Institute of Numerical Mathematics, Russian Academy of Sciences,

Gubkina 8, Moscow 117333, Russia

The mosaic-skeleton method was bred in a simple observation that rather large
blocks in very large matrices coming from integral formulations can be
approximated accurately by a sum of just few skeletons (some say dyads or
rank-one matrices). These blocks might correspond to a region where the
kernel is smooth enough, and anyway it can be a region where the kernel is
approximated by a short sum of separable functions (in other words, functional
skeletons). The mosaic-skeleton approximations are easy to result in fast
approximate matrix-vector multiplication algorithms close by nature to those
of multipole, interpolation, and wavelet-based approaches. All the said
techniques involve some hierarcy of interface regions and function
approximants. What the mosaic-skeleton method differs in from others is a
matrix analysis view on largely the same problem. Such a view can be very
useful due to the generality of matrix theory approaches. Since the effect of
approximations is like that of having small-rank matrices, we find it
pertinent to say about *mosaic ranks* of a matrix which turn to be pretty
small for many nonsingular matrices.

We cover a wide class of applications. In particular, following Brandt, we
propose to call *f(x,y)* an *asymptotically smooth* function if
there exist *c,d > 0* and a real number *g* such that

General matrix approximation algorithms were applied to integral equations of lots of applications from the classical potential flow or electrostatic problems to thermal analysis for stratified media and electromagnetic scattering problems. Selected numerical results will be presented.