LAP: A Linearize and Project Method for Solving Inverse Problems with Coupled Variables

Herring, James L.; Nagy, James G.; Ruthotto, Lars

doi:10.1007/bf03549661

Cited by 7 publications

(10 citation statements)

References 30 publications

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“…A generalization of the variable projection approach has been considered in [52], which deals with two separate classes of variables without requiring one class to be linear; see also more recent contributions on the generalization of VarPro in e.g. [1,54,28,61]. We would also like to mention the simplification of the Jacobian matrix in [51], and the algorithm of [42], which resembles the variable projection approach in some sense; see a comparison of these algorithms with the variable projection method in [22].…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks

Dong,

Yang

2022

Preprint

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We present a method for solving linear and nonlinear partial differential equations (PDE) based on the variable projection framework and artificial neural networks. For linear PDEs, enforcing the boundary/initial value problem on the collocation points gives rise to a separable nonlinear least squares problem about the network coefficients. We reformulate this problem by the variable projection approach to eliminate the linear output-layer coefficients, leading to a reduced problem about the hidden-layer coefficients only. The reduced problem is solved first by the nonlinear least squares method to determine the hidden-layer coefficients, and then the output-layer coefficients are computed by the linear least squares method. For nonlinear PDEs, enforcing the boundary/initial value problem on the collocation points gives rise to a nonlinear least squares problem that is not separable, which precludes the variable projection strategy for such problems. To enable the variable projection approach for nonlinear PDEs, we first linearize the problem with a Newton iteration, using a particular linearization formulated in terms of the updated approximation field. The linearized system is solved by the variable projection framework together with artificial neural networks. Upon convergence of the Newton iteration, the neural-network coefficients provide the representation of the solution field to the original nonlinear problem. We present ample numerical examples with linear and nonlinear PDEs to demonstrate the performance of the method developed herein. For smooth field solutions, the errors of the current method decrease exponentially as the number of collocation points or the number of output-layer coefficients increases. We compare extensively the current method with the extreme learning machine (ELM) method from a previous work. Under identical conditions and network configurations, the current method exhibits an accuracy significantly superior to the ELM method.

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Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks

Dong,

Yang

2022

Preprint

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“…To solve the tightly coupled (x, w)-subproblem (3.5), we develop a variation of the Linearize And Project (LAP) method proposed by Herring et al [9]. The LAP method is efficient for inverse problems with multiple, tightly coupled blocks of variables such as the problem under consideration.…”

Section: Lap Methodmentioning

confidence: 99%

“…In this paper, we present the LAP method based on the normal equation approach instead of the least squares approach presented in the original paper [9]. First, we consider the unconstrained problem where C x = R n 2 , Ĉw = R p .…”

Section: Lap Methodmentioning

confidence: 99%

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An ADMM-LAP method for total variation myopic deconvolution of adaptive optics retinal images

Chen,

Herring,

Nagy

et al. 2020

Preprint

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Adaptive optics (AO) corrected flood imaging of the retina is a popular technique for studying the retinal structure and function in the living eye. However, the raw retinal images are usually of poor contrast and the interpretation of such images requires image deconvolution. Different from standard deconvolution problems where the point spread function (PSF) is completely known, the PSF in these retinal imaging problems is only partially known which leads to the more complicated blind deconvolution problem. In this paper, we propose an efficient numerical scheme for solving this blind deconvolution problem with total variational (TV) regularization. First, we apply the alternating direction method of multipliers (ADMM) to tackle the TV regularizer. Specifically, we reformulate the TV problem as an equivalent equality constrained problem where the objective function is separable, and then minimize the augmented Lagrangian function by alternating between two (separated) blocks of unknowns to obtain the solution. Due to the structure of the retinal images, the subproblems with respect to the fidelity term appearing within each ADMM iteration are tightly coupled and a variation of the Linearize And Project (LAP) method is designed to solve these subproblems efficiently. The proposed method is called the ADMM-LAP method. Both the theoretical complexity analysis and numerical results are provided to demonstrate the efficiency of the ADMM-LAP method.

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“…Eq. ( 1) is a separable nonlinear least squares problem (Gan et al, 2018;Herring et al, 2018). We confront it by iteratively addressing the subproblems:…”

Section: Aligned Reconstructionmentioning

confidence: 99%

Motion‐corrected MRI with DISORDER: Distributed and incoherent sample orders for reconstruction deblurring using encoding redundancy

Cordero‐Grande

Ferrazzi

Teixeira

et al. 2020

Magnetic Resonance in Med

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Purpose: To enable rigid body motion-tolerant parallel volumetric magnetic resonance imaging by retrospective head motion correction on a variety of spatiotemporal scales and imaging sequences. Theory and methods: Tolerance against rigid body motion is based on distributed and incoherent sampling orders for boosting a joint retrospective motion estimation and reconstruction framework. Motion resilience stems from the encoding redundancy in the data, as generally provided by the coil array. Hence, it does not require external sensors, navigators or training data, so the methodology is readily applicable to sequences using 3D encodings. Results: Simulations are performed showing full inter-shot corrections for usual levels of in vivo motion, large number of shots, standard levels of noise and moderate acceleration factors. Feasibility of inter-and intra-shot corrections is shown under controlled motion in vivo. Practical efficacy is illustrated by high-quality results in most corrupted of 208 volumes from a series of 26 clinical pediatric examinations collected using standard protocols. Conclusions: The proposed framework addresses the rigid motion problem in volumetric anatomical brain scans with sufficient encoding redundancy which has enabled reliable pediatric examinations without sedation. K E Y W O R D S distributed and incoherent sampling, image reconstruction, magnetic resonance, motion correction, parallel imaging 714 | CORDERO-GRANDE Et Al.

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LAP: A Linearize and Project Method for Solving Inverse Problems with Coupled Variables

Cited by 7 publications

References 30 publications

Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks

Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks

An ADMM-LAP method for total variation myopic deconvolution of adaptive optics retinal images

Motion‐corrected MRI with DISORDER: Distributed and incoherent sample orders for reconstruction deblurring using encoding redundancy

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