Mitigating the impacts of PIC noise in gradient-based device optimization

We have identified and mitigated issues arising from PIC noise sources in the computation of adjoint-based gradients of an objective function with respect to device geometry. The geometry can be parameterized by $N$ degrees of freedom, and the gradient of an objective function with respect to these parameters is called the 'shape gradient'. The brute-force finite difference approach for computing a shape gradient is too computationally expensive because it requires $N + 1$ evaluations of the objective function, and each evaluation of the objective function requires a full device simulation. However, a recently developed adjoint technique enables the efficient computation of the shape gradient for a particular objective function using only two simulations [1]. In its original formulation, this technique considered the steady state of a prescribed emission model with a fixed set of emitted particles. It was shown that existing particle-in-cell (PIC) codes can be used for these simulations. But, this inherently introduces discrete particle noise into the simulations, which propagates noise into the gradient calculations. A further issue occurs in the optimization of nanoscale vacuum electron devices where the emission depends self-consistently on the applied electric field, and thus the emitted particles change each timestep, introducing additional particle noise. This application is not possible with the original method. We have characterized these noise sources, and developed techniques to mitigate them. In particular, we have developed a procedure to enable the use of adjoint-based gradient calculations with self-consistent emission models.

[1]: Phys. Plasmas 26, 013109 (2019)