Model reduction in state identification problems with an application to determination of thermal parameters

Large-dimensional parameter estimation problems are often computationally unstable and are therefore characterized as ill-posed inverse problems. Inverse problems tolerate measurement and modelling errors poorly which usually calls for accurate computational implementations of the underlying models. These implementations often turn out to be computationally too demanding for a specific application, especially in case of time-varying problems. The so-called approximation error approach has recently been developed to cope with both modelling and numerical discretization errors. This approach has been applied to both stationary (time-invariant) and nonstationary problems. Given a fixed available computational capacity, the employment of the approximation error approach usually yields significantly better estimates than with a conventional error model. In addition, the error estimates are more feasible than with a conventional error model. In this paper we extend the previous results and provide computationally efficient forms for the extended Kalman filters for large-dimensional state identification problems. We apply the approach to the determination of distributed thermal parameters of tissue. In the measurement setting the tissue is heated with focused ultrasound and the temperature evolution is observed through magnetic resonance imaging.