Adjoint lattice Boltzmann equation for parameter identification

The lattice Boltzmann equation is briefly introduced using moments to clearly separate the propagation and collision steps in the dynamics. In order to identify unknown parameters we introduce a cost function and adapt control theory to the lattice Boltzmann equation to get expressions for the derivatives of the cost function vs. parameters. This leads to an equivalent of the adjoint method with the definition of an adjoint lattice Boltzmann equation.To verify the general expressions for the derivatives, we consider two elementary situations: a linearized Poiseuille flow to show that the method can be used to optimize parameters, and a nonlinear situation in which a transverse shear wave is advected by a mean uniform flow. We indicate in the conclusion how the method can be used for more realistic situations.