The average magnetic field over the cross-section of a solenoid is estimated by…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction

Problem

The average magnetic field over the cross-section of a solenoid is estimated by sampling points uniformly over the cross-section and averaging the local field. The cross-section is split into two strata of unequal area fraction: a core with fraction p1=0.25p_1=0.25 of the area, where winding irregularities give a local field standard deviation σ1=0.02 T\sigma_1=0.02\ \mathrm{T}, and an outer shell with fraction p2=0.75p_2=0.75, where σ2=0.08 T\sigma_2=0.08\ \mathrm{T}. A stratified estimator draws NkN_k independent points in stratum kk (uniformly within that stratum), averages the field there to get μ^k\hat\mu_k, and combines them as I^=∑kpkμ^k\hat I=\sum_k p_k\hat\mu_k, using in total N=N1+N2=2000N=N_1+N_2=2000 sample points. ((a)) Show that I^\hat I is unbiased for the true cross-sectional mean field I=∑kpkμkI=\sum_k p_k\mu_k, and derive its variance var(I^)=∑kpk2σk2/Nk\mathrm{var}(\hat I)=\sum_k p_k^2\sigma_k^2/N_k for general strata probabilities pkp_k (not necessarily equal) and general allocations NkN_k with ∑kNk=N\sum_k N_k=N fixed. ((b)) Find the allocation N1,N2N_1,N_2 (as whole numbers of points, to the nearest integer, summing to N=2000N=2000) that minimises this variance, and show that the minimum variance is (∑kpkσk)2/N\big(\sum_k p_k\sigma_k\big)^2/N. ((c)) Compute the variance for proportional allocation (Nk=pkNN_k=p_kN) and for the optimal allocation found in part (b), and give the percentage reduction in variance achieved by using the optimal allocation instead of proportional allocation.

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