Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions


A source identification algorithm for systems of nonlinear ordinary differential equations of production-destruction type is applied to an inverse problem for a discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of particle concentrations of a specific size. Based on an ensemble of adjoint equation solutions, a sensitivity operator is constructed that relates 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 with \(r\)-pseudoinverse matrices is used. The efficiency and properties of the algorithm are studied numerically.

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The authors would like to thank E.A. Pyanova and E.A. Tsvetova for their helpful comments and advice.


This work (the development and investigation of the algorithms) was supported by the Russian Foundation for Basic Research, project no. 17-71-10184.

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Correspondence to A. V. Penenko or A. B. Salimova.

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Penenko, A.V., Salimova, A.B. Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions. Numer. Analys. Appl. 13, 152–164 (2020).

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  • Smoluchowski equation
  • inverse source problem
  • Newton--Kantorovich method
  • adjoint equations
  • sensitivity operator