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Numerical methods for quasi-stationary distributions.

Sara Oliver-Bonafoux1, Javier Aguilar1,2,3, Tobias Galla1

  • 1Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca, Spain.

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Summary
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We present improved numerical methods for computing the quasi-stationary distribution in stochastic processes. An iterative algorithm excels with simple boundaries, while a Monte Carlo method is better for complex ones.

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Area of Science:

  • * Stochastic processes and computational mathematics.
  • * Numerical analysis and algorithm development.

Background:

  • * The quasi-stationary distribution is crucial for understanding long-term behavior in processes before absorption.
  • * Existing numerical methods for computing this distribution have limitations.

Purpose of the Study:

  • * To generalize and enhance two established numerical methods for computing the quasi-stationary distribution.
  • * To analyze implementation details, accuracy, and efficiency of these methods.

Main Methods:

  • * Generalization of an iterative algorithm for solving nonlinear equations defining the quasi-stationary distribution to accommodate diverse Markov processes.
  • * Development of a single-trajectory Monte Carlo method with resetting for quasi-stationary distribution computation.

Main Results:

  • * The iterative algorithm is generally preferred for problems with simple boundaries.
  • * The Monte Carlo method with resetting is more suitable for complex boundaries.
  • * Both methods show varying accuracy and efficiency depending on the problem's complexity.

Conclusions:

  • * Enhanced numerical techniques offer improved computation of the quasi-stationary distribution.
  • * Method selection depends on the specific characteristics of the stochastic process and its boundaries.