Convexification for a Three-Dimensional Inverse Scattering Problem with the Moving Point Source

Khoa, Vo Anh; Klibanov, Michael V.; Nguyen, Loc H.

doi:10.1137/19m1303101

Cited by 36 publications

(40 citation statements)

References 31 publications

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“…In some recent previous works [13,14,23,31], the convergence analysis of the convexification method has been studied under the assumption that a function generated by the total field is expanded by a truncated Fourier series. The Fourier basis {Φ n } ∞ n=1 we exploit for the convexification method is a special Fourier basis that has been recently introduced in [18].…”

Section: R E T R a C T E Dmentioning

confidence: 99%

“…The Fourier basis {Φ n } ∞ n=1 we exploit for the convexification method is a special Fourier basis that has been recently introduced in [18]. This basis plays an important role in the study of the convexification method in [13,14,23,31]. The most important property of this basis is that the set of derivatives {Φ n } ∞ n=1 of the basis {Φ n } ∞ n=1 is linearly independent and dense in the L 2 sense.…”

Section: R E T R a C T E Dmentioning

confidence: 99%

“…However, the numerical implementation of the method faced some obstacles until its recent improvement [1] in 2017. This work has been continued by a number of more recent publications [14,18,[21][22][23][24], which address both convergence analysis and numerical results. In particular, the verification of the convexification method on experimental data has been done in [22,23].…”

Section: Introductionmentioning

confidence: 98%

“…In addition, the uniqueness theorem of an approximate problem can also be established; see e.g. [14,Theorem 3.2]. This inverse problem belongs to a wider class of coefficient inverse scattering problems that occur in various applications including non-destructive testing, explosive detection, medical imaging, radar imaging and geophysical exploration.…”

Section: Introductionmentioning

confidence: 99%

“…Typically, the GCNMs solve a coefficient inverse scattering problem using multifrequency data for a fixed direction of the incident plane wave. Most recently, it has been also studied for data associated with many locations of the point source but at a fixed single frequency [14]. We also want to mention that the GCNMs always consider non-over-determined data.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Convergence of a series associated with the convexification method for coefficient inverse problems

Klibanov

Nguyen

2020

Journal of Inverse and Ill-Posed Problems

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This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^{2} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^{1}-norm for a sequence of L^{2}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

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Section: R E T R a C T E Dmentioning

confidence: 99%

Section: R E T R a C T E Dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of a series associated with the convexification method for coefficient inverse problems

Klibanov

Nguyen

2020

Journal of Inverse and Ill-Posed Problems

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Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

Khoa

Nhan

2020

Math Methods in App Sciences

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In the recent developments of regularization theory for inverse and ill‐posed problems, a variational quasi‐reversibility (QR) method has been designed to solve a class of time‐reversed quasi‐linear parabolic problems. Known as a PDE‐based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward‐like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second‐order initial value problem for a wave‐type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman‐like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.

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The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

Nguyen

et al. 2021

J Sci Comput

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Convexification for a Three-Dimensional Inverse Scattering Problem with the Moving Point Source

Cited by 36 publications

References 31 publications

Convergence of a series associated with the convexification method for coefficient inverse problems

Convergence of a series associated with the convexification method for coefficient inverse problems

Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

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