Reconstruction of the collision kernel in the nonlinear Boltzmann equation

We consider an inverse problem for the Boltzmann equation with nonlinear collision operator in dimensions $n\ge 2$. We show that the kinetic collision kernel can be uniquely determined from the incoming-to-outgoing mappings on the boundary of the domain provided that the kernel satisfies a monotonicity condition. Furthermore, a reconstruction formula is also derived. The key methodology is based on the higher-order linearization scheme to reduce a nonlinear equation into simpler linear equations by introducing multiple small parameters into the original equation.