A numerical method for inverse source problems for Poisson and Helmholtz equations

Hamad, Ayman; Tadi, M.

doi:10.1016/j.physleta.2016.08.057

Cited by 13 publications

(3 citation statements)

References 20 publications

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“…Many algorithms for source identification problems analogous to the system (1.1)-(1.3) have been developed. For example, Hon used the Green function to determine the intensities and location of unknown point sources for Poisson equations from scattered boundary measurements [18], while an alternative iterative correction algorithm was employed to identify the source for two elliptic systems with partial data at the boundary [16]. In [1], the reduced space approach was applied to detect the piecewise constant sources from different observation data.…”

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%

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

Zhang,

Liu

2023

Inverse Problems

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“…An algebraic algorithm to identify the number, locations and intensities of the point sources from boundary measurements for the Helmholtz equation in an interior domain was developed in [18]. An iterative method for numerical reconstruction of the unknown source function in Poisson's and Helmholtz equations by means of measurements collected at the boundary was presented in [19]. In [20], linear integral transforms in Hilbert spaces were introduced and inversion formulas for inverse-source problems in the Helmholtz equation were provided.…”

Section: Introductionmentioning

confidence: 99%

Detecting Line Sources inside Cylinders by Analytical Algorithms

Lazaridis

Tsitsas

2023

Mathematics

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Inverse problems for line sources radiating inside a homogeneous magneto-dielectric cylinder are investigated. The developed algorithms concern the determination of the location and the current of each source. These algorithms are mostly analytical and are based on proper exploitation of the moments obtained by integrating the product of the total field on the cylindrical boundary with complex exponential functions. The information on the unknown parameters of the problem is encoded in these moments, and hence all parameters can be recovered by means of relatively simple explicit expressions. The cases of one and two sources are considered and analyzed. Under certain conditions, the permittivity and permeability of the cylinder are also recovered. The results from two types of numerical experiments are presented: (i) for a single source, the effect of noise on the boundary data is studied, (ii) for two sources, the pertinent nonlinear system of equations is solved numerically and the accuracy of the derived solution is discussed.

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“…Source identification problems are widely investigated in many research fields [1][2][3][4][5][6][7][8][9], and they are modeled by boundary value problems for which the analysis of the associated forward problem and its corresponding inverse problem must be considered, see, e.g., [10,11]. The latter problem involves determining the source that yields the measurement on the boundary of the region [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Stable Identification of Sources Located on Interface of Nonhomogeneous Media

et al. 2021

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This paper presents a stable method for the identification of sources located on the separation interface of two homogeneous media (where one of them is contained by the other one), from measurement yielded by those sources on the exterior boundary of the media. This is an ill-posed problem because numerical instability is presented, i.e., minimal errors in the measurement can result in significant changes in the solution. To obtain the proposed stable method the identification problem is categorized into three subproblems, two of which present numerical instability and regularization methods must be applied to obtain their solution in a stable form. To manage the numerical instability due to the ill-posedness of these subproblems, the Tikhonov regularization and sequential smoothing methods are used. We illustrate this methodology in a circular and irregular region to demonstrate the feasibility of the proposed method, which yields convergent and stable solutions for input data with and without noise.

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A numerical method for inverse source problems for Poisson and Helmholtz equations

Cited by 13 publications

References 20 publications

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

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

Detecting Line Sources inside Cylinders by Analytical Algorithms

Stable Identification of Sources Located on Interface of Nonhomogeneous Media

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