# Basis Function Sampling¶

## Introduction¶

The Basis Function enhanced sampling method is a variant of the Continuous Wang-Landau Sampling method developed by Whitmer et al., 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.

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

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 Legendre, Chebyshev, and Fourier 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 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) 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:

• cyclefrequency
• 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 result in an undesired ringing phenomena.
• 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 $$1\mathrm{e}{-3}$$.

## Tutorial¶

This tutorial will provide a reference for running BFS in SSAGES. There are multiple examples provided in the Examples/User directory of SSAGES, but this tutorial will cover the Alanine Dipeptide example. In the ADP subdirectory of the Examples/User section there should be a LAMMPS input file (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 options. In order to do so,

2. Do:
make yes-RIGID
make yes-MOLECULE

1. Go to your build folder and make.

Use the following command to run the example:

mpiexec -np 2 ./ssages ADP_BFS_2walkers.json


This should prompt SSAGES to begin an alanine dipeptide run. 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 histgram 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.

## Developer¶

Joshua Moller Julian Helfferich