Question about "promolecular" tricks

[marco-2023]

To my understanding, the promolecular trick (oversimplified) base idea is:

For numerical integration on a grid, we approximate:

$$ \int f(\mathbf{x}) d\mathbf{x} \approx \sum_i w_i f(\mathbf{x}_i) $$

Now assume we introduce a reference function $\tilde{f}(\mathbf{x})$ such that:

• $\tilde{f}(\mathbf{x})$ is close to $f(\mathbf{x})$
• $\int \tilde{f}(\mathbf{x}), d\mathbf{x}$ is analytically known

We decompose:

$$ f(\mathbf{x}) = \tilde{f}(\mathbf{x}) + \left(f(\mathbf{x}) - \tilde{f}(\mathbf{x})\right) $$

and split the integral:

$$ \int f(\mathbf{x}) d\mathbf{x}=\int \tilde{f}(\mathbf{x}) d\mathbf{x} + \int \left(f(\mathbf{x}) - \tilde{f}(\mathbf{x})\right) d\mathbf{x} $$

We approximate only the residual numerically:

$$ \int f(\mathbf{x}), d\mathbf{x} \approx \int \tilde{f}(\mathbf{x}), d\mathbf{x} + \sum_i w_i \left(f(\mathbf{x}_i) - \tilde{f}(\mathbf{x}_i)\right) $$

Then we play with localizing the domains, etc.

@PaulWAyers, is there a reason why we don't extend this to all grids (such that the user can optionally provide a callable in the grid domain for $\tilde{f}$)? Would this be worth the effort?

[PaulWAyers]

I think it makes sense. We can talk about it tomorrow, but I think this is the right idea. I was intending to focus on Gaussian fits (BFit) but any callable clearly works.

[PaulWAyers]

P.S. Note that the integral can be replaced with any linear operator, so derivatives work via the same protocol. @Ao-chuba is using the Laplacian/Poisson equation as a different linear operator.