arXiv subjects
Publications and source records attributed to Erich Novak.
This is an open letter that we sent to S. Ilie, G. Soederlind and R.M. Corless in August 2008.
We study the integration of functions with respect to an unknown density. We compare the simple Monte Carlo method (which is almost optimal for a certain large class of inputs) and compare it with the Metropolis algorithm (based on a suitable ball walk). Using MCMC we prove (for certain classes of inputs) that adaptive methods are much better than nonadaptive ones. Actually, the curse of dimension (for nonadaptive methods) can be broken by adaption.
We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable nonlinear frame widths. The main advantage of frames compared to Riesz basis, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains--also for the upper bounds. Key words: elliptic operator equation, worst case error, frames, nonlinear approximation, best n-term approximation, manifold width, Besov spaces on Lipschitz domains
We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large dimension, the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions. We also show, for any odd degree l=2k+1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.
We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings.
We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings. We identify those cases where optimal nonlinear approximation is better than optimal linear approximation.
We consider the computation of the mean of sequences in the quantum model of computation. We determine the query complexity in the case of sequences which satisfy a $p$-summability condition for $1\le p<2$. This settles a problem left open in Heinrich (2001).
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hoelder classes with d variables. The optimal orders for the complexity of deterministic and (general) randomized methods are known. We obtain the respective optimal orders for quantum algorithms and also for restricted Monte Carlo (only coin tossing instead of general random numbers). To summarize the results one can say that (1) there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if the smoothness is small; (2) there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if the smoothness is small.