Parameter Identification via Optimal Control for a Cahn–Hilliard-Chemotaxis System with a Variable Mobility

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Abstract

We consider the inverse problem of identifying parameters in a variant of the diffuse interface model for tumour growth proposed by Garcke et al. (Math Models Methods Appl Sci 26(6):1095–1148, 2016). The model contains three constant parameters; namely the tumour growth rate, the chemotaxis parameter and the nutrient consumption rate. We study the inverse problem from the viewpoint of PDE-constrained optimal control theory and establish first order optimality conditions. A chief difficulty in the theoretical analysis lies in proving high order continuous dependence of the strong solutions on the parameters, in order to show the solution map is continuously Fréchet differentiable when the model has a variable mobility. Due to technical restrictions, our results hold only in two dimensions for sufficiently smooth domains. Analogous results for polygonal domains are also shown for the case of constant mobilities. Finally, we propose a discrete scheme for the numerical simulation of the tumour model and solve the inverse problem using a trust-region Gauss–Newton approach.

Keywords

Cahn–Hilliard equation Chemotaxis Parameter identification Optimal control Variable mobility 

Mathematics Subject Classification

35Q92 35R30 49J20 49J50 65M32 92B05 92C17 

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Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany
  2. 2.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong

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