# American Institute of Mathematical Sciences

June  2019, 13(3): 635-652. doi: 10.3934/ipi.2019029

## Inverse random source problem for biharmonic equation in two dimensions

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

* Corresponding author: xxu@zju.edu.cn

Received  May 2018 Revised  December 2018 Published  March 2019

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

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.

Citation: Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
##### References:

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##### References:
The model for the two-dimensional biharmonic equation
The mesh generation under the polar coordination
The solution to direct problem with random source
Inverse stiffness D(The exact solution is 0.05)
The left subfigure is the L-curve for $g$ and the right subfigure is the inverse mean(The dotted plots are accurate values)
The left subfigure is the L-curve for $h^2$ and the right subfigure is the inverse variance(The dotted plots are accurate values)
The left subfigure is the L-curve for $g$ and the right subfigure is the inverse mean(The dotted plots are accurate values)
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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