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9. Convergence and Monte Carlo Errors

Understanding convergence to equilibrium using variation distance and eigenvalue problems in Markov chains. Learn about error measurement, estimation, and correction in Monte Carlo simulations.

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9. Convergence and Monte Carlo Errors

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  1. 9. Convergence and Monte Carlo Errors

  2. Measuring Convergence to Equilibrium • Variation distance where P1 and P2 are two probability distributions, A is a set of states, i is a single state.

  3. Eigenvalue Problem • Consider the matrix S defined by [S]ij = pi½ W(i->j) pj-½ then S is real and symmetric and eigenvalues of S satisfy |n| ≤ 1 • One of the eigenvalue must be 0=1 with eigenvector pj½.

  4. Spectrum Decomposition • Then we have UTSU = Λ, or S = U Λ UT where Λ is a diagonal matrix with diagonal elements k and U is orthonormal matrix, U UT = I. • W can be expressed in U, P, and Λ as W = P-½UΛUTP½

  5. Evolution in terms of eigen-states • Pn= P0Wn = P0 P-½UΛUTP½ P-½UΛUTP½… = P0 P-½UΛnUTP½ • In component form, this means Pn(j) = ∑iP0(i) pi-½pj½∑kkn uikujk

  6. Discussion • In the limit n goes to ∞, Pn(j) ≈ ∑iP0(i) pi-½pj½ ui0uj0 = pj • The leading correction to the limit is Pn(j) ≈ pj + a 1n = pj + a e-n/

  7. Exponential Correlation Time • We define  by the next largest eigenvalue  = - 1/log 1 This number characterizes the theoretical rate of convergence in a Markov chain.

  8. Measuring Error • Let Qt be some quantity of interest at time step t, then sample average is QN = (1/N) ∑tQt • We treat QN as a random variable. By central limit theorem, QN is normal distributed with a mean <QN>=<Q> and variance σN2 = <QN2>-<QN>2. <…> standards for average over the exact distribution.

  9. Confidence Interval • The chance that the actual mean <Q> is in the interval [ QN – σN, QN + σN ] is about 68 percents. • σN cannot be computed (exactly) in a single MC run of length N.

  10. Estimating Variance The calculation of var(Q) = <Q2>-<Q>2 and int can be done in a single run of length N.

  11. Error Formula • The above derivation gives the famous error estimate in Monte Carlo as: where var(Q) = <Q2>-<Q>2 can be estimated by sample variance of Qt.

  12. Time-Dependent Correlation function and integrated correlation time • We define and

  13. Circular Buffer for Calculating f(t) We store the values of Qs of the previous M-1 times and the current value Qt Qs Qt-1 Previous time t-1 Qt, Current time t Earliest time t-(M-1)

  14. An Example of f(t) Time-dependent correlation function for 3D Ising at Tc on a 163 lattice; Swendsen-Wang dynamics. From J S Wang, Physica A 164 (1990) 240.

  15. Efficient Method for Computing int We compute int by the formula int = NσN2/var(Q) For small value N and then extrapolating N to ∞. From J S Wang, O Kozan and R H Swendsen, Phys Rev E 66 (2002) 057101.

  16. Exponential and integrated correlation times where 1 < 1 is the second largest eigenvalue of W matrix. This result says that exponential correlation time  (=-1/log1) is related to the largest integrated correlation time.

  17. Critical Slowing Down  The correlation time becomes large near Tc. For a finite system (Tc)  Lz, with dynamical critical exponent z ≈ 2 for local moves Tc T

  18. Relaxation towards Equilibrium Magnetization m Schematic curves of relaxation of the total magnetization as a function of time. At Tc relaxation is slow, described by power law: m t -β/(zν) T < Tc T = Tc T > Tc Time t

  19. Jackknife Method • Let n be the number of independent samples • Let c be some estimate using all n samples • Let ci be the same estimate but using n-1 samples, with i-th sample removed • Then Jackknife error estimate is

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