  The operation of machine tools often demands a highly accurate knowledge of
the tool center point's (TCP) position. The displacement of the TCP over time
can be inferred from thermal models, which comprise a set of geometrically
coupled heat equations. Each of these equations represents the temperature in
part of the machine, and they are often formulated on complicated geometries.
  The accuracy of the TCP prediction depends highly on the accuracy of the
model parameters, such as heat exchange parameters, and the initial
temperature. Thus it is of utmost interest to determine the influence of these
parameters on the TCP displacement prediction. In turn, the accuracy of the
parameter estimate is essentially determined by the measurement accuracy and
the sensor placement.
  Determining the accuracy of a given sensor configuration is a key
prerequisite of optimal sensor placement. We develop here a thermal model for a
particular machine tool. On top of this model we propose two numerical
algorithms to evaluate any given thermal sensor configuration with respect to
its accuracy. We compute the posterior variances from the posterior covariance
matrix with respect to an uncertain initial temperature field. The full matrix
is dense and potentially very large, depending on the model size. Thus, we
apply a low-rank method to approximate relevant entries, i.e. the variances on
its diagonal. We first present a straightforward way to compute this
approximation which requires computation of the model sensitivities with with
respect to the initial values. Additionally, we present a low-rank tensor
method which exploits the underlying system structure. We compare the
efficiency of both algorithms with respect to runtime and memory requirements
and discuss their respective advantages with regard to optimal sensor placement
problems.
