Generalized Hybrid Iterative Methods for Large-Scale Bayesian Inverse Problems

Chung, Julianne; Saibaba, Arvind K.

doi:10.1137/16m1081968

Cited by 31 publications

(58 citation statements)

References 44 publications

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“…For genLSQR and genHyBR, we used Q = Q t ⊗ Q s where Q t and Q s correspond to C T (·) = C 1.5,.3 (·) and C S (·) = C .5,.007 (·) respectively. Relative errors per iteration provided in Figure 5 reveal similar behavior as that described in [8]. In particular, LSQR and genLSQR are plagued by semiconvergence (i.e., the "U"-shaped error curve that results from noise contamination during inversion), which can be avoided in the hybrid variants with the selection of the optimal regularization parameter.…”

Section: Space-time Image Deblurringmentioning

confidence: 52%

“…where the gen-GK relations were used to obtain the equivalence. Variants of this formulation, e.g., for LSMR, could be used as well [8]. After computing a solution to the projected problem, an approximate MAP estimate can be recovered by undoing the change of variables,…”

Section: Generalized Hybrid Methodsmentioning

confidence: 99%

“…where, now, s k ∈ µ + QS k . For fixed λ and in exact arithmetic, iterates of the genLSQR approach are mathematically equivalent to some pre-existing solvers (e.g., filtered GSVD solutions and priorconditioned solutions) [8]. However, if λ is not known a priori, hybrid methods can take advantage of the shift-invariance property of Krylov subspaces and select λ adaptively and automatically by utilizing well-known SVD-based regularization parameter selection schemes [22,20,51] for the projected problem (15), since B k is only of size (k+1)×k.…”

Section: Generalized Hybrid Methodsmentioning

confidence: 99%

“…Hybrid iterative methods for approximating x(λ) were described in [8] for generic A, R, and Q. For completeness, we give a brief description of these methods in Section 2.2 and refer the interested reader to the paper for more details.…”

Section: Problem Formulationsmentioning

confidence: 99%

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Efficient generalized Golub–Kahan based methods for dynamic inverse problems

et al. 2018

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We consider efficient methods for computing solutions to and estimating uncertainties in dynamic inverse problems, where the parameters of interest may change during the measurement procedure. Compared to static inverse problems, incorporating prior information in both space and time in a Bayesian framework can become computationally intensive, in part, due to the large number of unknown parameters. In these problems, explicit computation of the square root and/or inverse of the prior covariance matrix is not possible. In this work, we develop efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. We demonstrate that these methods can be more flexible than standard methods and develop efficient implementations that can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples from photoacoustic tomography, deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of the described approaches. Specifically, in passive seismic tomography, we demonstrate our approach on both synthetic and real data. To demonstrate the scalability of our algorithm, we solve a dynamic inverse problem with approximately 43, 000 measurements and 7.8 million unknowns in under 40 seconds on a standard desktop.

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Section: Space-time Image Deblurringmentioning

confidence: 52%

Section: Generalized Hybrid Methodsmentioning

confidence: 99%

Section: Generalized Hybrid Methodsmentioning

confidence: 99%

Section: Problem Formulationsmentioning

confidence: 99%

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Efficient generalized Golub–Kahan based methods for dynamic inverse problems

et al. 2018

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“…is A-weighted pseudoinverse of L and L † A = L † when p ≥ n; see [10] for details. This is computationally viable and attractive if not much effort is needed by applying L † A , e.g., when L is banded with small bandwidth and has a known null space; we refer the reader to, e.g., [4,5,8] for some available algorithms and codes. In many practical applications, however, such transformation is computationally unfeasible.…”

mentioning

confidence: 99%

Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization

Jia

Yang

2018

Inverse Problems

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Based on the joint bidiagonalization process of a large matrix pair {A, L}, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: min Lx subject to x ∈ S = {x| Ax − b ≤ τ e } with a Gaussian white noise e and τ > 1 slightly, where L is a regularization matrix. Our algorithm is different from the hybrid one proposed by Kilmer et al., which is based on the same process but solves the generalform Tikhonov regularization problem: minx Ax − b 2 + λ 2 Lx 2 . We prove that the iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly. This result and the analysis on it show that the method must have the desired semi-convergence property and get insight into the regularizing effects of the method. We use the L-curve criterion or the discrepancy principle to determine k * . The algorithm is simple and effective, and numerical experiments illustrate that it often computes more accurate regularized solutions than the hybrid one.

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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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Generalized Hybrid Iterative Methods for Large-Scale Bayesian Inverse Problems

Cited by 31 publications

References 44 publications

Efficient generalized Golub–Kahan based methods for dynamic inverse problems

Efficient generalized Golub–Kahan based methods for dynamic inverse problems

Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization

Numerical linear algebra in data assimilation

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