  In the framework of solid mechanics, the task of deriving material parameters
from experimental data has recently re-emerged with the progress in full-field
measurement capabilities and the renewed advances of machine learning. In this
context, new methods such as the virtual fields method and physics-informed
neural networks have been developed as alternatives to the already established
least-squares and finite element-based approaches. Moreover, model discovery
problems are starting to emerge and can also be addressed in a parameter
estimation framework. These developments call for a new unified perspective,
which is able to cover both traditional parameter estimation methods and novel
approaches in which the state variables or the model structure itself are
inferred as well. Adopting concepts discussed in the inverse problems
community, we distinguish between all-at-once and reduced approaches. With this
general framework, we are able to structure a large portion of the literature
on parameter estimation in computational mechanics - and we can identify
combinations that have not yet been addressed, two of which are proposed in
this paper. We also discuss statistical approaches to quantify the uncertainty
related to the estimated parameters, and we propose a novel two-step procedure
for identification of complex material models based on both frequentist and
Bayesian principles. Finally, we illustrate and compare several of the
aforementioned methods with mechanical benchmarks based on synthetic and real
data.
