Inverse random source problem for biharmonic equation in two dimensions

Yuxuan Gong; Xiang Xu

doi:10.3934/ipi.2019029

Article Contents

2019, Volume 13, Issue 3: 635-652. Doi: 10.3934/ipi.2019029

This issue Previous Article Causal holography in application to the inverse scattering problems Next Article A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation

Inverse random source problem for biharmonic equation in two dimensions

Yuxuan Gong and
Xiang Xu^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: xxu@zju.edu.cn
^* Corresponding author: xxu@zju.edu.cn

Received: April 30, 2018

Revised: November 30, 2018

Published: March 2019

The authors are partly supported by NSFC grant 11471284, 11421110002, 11621101, 91630309 and the Fundamental Research Funds for the Central Universities.

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Abstract

The establishment of relevant model and solving an inverse random source problem are one of the main tools for analyzing mechanical properties of elastic materials. In this paper, we study an inverse random source problem for biharmonic equation in two dimension. Under some regularity assumptions on the structure of random source, the well-posedness of the forward problem is established. Moreover, based on the explicit solution of the forward problem, we can solve the corresponding inverse random source problem via two transformed integral equations. Numerical examples are presented to illustrate the validity and effectiveness of the proposed inversion method.

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Mathematics Subject Classification: Primary: 45Q05, 60H15.

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Figure 1. The model for the two-dimensional biharmonic equation

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Figure 2. The mesh generation under the polar coordination

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Figure 3. The solution to direct problem with random source

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Figure 4. Inverse stiffness D(The exact solution is 0.05)

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Figure 5. The left subfigure is the L-curve for $ g $ and the right subfigure is the inverse mean(The dotted plots are accurate values)

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Figure 6. The left subfigure is the L-curve for $ h^2 $ and the right subfigure is the inverse variance(The dotted plots are accurate values)

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Figure 7. The left subfigure is the L-curve for $ g $ and the right subfigure is the inverse mean(The dotted plots are accurate values)

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Figure 8. The left subfigure is the L-curve for $ h^2 $ and the right subfigure is the inverse variance(The dotted plots are accurate values)

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