  We develop an optimization-based approach to the problem of reconstructing
temperature-dependent material properties in complex thermo-fluid systems
described by the equations for the conservation of mass, momentum and energy.
Our goal is to estimate the temperature dependence of the viscosity coefficient
in the momentum equation based on some noisy temperature measurements, where
the temperature is governed by a separate energy equation. We show that an
elegant and computationally efficient solution of this inverse problem is
obtained by formulating it as a PDE-constrained optimization problem which can
be solved with a gradient-based descent method. A key element of the proposed
approach, the cost functional gradients are characterized by mathematical
structure quite different than in typical problems of PDE-constrained
optimization and are expressed in terms of integrals defined over the level
sets of the temperature field. Advanced techniques of integration on manifolds
are required to evaluate numerically such gradients, and we systematically
compare three different methods. As a model system we consider a
two-dimensional unsteady flow in a lid-driven cavity with heat transfer, and
present a number of computational tests to validate our approach and illustrate
its performance.
