H.B. Keller Colloquium

We present an adjoint method for the spatially homogeneous, nonlinear Boltzmann equation, for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. The discretization (in time and velocity) is the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian. After a forward (in time) solution of DSMC, the adjoint variables are found by a backwards solver. The adjoint variable is equal to a velocity derivative of an objective function. Numerical tests show that this gives accurate velocity derivatives and can be used for optimization of the Boltzmann equation. For collision models, DSMC requires the use of an acceptance/rejection (AR) step. Discontinuities in the AR step lead to a new term, involving the so-called "score function". This is joint work with Yunan Yang and Denis Silantyev.