* HIGHLY OSCILLATORY PROBLEMS
Part my research in recent years deals with numerical integrators for highly oscillatory problems. The aim is to solve numerically ordinary or partial differential equations in situations where the solution is oscillatory and presents widely different time-scales. For instance in molecular dynamics, the configuration of the molecule evolves slowly (say over milliseconds) and simultaneously the atoms are subject to fast vibrations (with periods of the order of a few femtoseconds). The challenge is to mimic the slow dynamics without keeping track accurately of the fast motions (which requires an incredible amount of compuational work). Unfortunately most numerical techniques are unable to meet this challenge and do not provide useful results unless the time-step of the simulation is chosen small with respect to the fast time scales.
In particular I have been working in the so-called Mollified Impulse methods, a new class of methods meant to be applied with long-time steps. These were introduced by Bob Skeel, Bosco Garcia-Archilla and myself back in 1998. I have also been much interested in the Heterogenous Multiscale Methods introduced by E, Engquist and thier coworkers. I wrote an article on how to analyze them via the modulated Fourier expansion of Hairer and Lubich. I have also worked (with Mari Paz Calvo) on applications to mechanical problems (including Differential-Algebraic models) and algorithms with variable-steps.
** B-SERIES, WORD SERIES AND OTHER FORMAL TECHNIQUES
The analysis of numerical integrators for highly oscillatory problems led Phillipe Chartier, Ander Murua and myself to a remarkable discovery. By using B-series (formal series first introduced to analyze numerical integrators) we were able to discover a new way of averaging dynamical systems to any order of accuracy; this is therefore an application outside numerical mathematics of the modern tools of numerical analysis. In the new approach the averaged system is written down without any need to perform successive changes of variables. The averaged system is written down by computing recursively scalar coefficients that are universal, in the sense that they do not change with the specific system being averaged.
In turn the study of the method of averaging resulted in the introduction by Ander Murua and myself of word series; these are formal series similar to B-series but simpler to handle. Word series are very useful to analyze splitting integrators and to manipulate dynamical systems (for instance to reduce them to normal form or to find formal invariants of motion). My PhD student Alfonso Alamo investigated the use of word series in the analysis of splitting integrators for stochastic differential equations.
*** STOCHASTIC ANALYSIS/MONTE CARLO METHODS
In recent years I have become very interested in probability and stochastic processes. I have been particularly concerned with the hybrid or Hamiltonian Monte Carlo method, a sampling technique widely used in Bayesian statistics. Since both probability and the numerical integration of Hamiltonian dynamics play an important role in the technique, it provides an ideal topic for me to work on. I am co-operating with several people, as it may be seen from my list of publications. With Nawaf Bou-Rabee I have prepared a long paper on the Hamiltonian Monte Carlo method to appear in Acta Numerica in 2018.