  Using observation data to estimate unknown parameters in computational models
is broadly important. This task is often challenging because solutions are
non-unique due to the complexity of the model and limited observation data.
However, the parameters or states of the model are often known to satisfy
additional constraints beyond the model. Thus, we propose an approach to
improve parameter estimation in such inverse problems by incorporating
constraints in a Bayesian inference framework. Constraints are imposed by
constructing a likelihood function based on fitness of the solution to the
constraints. The posterior distribution of the parameters conditioned on (1)
the observed data and (2) satisfaction of the constraints is obtained, and the
estimate of the parameters is given by the maximum a posteriori estimation or
posterior mean. Both equality and inequality constraints can be considered by
this framework, and the strictness of the constraints can be controlled by
constraint uncertainty denoting a confidence on its correctness. Furthermore,
we extend this framework to an approximate Bayesian inference framework in
terms of the ensemble Kalman filter method, where the constraint is imposed by
re-weighing the ensemble members based on the likelihood function. A synthetic
model is presented to demonstrate the effectiveness of the proposed method and
in both the exact Bayesian inference and ensemble Kalman filter scenarios,
numerical simulations show that imposing constraints using the method presented
improves identification of the true parameter solution among multiple local
minima.
