Overlapping domain decomposition methods
 for linear inverse problems

Jiang, Daijun; Feng, Hui; Zou, Jun

doi:10.3934/ipi.2015.9.163

Cited by 10 publications

(6 citation statements)

References 17 publications

(27 reference statements)

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“…However, the flavor of this method is less familiar among the researchers of inverse problems for partial differential equations. Recently, the iterative thresholding algorithm was utilized in Jiang, Feng and Zou [17] to treat inverse problems for elliptic and parabolic equations. Very recently, based on the theoretical stability of Lipschitz type, we develop a similar iterative method for an inverse source problem in the three-dimensional time cone model in [24].…”

Section: Introductionmentioning

confidence: 99%

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Jiang

Liu

Yamamoto

2017

Journal of Differential Equations

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In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.

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Section: Introductionmentioning

confidence: 99%

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Jiang

Liu

Yamamoto

2017

Journal of Differential Equations

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“…here K > 0 is a tuning parameter, it acts as a weight between the previous step and the iterative update. The iteration (14) coincides with the iterative thresholding algorithm, which can be derived from the minimization problem of a surrogate functional. In their papers, Jiang et al [10] and Daubechies et al [22] introduce a surrogate functional that we exploit to discuss the choice of K guaranteeing the convergence.…”

Section: Mathematical Foundations and Proposed Algorithm A Mathematic...mentioning

confidence: 99%

“…In the line 4, g i denotes the gradient of E i w.r.t w computed in step i. The parameters w are updated according to (14) as shown in the line 5. The iteration is stopped when w i converged.…”

Section: B Proposed Algorithmmentioning

confidence: 99%

“…Iterative thresholding algorithm with its other generalizations have proven their feasibility, mainly in many image processing applications (see [11], [12], and [13]). In addition, this algorithm has been used by inverse problems' researchers for partial differential equations [14], [15], [16]. We will use the classical Tikhonov regularization to reformulate the objective function to be minimized.…”

Section: Introductionmentioning

confidence: 99%

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Itao: A New Iterative Thresholding Algorithm Based Optimizer for Deep Neural Networks

et al. 2022

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In this paper, we propose a new iterative thresholding algorithm based optimizer (Itao) for deep neural networks. It is a first-order gradient-based algorithm with Tikhonov regularization for stochastic objective functions. It is fast and straightforward to implement. It acts on the parameters and their gradients, with respect to the objective function, in only one step in the backpropagation system when training a neural network. This reduces the learning time and makes it well suited for neural networks with large parameters and/or large datasets. We have experimented this algorithm on several types of loss functions such as mean squared error, mean absolute error and categorical crossentropy. Different types of models such as regression and classification are studied. The robustness of this optimizer against the noisy labels is also verified. Many of the empirical results of conducted experiments in this study show that our optimizer works well in practice. It can outperform other state-of-the-art optimizers in terms of accuracy or at least give the same results in addition to the reduction of learning time. INDEX TERMSIterative thresholding algorithm based optimizer (Itao), deep learning, optimizer algorithm, neural networks.

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“…For the abstract formulation and convergence analysis of the algorithm, we refer to [9,10,31]. Attracted by its efficiency and robustness in many image processing problems, [18] first utilized the iterative thresholding algorithm to solve inverse problems for elliptic and parabolic equations. In [20,21,28], similar iteration methods were implemented to treat inverse source problems for hyperbolic-type equations with different types of observation data.…”

Section: Introductionmentioning

confidence: 99%

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

Liu

Yamamoto

2018

Inverse Problems

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In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with both partial boundary and interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.

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Overlapping domain decomposition methods for linear inverse problems

Cited by 10 publications

References 17 publications

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Itao: A New Iterative Thresholding Algorithm Based Optimizer for Deep Neural Networks

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

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