Inverse Scattering for the Biharmonic Wave Equation with a Random Potential

Li, Peijun; Wang, Xu

doi:10.1137/22m1538399

SIAM J. Math. Anal.

2024

DOI: 10.1137/22m1538399

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Inverse Scattering for the Biharmonic Wave Equation with a Random Potential

Peijun Li,

Xu Wang

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2024

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Cited by 4 publications

References 28 publications

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Analysis and computation of an inverse source problem for the biharmonic wave equation

Chang,

Guo,

Yin

et al. 2024

Inverse Problems

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In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on the Fourier series expansion by multi-wavenumber boundary measurements. The stability of the proposed method is analyzed and numerical experiments are presented to verify the accuracy and efficiency.

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Analysis and computation of an inverse source problem for the biharmonic wave equation

Chang,

Guo,

Yin

et al. 2024

Inverse Problems

View full text Add to dashboard Cite

show abstract

Inverse problems for the fractional diffusion equation driven by fractional Brownian sheet

Lei,

Fan,

2024

Phys. Scr.

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This paper explores the inverse problems of the space-time fractional diffusion equation driven by a space-time fractional Brownian sheet, including spatial stochastic source term, spatial deterministic source term, potential, and fractional order. In order to deal with the stochastic integral of fractional Brownian sheet with space-time Hurst indexes $\mathcal{H}_1 , \mathcal{H}_2 \in (0,1)$, we construct a unified Itô isometry framework for the first time, upon which we establish the regularity of the solution to the direct problem. Then, we ingeniously employ the generalized fractional Grönwall inequality, Sobolev embedding, and combine the statistical characteristics of the outward normal derivative of the weak solution on open subdomain at the boundary to achieve accurate recovery of the stochastic space-time source terms and prove the uniqueness of the space deterministic source term. Furthermore, by constructing a contraction mapping and exploiting the Banach fixed point theorem, we establish the uniqueness of the potential. Afterward, using the statistical characteristics of the solution at the terminal time, we establish the stability of the potential. Finally, leveraging the representation of the mild solution and the asymptotic properties of the Mittag-Leffler function, we examine the uniqueness and stability of fractional order at known terminal time. In scenario involving an unknown terminal time, we derive the uniqueness and stability of the terminal time.

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Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

Chang,

Guo,

Zhao

2024

SIAM J. Sci. Comput.

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Inverse Scattering for the Biharmonic Wave Equation with a Random Potential

Cited by 4 publications

References 28 publications

Analysis and computation of an inverse source problem for the biharmonic wave equation

Analysis and computation of an inverse source problem for the biharmonic wave equation

Inverse problems for the fractional diffusion equation driven by fractional Brownian sheet

Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

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