Inverse source problems without (pseudo) convexity assumptions

Victor Isakov; Shuai Lu

doi:10.3934/ipi.2018040

Article Contents

2018, Volume 12, Issue 4: 955-970. Doi: 10.3934/ipi.2018040

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Inverse source problems without (pseudo) convexity assumptions

Victor Isakov^1, and
Shuai Lu^2, ,

1.
Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA
2.
School of Mathematical Sciences, and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Shuai Lu
Corresponding author: Shuai Lu

Received: June 2017

Revised: January 2018

Published: August 2018

This research is supported in part by the Emylou Keith and Betty Dutcher Distinguished Professorship and the NSF grant DMS 15-14886. Shuai Lu is supported by NSFC No.11522108, 91630309, Shanghai Municipal Education Commission No.16SG01 and Special Funds for Major State Basic Research Projects of China (2015CB856003). The authors thank an anonymous referee for his careful reading of the manuscript and valuable remarks which greatly helped to improve the article.

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Abstract

We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 78A46.

Citation:

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Figure 1. Domain of the source problem. The source function is compactly supported in $\Omega\setminus\Omega_0$

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Figure 2. Annular domain of the interior inverse source problem. Solid red line is the observation $\Gamma$ and the domain $\Omega$ is $B_{2}(0.3, 0.3)\setminus \bar B_{0.5}(0, 0)$

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Figure 3. Annular domain with a piecewise constant source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wavenumber $k$ with the minimal relative error $0.1856$ at $k = 200$

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Figure 4. Annular domain with a mixed-point source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.1795$ at $k = 200$

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Figure 5. Annular domain with different observation sites for exact measurement data. $\Gamma = (0.3+2\cos\theta, 0.3+2\sin\theta)$, $\theta\in (0, \pi)$; $\Gamma_1 = (0.3+5\cos\theta, 0.3+5\sin\theta)$, $\theta\in(0, \frac{2}{5}\pi)$ and $\Gamma_2 = (0.3+10\cos\theta, 0.3+10\sin\theta)$, $\theta\in (0, \frac{1}{5}\pi)$. Left: the relative error slope versus the increasing wave number for the piecewise constant source; Right: the relative error slope versus the increasing wave number for the mixed-point source. The values at the tail of each line are the minimal relative errors with $k = 200$ or $k = 400$

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Figure 6. Rectangular domain of the interior inverse source problem. Solid red line is the observation site $\Gamma = \{0.45\}\times [-0.45, 0.45]$ and the blue shadowed domain $(0.5, 2.5)\times (-1, 1)$ is the source domain which is a subset of $\Omega$

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Figure 7. Rectangular domain with piecewise constant sources for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.3149$ at $k = 200$

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Figure 8. Error slopes for fixed observation site $\Gamma$ but different recovery domains. The values at the tail of each line are the minimal relative errors with $k = 200$ or $k = 400$

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Figure 9. Annular domain of the exterior inverse source problem. Solid red line is the observation site $\Gamma$ and shadowed domain is $\Omega$

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Figure 10. Annular domain with a piecewise constant source function for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.3923$ at $k = 200$

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Figure 11. Rectangular domain of the exterior inverse source problem. Solid red line is the observation site $\Gamma$ and shadowed domain is $\Omega$

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Figure 12. Rectangular domain with piecewise constant sources for exact measurement data. Left: the approximate source; Middle: the error between both sources; Right: relative error versus the wave number $k$ with the minimal relative error $0.1747$ at $k = 200$

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