Among the recent approaches developped in numerical analysis of parametric PDEs (Partial Differential Equations) I am interested in "reduced basis" descretization. This method, that writes the solution of a parametric EDP as a linear combination of the same EDP corresponding to a different choice of parameters, was implemented recently e.g. by the team of Prof. A.T.Patera at MIT. With A.T.Patera and Yvon Maday we pursued a theoretical study of the convergence of this method in order to explain its amazing performances.
Parallel in time discretization schemes
Whereas microscopic simulations can be performed on larger and larger space scales, in particular, through the use of multiscale techniques and massively parallel computers very few methods are available to achieve similar results in the time domain. Since for space, through the use of parallel computers, larger and larger scales have become accessible in microscopic simulations, it could be seducing to transpose the same procedures in the time domain. Naturally, contrary to space, time is sequential and this precludes a priori the straightforward implementation of a parallel approach. The parareal discretization scheme (designed initially with JL Lions and YMaday) is a possible solution to this endeavor. It combines very precise simulations run in parallel on disjoint time segments with a coarse (approximate) simulation over the entire time span. This scheme has also been extended to the control of evolution equations.
A posteriori error analysis
The description of the state of quantum systems and the prediction of their properties is often intermediated by the ab initio computations on the wavefunction. This computations give rise to problems of too large size for direct, multi-purpose, approaches. Adapted approximation techniques are then used but very few theoretical works exists that can guarantee the quality of the results thus obtained. It appears useful thus to quantify the confidence to put in the result of a numerical computation; to this end we use methods known in scientific computing as a posteriori. These methods aim at constructing quantities computable only from the numerical solution obtained (thus the denomination of a posteriori) that give qualitative of quantitative indications on the computation that was performed. These techniques have been applied e.g. to the method of adiabatic variable and to the Hartree-Fock equation.
