of the

What connection is there between the

and the convergence and accuracy

of sampling algorithms?

Can we do better?

Each samples a specific distribution.

Reduce modification factor when is "flat".

Switch to when smaller than original formulation.

Samples the density of states directly.

From partition function, all thermodynamic equilibrium information is available.

Avoids energetic barriers.

Non-Markovian until converged.

Detailed balance correction:

In order to process

- An
**acceptance function**, , . - A
**microstate weighting function**, , . - A
**moveset function**, .

The moveset defines a directed graph with vertices , and edges . A sampling algorithm follows a random walk on given by .

A network of spin sites that interact via

where the sum is over all adjacent sites (typically a lattice).

Single spin flips define a moveset.

Other possible moves: double flips,

inversions, Glauber-type.

[1D, 2D, BarabÃ¡siâ€“Albert, Star]

What about graphs that follow other

degree distributions and correlations?

[1D, 2D, BarabÃ¡siâ€“Albert, Star]

Star graph,

1D periodic chain, cycle graph,

Relative convergence times

to

Group the microstates into isomorphically

different arrangements of spins;

e.g. all four (not five!) arrangements of

yet widely different convergence times.

Same moves (single-spin flips), different moveset graphs.

Stars gives rise to "ladder"-type moveset graphs,

cycles are more complicated.

What can we change?

Minimize spectrum of converged WL walks.

Minimize "round-trip" time between two extermal states.

eigenvalue spectrum

We can optimize a new move by minimizing and weighting the new move relative to the old ones.

Possible new moves, inversions, -spin flips,

bridges, and "cheats".

This changes the edges in the moveset graph.

Assume that optimized moves will carry over during

the non-Markovian phase of the algorithm.

Fix the moveset, now try to optimize the weights.

This leads to non-flat histograms.

Trebst sampling (minimize round-trip times)

Isochronal sampling (minimize and match RT times)

*Labels* flow of walkers from extermal states,

.

Expand steady-state current to first order,

Assume that the weights are slowly varying in energy,

Absorbing Markov Chains

mean/variance of absorbance times

WL, Trebst, and Isochronal weights

are not necessarily optimal.

Minimize not just round-trip between

all states, not just extermal states?

Consider not just mean round-trip times,

but higher moments (e.g. variance, skew)?

Quantify the sampling difficultly

by the moveset topology?

There is room for improvement in the optimal moveset,

small systems provide insight to larger state space.

Trebst sampling is an improvment over flat histograms,

but assumes smooth DOS.

Sampling at energy macrostates is coarse,

it possibly could be improved with better macrostate fidelity.