Wavelets: Algorithms and Applications (Yves Meyer)

analyzed. Roughly speaking, when the residual criterion is used, the error of an approximate solution is measured by how much it violates the equation rather than by its distance from the true solution. The worst case setting is the most reassuring setting since the cost and errors of algorithms are defined by their worst case performance. However, for some problems it is too pessimistic and it causes the respective worst case complexity to be huge. This is especially the case for multivariate problems. For ill-posed problems, the situation is even worse since the worst case complexity is infinite. To cope with such intractability (or unsolvability) analyzing the problems in other settings is in order. Therefore, in Chapter 7, average case and probabilistic settings are considered. In these settings, the space of functions is equipped with a probability measure (Gaussian measures are assumed through out the book). Then the error and cost of algorithms are measured by their expectations (average case setting) or probability of being too large (probabilistic setting) instead of by their maximal values. It turns out that in these two settings, worst case unsolvable or intractable problems become solvable or tractable. Another two settings, asymptotic and randomized, are considered in Chapter 8. Both settings are closely related to the worst case setting; however, in the asymptotic setting sequences of algorithms with increasing number of information pieces are studied. A winner is the one with the fastest rate of convergence. In the randomized setting, nondeterministic (random) information and algorithms (e.g., Monte Carlo methods) are allowed. am not quoting concrete results on complexities for various problems and/or settings; in particular, how much complexity reduction is gained by switching from the worst case setting to other settings. Doing so would be like telling the conclusion of a suspense novel to the reader who is about to start reading it. Hence, instead, let me strongly recommend this valuable book.