Reconstruction of GridFunction with vector measurement data

[Sangwoo Kang]

Dear all

Hello. I'm trying to solve the integral equation related to the Maxwell equation with BEM++ (3.3.4). 

More specifically, solving the inverse source problem from measured vector data. 

But I have some question to reconstruct the right-hand side with  Grid function with vector data:

1. In your website, the grid  can be reconstructed by coefficients by 

GridFunction(RWGspace, coefficients=c)

But when I input c as N x 3 ndarray (Number of probes X 3(x,y,z components)), I got the message that c should be N x 1 array. 

2. To handle the problem I make interpolation function (fun) with data. Then I reconstruct Grid function with 

GridFunction(RWGspace, fun=fun), but it's too slow

But it's too slow. 

3. I want to reconstruct RHS = <RWG_space, Measured Data>_{L2}

So I didn't input dual space when inputting the GridFucntion command. is it correct?

To solve the problem I use RHS.projections(). Does it mean the weak form of RHS?

Thank you in advance. 

Have a nice day.

Best regards,

Sangwoo Kang

[Matthew Scroggs]

Hello,

The `coefficients` input of a `GridFunction` should be the scalar coefficients to multiply each vector function by. You can find the definitions of the RWG and RT basis functions used in Bempp in this paper: https://www.sciencedirect.com/science/article/pii/S0898122117304789. You may then be able to work out the coefficients using your measured data.

The difference between coefficients and projections is explained in section 3.3.1 of this paper: https://arxiv.org/abs/1711.10607. Generally, you should use `projections` if you are using `strong_form` on your left-hand side (this gives a mass matrix preconditioned system); and you should use `coefficients` if you are using `weak_form` on your left-hand side.

Hope this helps,

Matthew

[Sangwoo Kang]

Hello, 

Thank you for your reply. 

Q1. Actually, I got good results using PMCHWT formula which is compared with FEKO data in the following link: 

         https://www.dropbox.com/s/ok1o1p97z5oci8q/DielectricPMCHWT-CalculateScatteredData_ComparisonWithFEKO.ipynb?dl=0

      But I obtained the same results when I use weak-form(LHS) and projections (RHS), not using coefficients. Why I got the correct results?

      Note that I'm using BEMPP 3.3.4

Q2. My main goal of the study is to solve the inverse source problem for given measured electric data. From the following relation

where L, K is boundary operator E and H in BEMPP. But I got poor results even with smaller tolerance. Do you have any idea about the reason for this phenomenon?       

 Here, I use GMRES solver with least-square sense because the matrix is not square

      

Q3. Is it be possible to import RWG-basis function data to bempp?

       From the documents of the web site of BEMPP, https://bempp.com/2017/07/12/import-and-export-of-gmsh-files/, it seems to be possible I think if I use "FileReader" class 

Thank you in advance. 

Best regards,

Sangwoo Kang    

2020년 2월 19일 수요일 오전 11시 57분 20초 UTC+1, Matthew Scroggs 님의 말: