I’ve registered to this event (registration is free!) and proposed a talk, see below. I look forward to participating - particularly since the venue is not far from my place- although it is too early to confirm yet.

The talk that I have proposed concerns flowpipe construction. Flowpipe construction consists of under or over-approximating the sets of states reachable by dynamical systems.

As it turns out, a method has been recently developed for the class of polynomial ODEs with uncertain initial states (see [1], abbreviated XFZ18under, and earlier references therein). This method consists of reducing the Hamilton-Jacobi-Bellman equation to a hierarchy of semidefinite programs.

In this notebook I’ve considered the problem of approximating the flowpipe of a system of polynomial ODEs using XFZ18under. This is a Julia implementation that relies on the JuMP ecosystem (JuMP, PolyJuMP, SumOfSquares, MathProgInterface, MathOptInterfaceMosek) and the JuliaAlgebra ecosystem (MultivariatePolynomials, DynamicPolynomials). The plots are generated with ImplicitEquations.

Below you’ll find the abstract of the contribution; the arXiv (but not definitive) version is 1801.09526.

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches.

Further information and examples about the software of this project, go to the github page of JuliaReach. In particular, to know more about existing and planned algorithms for reachability computations you can follow the project Reachability.jl. Contributions are welcome as well!

Within the SageMath mathematics software system, I’m looking forward to be a mentor of the project Polynomial optimisation and sums of squares, together with Dr. Dima Pasechnik. This projects touches the area of semidefinite programming (SDP) in Sage. See below the full description for further details.

Skills: algebra, Python, C/C++, Cython, linear/nonlinear optimisation, numerical analysis (MSc/PhD level)

Abstract: Optimisation problems with polynomial constraints are efficiently, in practice, solved by building an increasingly tight sequence of semidefinite programming (SDP) relaxations, one of them known as Lasserre hierarchy.

Description: While Sagemath already has an ability to solve SDPs, more work has to be done in particular to implement moment matrices for polynomials and sums of squares approximations of nonnegative polynomials, and a frontend allowing to define systems of polynomial inequalities using natural syntax, similar to what already can be done with systems of linear inequalities. Another related topic would be to interface to Sagemath more SDP solvers (currently only CVXOPT is available), and possibly prototype an arbitrary precision SDP solver to avoid typical numerical difficulties arising in sums of squares-based SDP relaxations, e.g. implementing a version of the algorithm from Klerk and Vallentin.

See also all the projects of the SageMath organization following this link.

I look forward to participate in the Univ. Grenoble Alpes team, hosted by GRICAD.

Groupe d’utilisateurs de Python pour discuter de tout ce qui tourne autour de Python mais aussi d’autres langages et technologies developeurs

The group has currently 282 members, and more than 30 meetups in a wide range of topics have taken place, about open source, programming in Python, software and web development, among others. See the group webpage for further information.

In Tuesday, October 10th, I will give an introductory talk on SageMath and the Scientific Python ecosystem. Find below the abstract of my talk, and here is the webpage of the meetup event.

Here is the abstract of my talk:

SageMath is a community-driven mathematics software system, built on top of many existing open-source packages such as NumPy, SciPy, matplotlib, Sympy, Maxima and many more, its mission being to provide a viable alternative to commercial software such as Matlab, Mathematica, etc. In this talk we will cover the basics of SageMath using Jupyter notebooks. The examples can be tried online in the collaborative computing environment CoCalc. In the final part of the talk, we will see how to easily integrate with Cython (an optimising static compiler), getting the combined power of Python and C under the same umbrella.

I look forward to attending the session Hierarchies of SDP Relaxations for Polynomial Systems. We will contribute the paper Semidefinite Characterization of Invariant Measures, joint work with V. Magron and D. Henrion.

This is the abstract of our contribution:

We consider the problem of characterizing measures which are invariant with respect to the dynamics of polynomial systems, under general semialgebraic set constraints. This characterization consists in numerical approximation of the moments and support of such invariant measures. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the characterization of the support of an invariant measure which is singular with respect to the Lebesgue measure. In both cases, our results apply for discrete-time and continuous-time polynomial systems. Each problem is handled through an adequate reformulation into a linear optimization problem over measures, solved in practice with two distinct hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations.

Under specific assumptions, the first moment hierarchy allows to extract a sequence of polynomials converging in L2-norm to the density of an absolutely continuous invariant measure. The second moment hierarchy allows to approximate as close as desired the support of a singular invariant measure with the level sets of the Christoffel functions associated to the moment matrices of this measure.

We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.

The conference will address all aspects of optimization and its applications, including:

• continuous optimization (smooth and nonsmooth)
• optimal control and calculus of variations
• optimization with PDE
• numerical methods for mathematical programming
• robust optimization
• mixed integer optimization
• differential inclusions and set-valued analysis
• stochastic optimization
• multicriteria optimization
• optimization techniques for industrial applications

This semester, I was the co-tutor (together with Goran Frehse), of the student Frédéric Viry, who worked in the project:

Reachability of high-dimensional dynamical systems through low-dimensional projections.

In this project, a new decomposition-based strategy is built for the reachability analysis of high-dimensional affine systems. Moreover, Frédéric developed a Julia module for efficient reachability analysis in low dimensions which uses support functions as set representations.

Check the general audience wiki (in French) here for additional info and references.

See also the github hosted carlin project.

Below you will find some extra information.

Content: In this course we study controllability and stabilization methods for infinite-dimensional systems described by partial differential equations (PDEs). Classical PDEs as the transport equation, the heat equation, and the wave equation are considered in detail. Moreover, other models as the Korteweg-de Vries equation and the Kuramoto-Sivashinsky equation are studied from a control viewpoint.

Schedule:

Monday 23
08:30-10:00
10:15-11:45

Tuesday 24
08:30-10:00
10h15:11h45

Wednesday 25
08:30-10:00
10:15-11h45
and
13:30-15:00
15:15-16:45

Thursday 26
08:30-10:00
10:15-11:45

I look forward to LaTeXing my handwritten notes! See the advances in this SageMathCloud public link.

During my stay at Faber I would like to contribute to the Sage mathematical software project. Free software projects such as Sage, rather than a “solo” activity, evolve by virtue of the sum of efforts of individuals (developers, in the jargon). This represents a great opportunity to link with experienced Sage developers and to foster a sense of community. I’ll focus on improving the geometry and mathematical optimization libraries, and to write documentation. I hope these efforts will advance my own research projects/teaching, and also to be useful for people using Sage.

I look forward to these Sage days!

Olot seen from mount Montsacopa. Source: Wikipedia