Mathematical vs. Physical parameters, and prior widths.

[Lee-Ping Wang]

> VAV writes: What is the difference between "mathematical" and "physical" parameters? Is there some internal transformation between the two?

This is an important issue that I always have a bit of trouble describing. The difference is mostly rescaling, and it's closely tied to the numerical value for the prior width.

The physical parameters are what actually gets printed into the force field files, so they have physical units. It's impractical to optimize these parameters directly because their scales are so different (e.g. in Gromacs, bond force constants are on the order of 10^5 while the bond length is on the order of 10^-1.) The different scales of these parameters could lead to a badly conditioned Hessian matrix and problems with matrix inversion.

Thus, it makes sense for the optimization to work with variables that are order 1. These variables (the "mathematical parameters") are related to the physical parameters using a simple transformation:

k_phys = k_phys(0) + scale_factor * k_math ,

where:

- k_phys is the physical parameter that's being printed to the force field during the optimization
- k_phys0 is the original value that's read in from the initial force field
- k_math is the mathematical parameter that's stored internally
- scale_factor is a user-specified scale factor that transforms the mathematical parameters to physical parameters

Thus, in an un-regularized optimization the scale factor simply improves the condition number of the Hessian matrix and doesn't change the final answer. The important connection is that the scale factor is also the prior width in a regularized optimization.

During a regularized optimization, we have the normal objective function plus a penalty function:

X_reg(k_math) = X(k_phys(scale_factor, k_math)) + P(k_math)

where:

- X_reg is the regularized objective function
- X is the un-regularized objective function
- P is the penalty function

The un-regularized objective function has no explicit dependence on the mathematical parameters (the dependence is implicit through the physical parameters), because the mathematical parameters are internal variables that never get printed to the force field files. However, because the penalty function is isotropic in the mathematical parameters (i.e. treats them all the same), the choice of the scale factor (prior width) strongly affects how much the physical parameters are allowed to vary. If we imagine that the penalty function restrains the mathematical parameters to be < 1, then the physical parameters will be restrained to within one scale factor (prior width) of its starting value.

One final detail is that I make some small additions to the mathematical -> physical transformation to enforce things like charge neutrality and traceless quadrupoles. I'm also playing with some problems where the transformation is exponential.