A Wavelet Collocation Method for Solving PDEs Rich Vuduc U.C. Berkeley Math 228B richie@cs.berkeley.edu May 11, 2001 Abstract This report provides an overview of a recent paper by Vasilyev and Bowman [J. Comp. Phys., 165:660–693, 2000]. The paper discusses the use of so-called second-generation wavelet bases in a method-of-lines approach to the numerical solution of partial differential equations. The focus of this review is on the paper’s justification of its adap- tive grid method based on the approximating properties of a wavelet basis. Specifically, I (attempt to) explain the key ideas from wavelet approximation theory that give a sense for why wavelets lead to sparse approximations. Furthermore, I repeat a number of the paper’s ex- periments that emphasize the intuitive idea. Finally, I provide a list of references to related work on the use of wavelets for PDE problems that might make for interesting future class readings. 1 Overview Vasilyev and Bowman [VB00] proposed an algorithm for numerically solving partial differential equations that is essentially the method-of-lines approach with a wavelet analysis to reduce the size of the computational grid. The con- text is an arbitrary one-dimensional time-dependent non-linear PDE with general boundary conditions: u = u(x,t) u t = F (x,t,u,∂u,... ) 0 = Φ(x,t,u,∂u,... ) The basic procedure consists of the following steps. Assume that at time t we have some (possibly irregular) spatial discretization identified by the N points {x J k }, where 0 ≤ k<N and N is some integer multiple of 2 J . 1