Multiscale problems and methods in numerical simulations : lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 9-15, 2001 / C. Canuto

Cohen illustrates the difference between linear and nonlinear approximations by means of a simple example. He then surveys theoretical and computational aspects of nonlinear approximations applied to data compression, statistical estimation, and numerical simulation by wavelet thresholding or adaptive partitions. Multiscale appears in the context of space refinement for wavelets. Dahmen surveys multiscale and wavelet methods for operator equations. Examples are given of sparse and quasi-sparse representations of functions and preconditioning. The main features of wavelet bases are summarized. Construction and analysis principles are presented for multiscale decompositions. An application to a scalar second-order elliptic boundary value problem leads to boundary integral equations and saddle points problems. Many aspects of adaptive wavelet schemes are treated in some detail. Bramble surveys multilevel methods in finite elements. A model problem in Sobolev spaces is approximated by finite elements. This leads to a two-level multigrid method. An abstract multilevel algorithm is described and analyzed under some regularity assumptions. An analysis under less stringent assumptions requires non-nested spaces and varying forms. It is shown that the multilevel framework provides computationally efficient realizations of norms on Sobolev scales. (MathSciNet)