Solving differential equations fast, precisely, and reliably

Principal Investigator: Joris van der Hoeven

ODELIX is financed by the European Union through an ERC-2023-ADG grant, number 101142171

The project is hosted by CNRS and École polytechnique at the LIX laboratory

Being a language of nature, differential equations are ubiquitous in science and technology. Solving them is a fundamental computational task with a long and rich history. Applications usually require approximate solutions, which can be computed using numerical methods such as Runge-Kutta schemes. Alternatively, one may search for symbolic solutions, which have the advantage of presenting the solutions in an exact and more intelligible way. However, such solutions do not always exist and may be hard to compute.

The present proposal aims at making the resolution of differential equations both faster and more reliable. We will undertake a systematic analysis of the cost to compute both numeric and symbolic solutions, as a function of the required precision, special properties of the equation and its solutions, and hardware specifics of the computer. This includes the cost to certify approximate numeric solutions, e.g. through the computation of provable error bounds. In order to compute symbolic solutions more efficiently, we will develop a new theory that relies on two techniques from computer algebra that were improved significantly in the past decade: numerical homotopy continuation and sparse interpolation.

Theoretical progress on the above problems will be accompanied by open source implementations. For this purpose, we will also implement several high performance libraries of independent interest: non-conventional medium precision arithmetic, reliable homotopy continuation, sparse interpolation, faster-than-just-in-time compilation, etc. Altogether, these implementations will validate the correctness and efficiency of our approach. They should also allow us to tackle problems from applications that are currently out of reach.

The Odelix project is structured into four main axes:

Axis 1: numerical resolution of ODEs

We wish to understand the computational complexity of solving systems of ODEs as a function of various parameters: the required precision, the size and dimension of the system, its stability and numerical conditioning, etc. This knowledge will be applied in order to develop more efficient and reliable solvers that will be distributed as a free software library.

Axis 2: symbolic resolution of ODEs

This axis concerns the symbolic resolution of systems of ODEs, using techniques from differential algebra, homotopy continuation, and sparse interpolation. One major aim is to develop resolution and elimination methods with a lower computational complexity.

Axis 3: asymptotic resolution of ODEs

With his long term collaborators Matthias Aschenbrenner and Lou van den Dries, the principal investigator developed the model theory of asymptotic differential algebra and applications. Within the Odelix project, we wish to develop effective counterparts of this work.

Axis 4: performance support libraries and interfaces

The success of our work on the above three axes will depend on algorithmic improvements and software implementations that need not explicitly involve any differential equations. In particular, this concerns multiple precision arithmetic, reliable computing, HPC implementations, straight-line programs, etc. The required developments are grouped together in this last axis.