Acceleration of covariance-based analysis by Jacobian simplification

[luisfabib]

Computation of covariance-based uncertainties requires the estimation of the Jacobian at the found solution. For non-linear parameters this has to be estimated numerically. Until now DeerLab was using the numdifftools.Jacobian function to do this.

The functions does a series of finite difference-based calculations that yield very accurate Jacobians. This function, however, is very costly due to the large number of function evaluations, and has become a bottleneck for most DeerLab fit functions. For example, a simple 4-pulse DEER fit with fitsignal might take 8s to run, from which 7s correspond to the Jacobian estimation.

To prevent this I have substituted the Jacobian estimation using numdifftools by a simple one-step finite-difference approximation. While it has the potential to be less accurate...

• ... the covariance based approach has many other approximations which lead to larger inaccuracies in the uncertainty than those caused by inaccurate Jacobians.
• ... for a set of tests, the simple one-step finite-difference yielded numerically equal results to the numdifftools results for DEER models.
  Therefore, I have removed the dependencies on numdifftools altogether and introduced the simpler version.

This has led to runtime accelerations of the fit functions by factors x2-10. For example, the 4-pulse DEER fit which might have taken 8s now requires just 1s, with the NNLS problem being the bottleneck as should be.