  We present an algorithm for compressing the radiosity view factor model
commonly used in radiation heat transfer and computer graphics. We use a format
inspired by the hierarchical off-diagonal low rank format, where elements are
recursively partitioned using a quadtree or octree and blocks are compressed
using a sparse singular value decomposition -- the hierarchical matrix is
assembled using dynamic programming. The motivating application is
time-dependent thermal modeling on vast planetary surfaces, with a focus on
permanently shadowed craters which receive energy through indirect irradiance.
In this setting, shape models are comprised of a large number of triangular
facets which conform to a rough surface. At each time step, a quadratic number
of triangle-to-triangle scattered fluxes must be summed; that is, as the sun
moves through the sky, we must solve the same view factor system of equations
for a potentially unlimited number of time-varying righthand sides. We first
conduct numerical experiments with a synthetic spherical cap-shaped crater,
where the equilibrium temperature is analytically available. We also test our
implementation with triangle meshes of planetary surfaces derived from digital
elevation models recovered by orbiting spacecrafts. Our results indicate that
the compressed view factor matrix can be assembled in quadratic time, which is
comparable to the time it takes to assemble the full view matrix itself. Memory
requirements during assembly are reduced by a large factor. Finally, for a
range of compression tolerances, the size of the compressed view factor matrix
and the speed of the resulting matrix vector product both scale linearly (as
opposed to quadratically for the full matrix), resulting in orders of magnitude
savings in processing time and memory space.
