  Can Monte Carlo (MC) solvers be directly used in gradient-based methods for
PDE-constrained optimization problems? In these problems, a gradient of the
loss function is typically presented as a product of two PDE solutions, one for
the forward equation and the other for the adjoint. When MC solvers are used,
the numerical solutions are Dirac measures. As such, one immediately faces the
difficulty in explaining the multiplication of two measures. This suggests that
MC solvers are naturally incompatible with gradient-based optimization under
PDE constraints. In this paper, we study two different strategies to overcome
the difficulty. One is to adopt the Discrete-Then-Optimize technique and
conduct the full optimization on the algebraic system, avoiding the Dirac
measures. The second strategy stays within the Optimize-Then-Discretize
framework. We propose a correlated simulation where, instead of using MC
solvers separately for both forward and adjoint problems, we recycle the
samples in the forward simulation in the adjoint solver. This frames the
adjoint solution as a test function, and hence allows a rigorous convergence
analysis. The investigation is presented through the lens of the radiative
transfer equation, either in the inverse setting from optical imaging or in the
optimal control framework. We detail the algorithm development, convergence
analysis, and complexity cost. Numerical evidence is also presented to
demonstrate the claims.
