Bayes Meets Krylov: Statistically Inspired Preconditioners for CGLS

Calvetti, Daniela; Pitolli, Francesca; Somersalo, Erkki; Vantaggi, Barbara

doi:10.1137/15m1055061

Cited by 35 publications

(32 citation statements)

References 43 publications

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“…As many numerical methods rely on linear solvers, such as the CG method, understanding the error incurred by these numerical methods is critical. Other works to recently highlight the value of statistical thinking in this application area includes Calvetti et al [2018].…”

Section: Probabilistic Numerical Methodsmentioning

confidence: 99%

A Bayesian Conjugate Gradient Method (with Discussion)

Cockayne¹,

Oates²,

Ipsen³

2019

Bayesian Anal.

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A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about, for example, the magnitude of the error. In this paper we propose a novel statistical model for this error, set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

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Section: Probabilistic Numerical Methodsmentioning

confidence: 99%

A Bayesian Conjugate Gradient Method (with Discussion)

Cockayne¹,

Oates²,

Ipsen³

2019

Bayesian Anal.

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“…To solve high dimensional problems the algorithm should be implemented in a more efficient way. For instance, the slight instabilities appearing near the initial point in Figure 7 can be avoided by solving the overdetermined linear system (17) by Krylov methods [24]. Moreover, the Jacobian matrix J G can be compressed by usual techniques [25].…”

Section: Discussionmentioning

confidence: 99%

A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Pitolli

2019

Algorithms

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Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.

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“…For the deterministic version of the ensemble Kalman filter [24], the ETKF which is illustrated in Algorithm 7, we again write the forecast error covariance in low rank form (33), with ensembles {x F k } r k=1 . It is then assumed that the state estimate x A is of the form x A = x F + X F w A , where w A is a vector of coefficients in the small dimensional ensemble subspace R r , and X F ∈ R n×r .…”

Section: Compute the Forecast Ensemble E F =  I (E A ) End Formentioning

confidence: 99%

Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

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Bayes Meets Krylov: Statistically Inspired Preconditioners for CGLS

Cited by 35 publications

References 43 publications

A Bayesian Conjugate Gradient Method (with Discussion)

A Bayesian Conjugate Gradient Method (with Discussion)

A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Numerical linear algebra in data assimilation

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