Port TMJets algorithm

[mforets]

In https://github.com/JuliaReach/Reachability.jl/pull/537 we added the TMJets continuous post, that relies on functionality from TaylorSeries, TaylorIntegration and TaylorModels.

For further development we'd like to handle the higher-level integration steps in Reachability.jl.

Moreover, some possible extensions (we should create specific issues for each of these):

• [x] port validated_integ into the TMJets continuous post (https://github.com/JuliaReach/Reachability.jl/issues/611)
• [ ] possibility to decrease the time-step; (currently we must play with orderT and absolute tolerance) to control it and thus the remainder of the integration before checking the safe region
• [ ] add domain splitting to improve the boxes of the flow pipe in the loop (currently we can only do once in the beginning of the computation)
• [ ] pass the invariant to the continuous post
• [x] make validated_integ to return the polynomials; then we can use different evaluation methods to enclose the polynomials
• define and use lazy taylor models
• different set representations for taylor model flowpipe-guard intersections (see Xin Chen's thesis)

---

CC: @lbenet, @dpsanders

[dpsanders]

What kind of lazy Taylor model do you have in mind? I already have a package LazyTaylorSeries.jl that has a proof of concept.

[mforets]

Thanks for pointing to your package, i added it to the list above!

What i had in mind was to make TalorModel <: LazySet, but why this can be useful? It could ease the way to define minkowski sums or arrays of minkowski sums lazily, lazy transformations of Taylor models, etc. I imagine that one can be interested in clustering taylor models for flowpipe computations (see e.g. Section 3.1 page 91, Frehse and Ray)

On the other hand, in LazySets we often assume that the sets are convex that is where the machinery of support functions shines. As a first step, i'd be interested to see how does overapproximations with taylor models with some of the existent lazysets play (eg. zonotopes or polytopes).