Understand "reduction_error_analysis" results

[CheLamVien]

Dear all,

I am new to MOR. I have a question regarding the difference between "errors" (Array of the norms of the model reduction errors w.r.t. all given test Parameters, reduced basis dimensions and norms) and "error_estimates" (Array of the model reduction error estimates w.r.t. all given test Parameters and reduced basis dimensions.). I could not find the code for how to calculate "error_estimates".

I ran the "thermalblock_simple.py" script and saw that the norm error is smaller than the error estimator, but I could not explain it.
Any explanations are highly appreciated.
Thank you so much for your time.

Best regards,
VC

[sdrave]

Hi @CheLamVien, I'm not sure if I understand your question correctly, but error_estimates contains a posteriori error estimates for the model reduction error contained in errors. Those estimates are always upper bounds and generally larger than the true error. (Good estimators have the property that the error is never overestimated by more than a fixed factor.) However, the crucial benefit of such an error estimator is that it can be computed from the reduced solution without knowing the actual full-order solution. To compute the true error, you to compute both the reduced-order and full-order solution.

In pyMOR, a posterior error estimates are computed using the estimate_error method of the reduced-order Model.

[CheLamVien]

Thank you so much for your explanation. I got it now.
I have another question; I see an option for NGsolve binding in PyMOR. I have followed the demo example . I would like to know if I can use the iterative solver in NGsolve.

Best regards,
Vien Che

[sdrave]

Yes, you should be able to select the used linear solver by setting solver_options of NGSolveMatrixOperator or setting the default_solver default of NGSolveMatrixOperator._prepare_apply. If you need more customization, this could be easily added to the bindings.

[CheLamVien]

Thank you again. Now, I can use the CGSolver and preconditioner in NGSolve. I also would like to use the complex number. Could you please give me some hints?
I really appreciate any help you can provide.

[CheLamVien]

I think I have figured out how to use the complex number. I got the error: raise ValueError(f'norm is complex (square = {norm_squared})'). I changed raise_complex=False. Is this a correct solution, and why must we raise this value? What is the reason behind it?
Thank you so much in advance.