A parallel multilevel domain decomposition method for source identification problems governed by elliptic equations

Deng, Xiaomao; Liao, Zi-Ju; Cai, Xiao‐Chuan

doi:10.1016/j.cam.2021.113441

Cited by 3 publications

(1 citation statement)

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“…It should be noticed that a new variational approach for recovering a general source in elliptic system has been considered using interior measurement data in [19], which is analogous to our model. In [8], a multilevel domain decomposition method was proposed for 2D steady source identification problem governed by elliptic equations. All of the above numerical schemes for recovering the unknown sources are based on the evaluations of f (x) on grid points of Ω.…”

Section: Introductionmentioning

confidence: 99%

Solving an inverse source problem by deep neural network method with convergence and error analysis

Zhang,

Liu

2023

Inverse Problems

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For the inverse source problem of an elliptic system using noisy internal measurement as inversion input, we approximate its solution by neural network function, which is obtained by optimizing an empirical loss function with appropriate regularizing terms. We analyze the convergence of the general loss from noisy inversion input data in Deep Galerkin Method by the regularizing empirical loss function. Based on the upper bound of the expected loss function by its regularizing empirical form, we establish the upper bound of the expected loss function at the minimizer of the regularizing empirical noisy loss function in terms of the number of sampling points as well as the noise level quantitatively, for suitably chosen regularizing parameters and regularizing terms. Then, by specifying the number of sampling points in terms of noise level of inversion input data, we establish the error orders representing the diﬀerence between the neural network solution and the exact one, under some a-priori restrictions on the source. Finally, we give numerical implementations for several examples to verify our theoretical results.

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Section: Introductionmentioning

confidence: 99%