Question about solving the inverse problem

[Yunfei Huang]

Hi Everyone, 

Now I have an inverse problem that is given the measured displacement u_i and our target to get the force field f_j.  If I unstand correctly that the linear system of dealii is A_{ij}u_j = f_i.   I am not sure this form whether I can use the L2 regularization argmin{||u-u_{measure}|| + \lamda||f|| }.  I have only had experience with A_(ij)f_j = u_i.  Then we have an analytical solution using SVD. However,  If we have  A_{ij}u_j = f_i, it looks like we do not have analytical solution.

Does anyone have any ideas on how to solve this inverse problem using dealii?

Best regards, 

Yunfei 

[blais...@gmail.com]

Dear Yunfei,

The linear system you solve depends on the weak form of the problem you are solving, which is established from the strong form of the PDE.

If you want to reconstruct the force from the displacement field, I think you would need to formulate the adjoint PDE problem to your displacement PDE. I think since the linear elasticity equation is a Poisson-ish equation, the equation would be self-adjoint and the adjoint system could be formed straightforwardly from the forward problem.

I hope that helps

Bruno

[Yunfei Huang]

Dear Bruno,

Thanks a lot for your reply. 

I am not so clear on this part. I would be grateful if you could share some documentation or an example to help me learn more about it.

Best regards,

Yunfei 

blais...@gmail.com <blais...@gmail.com> 于2025年8月26日周二 15:11写道：

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[blais...@gmail.com]

Dear Yunfei,

I think an article like this is what you need to get started:

https://www.sciencedirect.com/science/article/pii/S0045782521001468

Best regards

Bruno

[Yunfei Huang]

Dear Bruno, 

Thanks a lot for your paper.

Best regards.

Yunfei 

blais...@gmail.com <blais...@gmail.com> 于2025年9月1日周一 17:31写道：

To view this discussion visit https://groups.google.com/d/msgid/dealii/2c26fa58-a472-49a2-83f1-382e3ec34a7bn%40googlegroups.com.