  We introduce fast algorithms for generalized unnormalized optimal transport.
To handle densities with different total mass, we consider a dynamic model,
which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we
derive the corresponding $L^1$ generalized unnormalized Kantorovich formula. We
further show that the problem becomes a simple $L^1$ minimization which is
solved efficiently by a primal-dual algorithm. For $p=2$, we derive the $L^2$
generalized unnormalized Kantorovich formula, a new unnormalized Monge problem
and the corresponding Monge-Amp\`ere equation. Furthermore, we introduce a new
unconstrained optimization formulation of the problem. The associated gradient
flow is essentially related to an elliptic equation which can be solved
efficiently. Here the proposed gradient descent procedure together with the
Nesterov acceleration involves the Hamilton-Jacobi equation which arises from
the KKT conditions. Several numerical examples are presented to illustrate the
effectiveness of the proposed algorithms.
