A variational Monte Carlo simulation of the helium atom generates a correlated Markov…
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement
Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output
Problem
A variational Monte Carlo simulation of the helium atom generates a correlated Markov chain of local-energy samples EL(i) (in Hartree), one per Monte Carlo step. A first run of N=214=16384 steps has sample standard deviation σE=0.384Hartree, and blocking gives: b: 1, 2, 4, 8, 16, 32, 64, 128, 256 σ^b (Hartree): 0.0030, 0.0038, 0.0049, 0.0063, 0.0081, 0.0104, 0.0134, 0.0172, 0.0210 with σ^b still rising by more than 20% per doubling of b even at the largest block size shown.
((a)) Decide whether this blocking curve has reached a plateau, and state what this implies about quoting a statistical error bar from this run.
((b)) State, in one sentence, why an ordinary bootstrap of the raw series, or a block bootstrap using an unconverged block size, would give a misleading standard error here.
((c)) An independent run, sixty-four times longer, with N′=220=1048576 steps and the same σE=0.384Hartree, has a blocking curve that is level at σ^b=0.0240Hartree, within its own scatter, at b=8192, 16384 and 32768 (128, 64 and 32 blocks). Compute the statistical inefficiency s, the integrated autocorrelation time τint (in steps), and the effective sample size Neff for this longer run.
((d)) State what the first, short run means for assembling an uncertainty budget for the ground-state energy, and what must be done before that budget can be closed.