Source indentification for the Smoluchowski equation using an ensemble of the adjoint equation solutions

A source identification algorithm for the systems of nonlinear ordinary differential equations of the production-destruction type is applied to the inverse problem for the discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of the specific size particles concentration. Based on an ensemble of adjoint equations solutions, the sensitivity operator is constructed that links the perturbations of the sought for model parameters with perturbations of the measured values. This reduces the inverse problem to a family of quasilinear operator equations. To solve the equations, an algorithm of the Newton–Kantorovich type is used with $r$-pseudoinverse matrices. The efficiency and properties of the algorithm are numerically studied.