Reduced-order models in QMC

After the last post about multiscale modeling, let's explore how some of the methods and ideas might apply to improving QMC.

Possible sources:
- Reduced order models.  Usually applied to time-dependent problems.
- Dimensionality reduction.  From statistics and machine learning
- Low-rank matrix approximation.
- Compressed sensing - determine a sparse sampling before performing the sampling.  See this article for an application to wavefunctions.

Some of these approaches are applications to different domains but use similar underlying mathematics (e.g., apply Singular Value Decomposition (SVD) to an appropriate matrix).

The "order" in "reduced-order models" refers to the complexity of the model.  Some possibilities for QMC on how the complexity might be varied include: number of electrons, number of sample points in the integral, quality of the wavefunction, and the precision of the arithmetic.

Let's look at each of these in a little more detail:

1. Number of electrons
   A. Integrate out some electrons - yields pseudopotential
   B. Integrate out all electrons - yields intermolecular potentials between atoms/molecules

 To ultimately reduce the complexity of the model, most or all of the electrons need to be eliminated.  This is also the natural decomposition at an initial look - generate potential from electronic structure calculation.    However, this may be too large of jump of scales in some cases, and it may be useful to find intermediate models that still include some or all of the electrons.

2. Number of sample points in the integral
    A. Subset of points from an existing set of samples.  Not so useful for that particular run, but could be useful for a nearby system. Possible ways a system could be 'nearby':
         - Changing parameters for VMC optimization
         - Core electrons for psuedopotentials
         - Weak intermolecular interactions
   B. Generate new points faster or more accurately
        Quasi Monte Carlo methods (the other QMC) or other low discrepancy sequences.

Side note on pseudopotentials.  They serve two purposes:
   a. Reduce the number of particles that need to be handled
   b. Reduce the variance introduced by sampling the core electrons

  I suspect that (b) is more important than (a), at least for small problems.  Pseudopotentials complicate various QMC algorithms, so improved sampling that reduces the variance could be an effective alternative.

3. Quality of wavefunction
  In general we would like to have a systematic way of controlling the quality (computational complexity vs. error) of the wavefunction.   One use for less-accurate approximations to the wavefunction might be as a guide for faster generation of new sample points.

4. Precision of arithmetic
  The full model is really the position of all the electrons in space.   We might consider restricting the electrons to lattice points or reducing the precision in places.   For examples in machine learning, see here and here.