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 verification is no harder than computation in any setting. For IBC problems verification can be easier or harder than computation. We will show that the worst case complexity of verification for IBC problems is often infinite. We therefore switch to the probabilistic case and study the probabilistic complexity of verification as a function of the error tolerance " and the probability of failure ffi. We assume that the solution element is specified by a linear continuous functional defined on a Banach space equipped with a Gaussian measure. For fixed ffi and small ", the complexity of verification is zero, whereas for fixed " and small ffi the complexity of verification is essentially a function of only ffi and maybe exponentially harder than the ..