but indefinite matrix. Of the available methods with guaranteed convergence to the unique solution of this problem the easiest to implement, and perhaps the most widely used, is the alternating projections method. However, the rate of convergence of this method is at best linear, and it can require a large number of iterations to converge to within a given tolerance. We show that Anderson acceleration, a technique for accelerating the convergence of fixed-point iterations, can be applied to the alternating projections method and that in practice it brings a significant reduction in both the number of iterations and the computation time. We also show that Anderson acceleration remains effective, and indeed can provide even greater improvements, when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. Alternating projections is a general method for finding a point in the intersection of several sets and ours appears to be the first demonstration that this class of methods can benefit from Anderson acceleration.

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Section: Introductionmentioning

confidence: 84%

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Higham

Strabić

2015

Numer Algor

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“…From a mathematical perspective, Pulay's technique can be considered to be a multisecant type method [4] that represents a specific variant of Broyden's approach [13]. In the linear setting, the DIIS approach bears remarkable similarity to the Generalized Minimal Residual (GMRES) method [28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Restarted Pulay mixing for efficient and robust acceleration of fixed-point iterations

Pratapa

Suryanarayana

2015

Chemical Physics Letters

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a b s t r a c tWe present a variant of the restarted Pulay's Direct Inversion in the Iterative Subspace (DIIS) method for efficiently and robustly accelerating the convergence of fixed-point iterations. Specifically, we propose a simple modification of DIIS without any additional parameters, which we refer to as the r-Pulay method. We demonstrate the efficacy of r-Pulay in the context of the Jacobi iteration for solving large linear systems of equations, as well as in the Self Consistent Field (SCF) approach for Density Functional Theory (DFT) calculations. Overall, we find r-Pulay to be an attractive version of the restarted DIIS method.

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“…Similarly to other works on AR, it is possible to identify a connection between AAR and GMRES . Let us assume again that m = k at first.…”

Section: Aar Methodsmentioning

confidence: 99%

“…For the purpose of comparing the way full AAR and full GMRES work, we need to introduce some quantities that are employed to describe the behavior of the methods. Following the work of Potra and Engler, we introduce the stagnation index as

η_{G} = \min \{ℓ \in N : x_{G}^{ℓ} = x_{G}^{ℓ + 1}\},

where

{boldx}_{G}^{ℓ}

is the approximate solution computed by full GMRES at the ℓ th iteration. In the work of Wilkinson, the grade of r 0 ≠ 0 with respect to A is defined as

ν (A, r^{0}) = \max \{ℓ \in N : \dim (K_{ℓ} (A, r^{0})) = ℓ\},

with

{scriptK}_{p} false(A, {boldr}^{0} false)

being the p ‐dimensional Krylov subspace defined on matrix A and vector r 0 .…”

Section: Aar Methodsmentioning

confidence: 99%

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Convergence analysis of Anderson‐type acceleration of Richardson's iteration

Pasini

2019

Numerical Linear Algebra App

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Summary We consider Anderson extrapolation to accelerate the (stationary) Richardson iterative method for sparse linear systems. Using an Anderson mixing at periodic intervals, we assess how this benefits convergence to a prescribed accuracy. The method, named alternating Anderson–Richardson, has appealing properties for high‐performance computing, such as the potential to reduce communication and storage in comparison to more conventional linear solvers. We establish sufficient conditions for convergence, and we evaluate the performance of this technique in combination with various preconditioners through numerical examples. Furthermore, we propose an augmented version of this technique.

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A characterization of the behavior of the Anderson acceleration on linear problems

Cited by 54 publications

References 6 publications

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Restarted Pulay mixing for efficient and robust acceleration of fixed-point iterations

Convergence analysis of Anderson‐type acceleration of Richardson's iteration

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