The complexity of de nite elliptic problems with noisy dataTechnical Report CUCS - 035 - 96

We study the complexity of 2mth order deenite elliptic problems Lu = f (with homogeneous Dirichlet boundary conditions) over a d-dimensional domain , error being measured in the H m (()-norm. The problem elements f belong to the unit ball of W r;p ((), where p 2 2; 1] and r > d=p. Information consists of (possibly-adaptive) noisy evaluations of f or the coeecients of L. The absolute error in each noisy evaluation is at most. We nd that the nth minimal radius for this problem is proportional to n ?r=d + , and that a noisy nite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be eeciently implemented using multigrid techniques. Using these results, we nd tight bounds on the "-complexity (minimal cost of calculating an "-approximation) for this problem, said bounds depending on the cost c() of calculating a-noisy information value. As an example, if the cost of a-noisy evaluation is c() = ?s (for s > 0), then the complexity is proportional to (1=") d=r+s .