The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Implicit Solution of Hyperbolic Equations with Space-Time Adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.     

Quantum dynamics of gas-phase SN2 reactions.

Understanding the state-resolved dynamics of elementary chemical reactions involving polyatomic molecules, such as the well-known reaction mechanism of nucleophilic bimolecular substitution (SN2), is one of the principal goals in chemistry. In this Review, the progress in the quantum mechanical treatment of SN2 reactions in the gas phase is reviewed. The potential energy profile of this class of reactions is characterized by two relatively deep wells, which correspond to pre- and post-reaction chargedipole complexes. As a consequence, the complex-forming reaction is dominated by Feshbach resonances. Calculations in the energetic continuum constitute a major challenge because the high density of resonance states imposes considerable requirements on the convergence and the energetic resolution of the scattering data. However, the effort is rewarding because new insights into the details of multimode quantum dynamics of elementary chemical reactions can be obtained.     

Rate optimal estimation with the integration method in the presence of many covariates

For multivariate regressors, integrating the Nadaraya–Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.