My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
where the integrand, g(r), has been split into a score function, f(r), and an importance
function, P (r).
The importance function is restricted to the form of a
probability density: it must be non-negative and normalised to unity over the region
of integration. In regions where g is non-zero, P must also be non-zero.
A set of random vectors {r i } is then drawn from the distribution P (r). The
integral is estimated as
I ≈ 1 M
M∑
f(r i ). (3.5)
i=1
In the limit M → ∞, this expression for I is exact. For finite M, the standard
error in the estimate is
where the variance of f is
σ I =
σ f
√
M
, (3.6)
∫ [f(r) ] 2P
σf 2 = − µf (r) dr (3.7)
and µ f is the mean value; equation (3.4) shows that µ f = I. From equation (3.6),
it can be seen that the error scales as M −1/2 , and also that the choice of score and
importance functions is very important.
The optimum selection is the one that
minimises the variance; this is the solution of the equation
(
∫ )
δ
σ 2
δP (r)
f[P (r)] − λ P (r) dr = 0. (3.8)
Here, λ is the Lagrange multiplier associated with the normalisation constraint
∫
P (r) dr = 1. Writing σ
2
f in terms of g and P and carrying out the functional
differentiation gives
λ = − g2 (r)
P 2 (r) , (3.9)
which leads to the following result for the ideal form of P (r):
P (r) =
|g(r)|
∫
|g(r′ )| dr ′ . (3.10)
The corresponding score function is
f(r) = g(r)
|g(r)|
∫
|g(r ′ )| dr ′ . (3.11)
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