This IMA Volume in Mathematics and its Applications WAVE PROPAGATION IN COMPLEX MEDIA is based on the proceedings of two workshops: • Wavelets, multigrid and other fast algorithms (multipole, FFT) and their use in wave propagation and • Waves in random and other complex media. Both workshops were integral parts of the 1994-1995 IMA program on "Waves and Scattering." We would like to thank Gregory Beylkin, Robert Burridge, Ingrid Daubechies, Leonid Pastur, and George Papanicolaou for their excellent work as organizers of these meetings. We also take this opportunity to thank the National Science Foun­ dation (NSF), the Army Research Office (ARO, and the Office of Naval Research (ONR), whose financial support made these workshops possible. A vner Friedman Robert Gulliver v PREFACE During the last few years the numerical techniques for the solution of elliptic problems, in potential theory for example, have been drastically improved. Several so-called fast methods have been developed which re­ duce the required computing time many orders of magnitude over that of classical algorithms. The new methods include multigrid, fast Fourier transforms, multi pole methods and wavelet techniques. Wavelets have re­ cently been developed into a very useful tool in signal processing, the solu­ tion of integral equation, etc. Wavelet techniques should be quite useful in many wave propagation problems, especially in inhomogeneous and nonlin­ ear media where special features of the solution such as singularities might be tracked efficiently.