Ivancevic_Applied-Diff-Geom

1024 Applied Differential Geometry: A Modern Introductionestimated asˆψ(x i , t) =∑α∈alivew α , u i dt =1ˆψ(x i , t)N∑α∈alivew α dξ i α(t), (6.34)withw α = 1 N exp(−φ(xi α(t f ))/λ),where ‘alive’ denotes the subset of trajectories that do not get killed alongthe way by the † operation. The normalization 1/N ensures that the annihilationprocess is properly taken into account. Equation (6.34) states thatoptimal control at time t is obtained by averaging the initial directions ofthe noise component of the trajectories dξ i α(t), weighted by their success att f .The above sampling procedure can be quite inefficient, when many trajectoriesget annihilated. One of the simplest procedures to improve itis by importance sampling. We replace the diffusion process that yieldsρ(y i , t f |x i , t) by another diffusion process, that will yield ρ ′ (y i , t f |x i , t) =exp(−S ′ /λ). Then (6.31) becomes,∫ψ(x i , t) = [dx i ] x i exp (−S ′ /λ) exp (−(S − S ′ )/λ) .The idea is to chose ρ ′ such as to make the sampling of the path integralas efficient as possible. Following [Kappen (2006)], here we use the Laplaceapproximation, which is given by the k deterministic trajectories x β (t → t f )that minimize the ActionJ(x i , t) ≈ −λ logk∑exp(−S(x i β(t → t f )/λ).β=1The Laplace approximation ignores all fluctuations around the modes andbecomes exact in the limit λ → 0. The Laplace approximation can becomputed efficiently, requiring O(n 2 m 2 ) operations, where m is the numberof time discretization.For each Laplace trajectory, we can define a diffusion processes ρ ′ β accordingto (6.33) with b i (x i , t) = x i β (t). The estimators for ψ and ui aregiven again by (6.34), but with weightsw α = 1 N exp ( − ( S(x i α(t → t f )) − S ′ β(x i α(t → t f )) ) /λ ) .S is the original Action (6.32) and Sβ ′ is the new Action for the Laplaceguided diffusion. When there are multiple Laplace trajectories one should