# Basis Function Sampling¶

## Introduction¶

The Basis Function Sampling method is a variant of the Continuous Wang-Landau Sampling method developed by Whitmer et al. [25], which biases a PMF through the summation of Kronecker deltas. In this method, the Kronecker delta is approximated by projection of a locally biased histogram to a truncated set of orthogonal basis functions.

$\int_\Xi f_{i}(\vec{\xi}) f_{j}(\vec{\xi}) w(\vec{\xi}) d\vec{\xi} = \delta_{ij}c_{i}$

By projecting a basis set, the system resolves the same properties as the Kronecker deltas, but in a continuous and differentiable manner that lends well towards MD simulations. The current version of SSAGES has support for Chebyshev, Fourier, and Legendre polynomials. Each of these has their defined weight function $$w(\xi)$$ implemented specific to the method. Additionally, any combination of implemented basis sets can be used for any system. It is advised that a periodic basis set (e.g. Fourier) be used with a periodic CV, but it is not required.

The BFS method applies its bias in sweeps of $$N$$ through a histogram ($$H_{i}$$) that is updated at every $$j$$ microstate or timestep. This histogram is then modified to an unbiased partition function estimate ($$\tilde{H_{i}}$$) by exponentiation with the current bias potential ($$\Phi_{i}$$).

$\tilde{H}_{i}(\xi) = H_{i}(\xi) e^{\beta \Phi_{i}}$

A weight function has been added into this implementation ($$W(t_{j})$$) so that the user can define the effective strength of the applied bias. If not chosen, the weight is normalized to the length of the interval.

$Z_{i}(\xi) = \sum_{j} W(t_{j})\tilde{H_{j}}(\xi)$

This final estimate is then projected to the truncated basis set. After this set is evaluated, the coefficients of the basis set are evaluated. This process is iterated until the surface converges, which is determined by the overall update of the coefficients.

$\begin{split}\beta \Phi_{i+1}(\xi) &= \sum_j^N \alpha^i_j L_j(\xi)\\ \alpha^i_j &= \frac{2j + 1}{2} \int_{-1}^1 \log(Z_i(\xi))L_j(\xi)d\xi\end{split}$

## Options & Parameters¶

These are all the options that SSAGES provides for running Basis Function Sampling. In order to add BFS to the JSON file, the method should be labeled as "BFSMethod".

Basis Function Sampling requires the use of a basis set. These are defined by defining an object of “basis_functions”. These have the following properties:

type

Currently can either be Chebyshev, Fourier, or Legendre.

polynomial_order

Order of the polynomial. In the case of Chebyshev or Legendre, this results in an order of input value + 1 as the method takes the 0th order internally. For a Fourier series, the order is the total number of coefficients including the sine and cosine series.

upper_bound

Only exists for Chebyshev and Fourier series. This is the upper bound of the CV.

lower_bound

Only exists for Chebyshev and Fourier series. This is the lower bound of the CV.

CV_restraint_spring_constants

The strength of the springs keeping the system in bounds in a non-periodic system.

CV_restraint_maximums

The upper bounds of each CV in a non-periodic system.

CV_restraint_minimums

The lower bounds of each CV in a non-periodic system.

cycle_frequency

The frequency of updating the projection bias.

frequency

The frequency of each integration step. This should almost always be set to 1.

weight

The weight of each visited histogram step. Should be kept around the same value as the cycle_frequency (usually 0.1 times that).

Note

The system has a higher chance of exploding at higher weight values.

basis_filename

A suffix to name the output file. If not specified, the output will be basis.out.

temperature

The temperature of the simulation.

tolerance

Convergence criteria. The sum of the difference in subsequent updates of the coefficients squared must be less than this for convergence to work.

convergence_exit

A boolean option to let the user choose if the system should exit once the convergence is met.

## Required to Run BFS¶

In order to use the method properly a few things must be put in the JSON file. A grid is required to run Basis Function Sampling. Refer to the Grid section in order to understand options available for the grid implementation. The only inputs required to run the method:

• cycle_frequency

• frequency

• basis_functions

• temperature

## Example Input¶

"methods": [{
"type": "BFSMethod",
"basis_functions": [
{
"type": "Fourier",
"polynomial_order": 30,
"upper_bound": 3.14,
"lower_bound": -3.14
},
{
"type": "Fourier",
"polynomial_order": 30,
"upper_bound": 3.14,
"lower_bound": -3.14
}
],
"cvs": [0, 1],
"cycle_frequency": 100000,
"basis_filename": "example",
"frequency": 1,
"temperature": 300.0,
"weight": 1.0,
"tolerance": 1e-3,
"convergence_exit": true,
"grid": {
"lower": [-3.14, -3.14],
"upper": [3.14, 3.14],
"number_points": [100, 100],
"periodic": [true, true]
}
}]


## Guidelines for Running BFS¶

• It is generally a good idea to choose a lower order polynomial initially. Excessive number of polynomials may create an unwanted “ringing” effect that could result in much slower convergence.

• For higher order polynomials, the error in projection is less, but the number of bins must increase in order to accurately project the surface. This may also create an unwanted “ringing” effect.

• A good rule of thumb for these simulations is to do at least one order of magnitude more bins than polynomial order.

If the system that is to be used requires a non-periodic boundary condition, then it is typically a good idea to place the bounds approximately 0.1–0.2 units outside the grid boundaries.

The convergence_exit option is available if the user chooses to continue running past convergence, but a good heuristic for tolerance is around 0.001.

## Tutorial¶

This tutorial will provide a reference for running BFS in SSAGES. There are multiple examples provided in the Examples/User/BasicFunc directory of SSAGES, but this tutorial will cover the Alanine Dipeptide example.

In the Examples/User/BasicFunc/ADP subdirectory, there should be two LAMMPS input files (titled in.ADP_BFS_example{0,1}) and two JSON input files. Both of these files will work for SSAGES, but the one titled ADP_BFS_2walkers.json makes use of multiple walkers.

For LAMMPS to run the example, it must be made with rigid and molecule packages. In order to do so, issue the following commands from your build directory:

make yes-rigid
make yes-molecule
make


Use the following command to run the example:

mpiexec -np 2 ./ssages ADP_BFS_2walkers.json


This should prompt SSAGES to begin the simulation. If the run is successful, the console will output the current sweep number on each node. At this point, the user can elect to read the output information after each sweep.

basis.out

The basis.out file outputs in at least 3 columns. These columns refer to the CV values, the projected PMF from the basis set, and the log of the histogram. Depending on the number of CVs chosen for a simulation, the number of CV columns will also correspond. Only the first CV column should be labeled.

The important line for graphing purposes is the projected PMF, which is the basis set projection from taking the log of the biased histogram. The biased histogram is printed so that it can be read in for doing restart runs (subject to change). For plotting the PMF, a simple plotting tool over the CV value and projected PMF columns will result in the free energy surface of the simulation. The free energy surface will return a crude estimate within the first few sweeps, and then will take a longer period of time to retrieve the fully converged surface. A reference image of the converged alanine dipeptide example is provided in the same directory as the LAMMPS and JSON input files.

restart.out

This holds all the coefficient values after each bias projection update, as well as the biased histogram. This file is entirely used for restart runs.

## Developers¶

• Joshua Moller

• Julian Helfferich