. Author manuscript; available in PMC: 2017 Mar 3.

Published in final edited form as: Multiscale Model Simul. 2016 Mar 3;14(1):301–322. doi: 10.1137/15M102157X

Algorithm 1.

Exact Milestoning algorithm.

Input: Milestones

M = \cup_{j = 1}^{m} M_{j}

, initial guess ξ, and tolerance ε > 0 for the absolute error in the MFPT.

Output: Estimates for μ, local MFPTs 𝔼^μ[τ_M], and overall MFPT 𝔼^ρ[τ_P].

μ⁽⁰⁾ ← ξ

T⁽⁰⁾ ← +∞

for all n = 1, 2, . . . do

for i = 1 to m do

Estimate

a_{i j}^{(n)}

and

E^{μ_{i}^{(n - 1)}} [τ_{M}]

A_{i j}^{(n)} \leftarrow a_{i j}^{(n)} (M_{j})

end for

Solve w^TA = w ^T (with

A = (A_{i j}^{(n)}) \in ℝ_{\geq 0}^{m \times m}

and

w = (w_{1}, \dots, w_{m}) \in ℝ_{\geq 0}^{m}

)

for j = 1 to m do

μ_{j}^{(n)} \leftarrow \sum_{i = 1}^{m} w_{i} a_{i j}^{(n)}

end for

Normalize μ⁽ⁿ⁾

T⁽ⁿ⁾ ← μ(P)⁻¹ 𝔼^{μ⁽ⁿ⁻¹⁾}[τ_M]

if |T⁽ⁿ⁾ − T⁽ⁿ⁻¹⁾| < ε then

break

end if

end for

return (μ⁽ⁿ⁾, 𝔼^{μ⁽ⁿ⁻¹⁾}[τ_M], T⁽ⁿ⁾)