Complexity of Verification and Computation for Ibc Problems

We analyze the complexity of verifying whether a given element is within " of the solution element. This may be contrasted with the complexity of computing an element that is within " of the solution element. For discrete problems with " = 0 veriication is no harder than computation in any setting. For IBC problems veriication can be easier or harder than computation. We will show that the worst case complexity of veriication for IBC problems is often innnite. We therefore switch to the probabilistic case and study the probabilistic complexity of veriication as a function of the error tolerance " and the probability of failure. We assume that the solution element is speciied by a linear continuous functional deened on a Banach space equipped with a Gaussian measure. For xed and small ", the complexity of veriication is zero, whereas for xed " and small the complexity of veriication is essentially a function of only and maybe exponentially harder than the complexity of computation.