What Is the Complexity of Solution-Restricted Operator Equations?

Abstract

We study the worst case complexity of operator equations Lu = f where L: G → X is a bounded linear injection of normed linear spaces. Past work on the complexity of such problems has generally required the class F of problem elements f to be the unit ball of X. However, there are many problems for which this choice of F yields unsatisfactory results. Mixed elliptic—hyperbolic problems are one example. the difficulty being that our technical tools are nor strong enough to give good complexity bounds. Ill-posed problems are another example. because we know that the complexity of computing finite-error approximations is infinite if F is a ball in X. In this paper, we pursue another idea. Rather than directly restrict the class F of problem elements f, we will consider problems that are solution-restricted: i.e., we restrict the class U of solution elements u. In particular, we assume that U is the unit hall of a normed linear space W that is densely, continuously embedded in G. The main idea is that our problem can now be reduced to the standard approximation problem of approximating the embedding of W into G.This allows us to characterize optimal information and algorithms for our problem..We use this idea to study three problems: the Tricomi problem (a mixed hyperbolic— elliptic problem arising in the study of transonic flow), the inverse finite Laplace transform (an ill-posed problem arising. e.g.. in geomathematics), and the backwards heat equation. We determine the problem complexity and derive nearly optimal algorithms for each of these problems.