This talk considers the problem of computing a linear approximation of quadratic
Wasserstein distance $W_2$. In particular, we compute an approximation of the
negative homogeneous weighted Sobolev norm whose connection to Wasserstein
distance follows from a classic linearization of a general Monge-Ampére
equation. We reduce the computational problem to solving an elliptic boundary
value problem involving the Witten Laplacian, which is a Schrödinger operator
of the form
$$
H = -\Delta + V,
$$
where $V$ is a potential that depends on $f$. We show that this connection leads
to efficient methods of calcuation and a number of interesting applications
including embedding and smoothing images.