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Michele Ceriotti1, Giovanni Bussi, Michele Parrinello

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We developed a new sampling method for ill-conditioned Boltzmann distributions. This approach uses heat-bath thermalization along conjugate directions, outperforming local updates for high condition numbers and offering stability advantages.

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Area of Science:

  • Computational Physics
  • Statistical Mechanics
  • Numerical Analysis

Background:

  • Sampling from Boltzmann distributions is crucial in statistical mechanics.
  • Ill-conditioned quadratic actions pose significant challenges for standard sampling algorithms.
  • Existing methods like local updates can be inefficient for systems with high condition numbers.

Purpose of the Study:

  • To introduce a novel and efficient sampling method for Boltzmann distributions with ill-conditioned quadratic actions.
  • To address the limitations of existing sampling techniques in high condition number scenarios.

Main Methods:

  • The proposed method employs heat-bath thermalization.
  • Thermalization is performed along conjugate directions generated by a conjugate-gradient procedure.
  • This approach targets specific modes of the quadratic action.

Main Results:

  • The new scheme demonstrates superior performance compared to local updates for matrices with very high condition numbers.
  • It effectively avoids the slowing down of modes associated with lower eigenvalues.
  • The method shows improved stability and greater flexibility for optimization compared to global heat-bath approaches.

Conclusions:

  • This conjugate-direction heat-bath method offers an efficient and stable solution for sampling ill-conditioned Boltzmann distributions.
  • It provides a valuable alternative to existing methods, particularly in challenging computational scenarios.
  • The technique allows for tailored optimizations, enhancing its applicability across various problems.