Benoît Richard
Interventions
The principle of Coulomb explosion imaging is simple: use a powerful x-ray pulse to slap a molecule so hard that it (literally) explodes into its constituting atoms. The fragments can then be measured to produce an indirect image of the molecule.
Currently, a key question in the Coulomb explosion community is what we can actually learn from these measurements. What information is preserved about the state of the molecule just before it exploded?
Our simulations show that if the detectors are perfect, we can use this method to image advanced properties of the molecule, namely the high dimensional shape of its ground-state quantum fluctuations. Unfortunately, we can't directly use the experimental data to this end, because it suffers from several key limitations. We circumvent them by fitting a high-dimensional distribution to the data using gradient-based optimization to perform the fit. Then we simply perform our analysis on the fitted model.
Putting everything together, we demonstrate that the experiment indeed succeeds in imaging subtle quantum properties of the studied molecule.
Floating point numbers are beautiful but are limited. If we push them hard enough, computations that use them become inaccurate, and they are only a very thin sample of all real numbers. In contrast, considering intervals on the real line provides strict guarantee and accuracy and allows to draw general conclusions on continuous subsets of the real line. With intervals, a range of rigorous numerical proofs of mathematical facts can be performed.F or example, the existence or absence of solutions to an equation can be established and proven. By defining all required basic arithmetic operations on intervals, they can be used in arbitrary code, extending their benefits to all sort of calculations.
The goal of IntervalArithmetics.jl is to bring this method to the Julia ecosystem, without sacrificing Julia's flexibility. To reach this objective, we must face a difficult challenge: how to guarantee that interval computations stay rigorous while seamlessly being performed in codes that are, in general, unaware of the existence of interval arithmetic?
In this talk, I will present the strategies that we have implemented for the 1.0 version of the package which allows safe and easy use of rigorous interval computations in Julia, and how it can be applied to root-finding problems.