Particle Sensitivity Analyses
An important class of optimization problems contain partial integral differential equations as constraints so that gradient-based methods require sensitivities for the various functionals involving the solution of these equations. Such problems include models for plasmas, radiation transport, low-density fluids where the partial integral differential equation is the Boltzmann equation or the Fokker-Planck equation when molecular motion is of interest. These equations embody a deterministic model for the aggregate behavior of particles where the solution represents the density of particles. In lieu of approximating the solution via numerical discretization of the partial integral differential equation, a large number of particle trajectories may be averaged via Monte Carlo sampling to approximate the functionals involving the density. The link between such a class of partial integral differential equations and particle trajectories is well-known and is a generalization of the relationship between the diffusion equation and Brownian motion.
Determining a sensitivity during Monte Carlo sampling for a partial integral differential equation is fraught with difficulty since a sample, a trajectory, is not differentiable. Our solution is to exploit tools from probability to develop the needed sensitivities; see for example the book [Glasserman, 2004, Chap.7] for applications to finance and [Asmussen and Glynn, 2007, Chap.VII] for a general treatment. The purpose of my presentation is to introduce the audience to this literature.
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