Consider the sequential optimization of a continuous, possibly non-convex, and expensive to evaluate objective function f . The problem can be cast as a Gaussian Process (GP) bandit where f lives in a reproducing kernel Hilbert space (RKHS). The state of the art analysis of several learning algorithms shows a significant gap between the lower and upper bounds on the simple regret performance. When N is the number of exploration trials and γ N is the maximal information gain, we prove an Õ( γ N /N ) bound on the simple regret performance of a pure exploration algorithm that is significantly tighter than the existing bounds. We show that this bound is order optimal up to logarithmic factors for the cases where a lower bound on regret is known. To establish these results, we prove novel and sharp confidence intervals for GP models applicable to RKHS elements which may be of broader interest.1 Zeroth-order feedback signifies observations from f in contrast to first-order feedback which refers to observations from gradient of f as e.g. in stochastic gradient descent [see, e.g., Agarwal et al., 2011, Vakili andZhao, 2019].Preprint. Under review.

We consider the domain decomposition method approach to solve the Helmholtz equation. A new double sweep based approach is presented valid for any type of interface boundary conditions and that benefits from the overlap. It makes use of a splitting of the local problems in the subdomain. Despite of the fact that a first order interface boundary conditions is used, the splitting double sweep method demonstrates good stability properties with respect to the number of subdomains and the frequency even for heterogeneous media. Convergence is improved when compared to the double sweep method for all of our test cases: waveguide, open cavity and wedge problems.

No abstract

No abstract