Hernan Leövey

Automatic evaluations of cross-derivatives

Cross–derivatives are mixed partial derivatives that are obtained by differentiating at most once in every coordinate direction. They are a computational tool in combinatorics and high– dimensional integration. Here we present two methods of computing exact values of all cross– derivatives at a given point both following the general philosophy of automatic differentiation. Implementation details are discussed and numerical results given.

Derivative‐Based Global Sensitivity Analysis: Upper Bounding of Sensitivities in Seismic‐Hazard Assessment Using Automatic Differentiation

Abstract Seismic‐hazard assessment is of great importance within the field of engineering seismology. Nowadays, it is common practice to define future seismic demands using probabilistic seismic‐hazard analysis (PSHA). Often it is neither obvious nor transparent how PSHA responds to changes in its inputs. In addition, PSHA relies on many uncertain inputs. Sensitivity analysis (SA) is concerned with the assessment and quantification of how changes in the model inputs affect the model response and how input uncertainties influence the distribution of the model response. Sensitivity studies are challenging primarily for computational reasons; hence, the development of efficient methods is of major importance. Powerful local (deterministic) methods widely used in other fields can make SA feasible, even for complex models with a large number of inputs; for example, automatic/algorithmic differentiation (AD)‐based adjoint methods. Recently developed derivative‐based global sensitivity measures can combine the advantages of such local SA methods with efficient sampling strategies facilitating quantitative global sensitivity analysis (GSA) for complex models. In our study, we propose and implement exactly this combination. It allows an upper bounding of the sensitivities involved in PSHA globally and, therefore, an identification of the noninfluential and the most important uncertain inputs. To the best of our knowledge, it is the first time that derivative‐based GSA measures are combined with AD in practice. In addition, we show that first‐order uncertainty propagation using the delta method can give satisfactory approximations of global sensitivity measures and allow a rough characterization of the model output distribution in the case of PSHA. An illustrative example is shown for the suggested derivative‐based GSA of a PSHA that uses stochastic ground‐motion simulations.

Quasi-Monte Carlo methods for linear two-stage stochastic programming problems

Quasi-Monte Carlo (QMC) algorithms are studied for generating scenarios to solve two-stage linear stochastic programming problems. Their integrands are piecewise linear-quadratic, but do not belong to the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to

A first look at quasi-Monte Carlo for lattice field theory problems

L_{2}

High dimensional integration of kinks and jumps—Smoothing by preintegration

L2. This implies that randomly shifted lattice rules may achieve the optimal rate of convergence

Improving Monte Carlo integration by symmetrization

O(n^{-1+\delta })