  If the rates, $K(x,y)$, at which particles of size $x$ coalesce with
particles of size $y$ is known, then the mean-field evolution of the
particle-size distribution of an ensemble of irreversibly coalescing particles
is described by the Smoluchowski equation. We study the corresponding inverse
problem which aims to determine the coalescence rates, $K(x,y)$ from
measurements of the particle size distribution. We assume that $K(x,y)$ is a
homogeneous function of its arguments, a case which occurs commonly in
practice. The problem of determining, $K(x,y)$, a function to two variables,
then reduces to a simpler problem of determining a function of a single
variable plus two exponents, $\mu$ and $\nu$, which characterise the scaling
properties of $K(x,y)$. The price of this simplification is that the resulting
least squares problem is nonlinear in the exponents $\mu$ and $\nu$. We
demonstrate the effectiveness of the method on a selection of coalescence
problems arising in polymer physics, cloud science and astrophysics. The
applications include examples in which the particle size distribution is
stationary owing to the presence of sources and sinks of particles and examples
in which the particle size distribution is undergoing self-similar relaxation
in time.
