Parameter Identification in a Random Environment Exemplified by a Multiscale Model for Crystal Growth

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We describe a framework for solving nonlinear inverse problems in a random environment. Such problems arise, for instance, in the identification of parameters in a stochastic process or in a differential equation where the parameters themselves are random variables. The corresponding inverse problems can be treated by Tikhonov regularization in a stochastic setup. Both the solution and the data in such inverse problems can be random variables. As an example, the inverse problem considered here concerns the identification of the parameter relating the nucleation rate to the temperature field in a mesoscale model for crystal growth. The derivation of the mesoscale model from a microscale model by geometric averages is outlined in the first sections. We formulate the corresponding inverse problem both for the “simply stochastic” case, which leads to a deterministic inverse problem, and for the “doubly stochastic” case yielding a stochastic inverse problem. We apply the stochastic version of the theory of Tikhonov regularization to prove convergence and convergence rates and outline how the stochastic regularization approach can be used to deal with scale-dependent modelling errors.