Adjoint with sparse measurements

Hi, I am trying to solve an inverse problem, where I have some sparse measurement within the domain. I try to optimize the initial condition to match with the measurement.

There are two approaches that I can think of/found:

  1. use point-wise evaluation and loss would be mse of a set of points evaluated at sensor location
  2. use pointsource function that are sum of delta functions at sensor locations

Method 2 seems to be a lot faster than method 1, especially given our case of a large domain and many sensors. However, the delta function is not perfect. I am wondering when I supervise on my observations, should I zero out nodes that I haven’t observed, so that those information will for-sure not be used. Or I shouldn’t zero out anything, and the delta function will help zero those node values - however in this case, since the pointsource is not perfect delta function, some information will leak out.

Any thoughts/input would be greatly appreciated.