Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

Russel Caflisch, Denis Silantyev, Yunan Yang

Research output: Contribution to journalArticlepeer-review


Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the DSMC method, it is difficult to efficiently solve the high-dimensional continuous adjoint equation obtained by the “optimize-then-discretize” approach. This challenge motivates us to propose an adjoint DSMC method following the “discretize-then-optimize” approach for Boltzmann-constrained optimization. We also analyze the properties of the two frameworks and their connections. Several numerical examples are presented to demonstrate their accuracy and efficiency.

Original languageEnglish (US)
Article number110404
JournalJournal of Computational Physics
StatePublished - Aug 15 2021


  • Adjoint-state method
  • Boltzmann equation
  • DSMC
  • Direct simulation Monte Carlo methods
  • Linear Boltzmann equation
  • Optimization

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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