$$
U(x, b) = \int \sigma(x^\prime, B |D) - \sigma(x^\prime, B |D \cup \{x\}) dx^\prime
$$
where $b \in [0, B]$ is the fidelity parameter.
Basically, this formulation reduces the uncertainty over the space.
Acquisition function is $U(x) / c(x)$ where $c(x)$ is the cost function
According to a reference, the greedy algorithm of $U(x)/c(x)$ becomes an approximation algorithm of the optimal solution in the knapsack problem with total cost constraint. (submodular optimization and the method is called cost-adjusted strategy)
Confusing points
it seems they are not doing Bayesian optimization, but they would like to just reduce the uncertainties of a simulator over the search space and that is the exactly reason why they are using $U$. apparently, The greedy algorithm on $U/c$ reduces the uncertainties very efficiently analytically.
Multi-fidelity experimental design for ice-sheet simulation
Main points
$$ U(x, b) = \int \sigma(x^\prime, B |D) - \sigma(x^\prime, B |D \cup \{x\}) dx^\prime $$
where $b \in [0, B]$ is the fidelity parameter.
Basically, this formulation reduces the uncertainty over the space.
Confusing points