# Do computational scientists think about the statistical aspects of using least squares?

Correct me if I'm wrong, but when least squares is used in the computational science community, it's typically not in the context of regression. It could be used to solve for gradients in discretized PDEs and for other things. In these kind contexts, is there any relevance to consider considering statistical aspects such as checking p values of the least squares coefficients, ensuring that the residuals $$\hat{y} - y_{observed}$$ are normally distributed, etc..?

I don't think I've ever seen the word "regression" brought up when using least squares in the scientific computing community, and that gave me the impression that computational scientists and statisticians use least squares in near orthogonal manners.

• There are people who specialize in uncertainty quantification and stochastic differential equations, if that's what you're asking about. In a typical realization of a convex optimization problem, people don't consider those issues. – Biswajit Banerjee May 9 at 22:27
• There’s no real need to do hypothesis tests and p-values on well established physical laws. Uncertainty quantification is important though. – Brian Borchers May 10 at 1:10
• There are situations in computational science when you want to see what model describes sufficiently well some experimental data, there you'd certainly want to do statistical tests. – Maxim Umansky May 10 at 13:38
• I think you are referring to the Galerkin Least Squares method for solving PDEs? I understand all of the words you are using in your question (and am from all of the relevant fields), but I think I am missing the background of your question when you say "when least squares is used in the computational science community, it's typically not in the context of regression. It could be used to solve for gradients in discretized PDEs and for other things." It would be useful if you pointed to a few papers or books that you have in mind when you make this claim. – Wolfgang Bangerth May 11 at 4:55