High performance resolution of linear differential equations at multiple precisions (M/F)

Job Description

Thesis Subject

The aim of this PhD is to develop high performance algorithms for the
efficient evaluation of holonomic functions, i.e. functions f that satisfy a
non-trivial linear differential equation L_r(z) f^(r)(z) + ⋯ + L_0(z) f(z) = 0 for
polynomials L_0, …, L_r ∈ ℂ[z]. The main focus will be on exploiting the
capabilities of modern processors as fully as possible. A broad spectrum of
target precisions will be investigated, from machine precision to high
precision, while also considering intermediate precisions that are interesting
for applications. The new algorithms will be implemented and tested within the
framework of the C++ library JIL [9]. Contingent on the initial results, the
algorithms may also be extended to more general differential equations.

This general goal involves several ingredients. First of all, the underlying
fixed or floating point arithmetic has to be chosen or designed with care. How
to represent medium precision fixed and floating point numbers? How to exploit
SIMD (single instruction multiple data) instructions to speed up evaluations?
For a specific target precision, how to design the most efficient
implementation dedicated to this precision? When needed, the candidate will
extend techniques that were proposed by van der Hoeven and Lecerf.

For small and medium precisions, naive evaluation strategies are often most
efficient, but it is non-trivial to implemented them in a way that best
exploits modern hardware. The idea is to automatically generate code for the
evaluation of holonomic functions in the form of straight-line programs (SLPs)
as provided by the JIL library. In order to make this integration smooth, it
will probably be necessary to extend JIL with new language features and new
compilation strategies dedicated to SLPs. Pushing the SLP framework to new
limits will be one of the theoretical and practical challenges for this PhD.

A special case of interest is that of elementary (e.g., trigonometric and
hyperbolic) and special (e.g., Bessel) functions. Many of these functions are
holonomic or can be expressed in terms of holonomic functions, but they also
satisfy more specific properties that allow for additional strategies.
Here the main scenario where progress can be expected is that of intermediate
precisions beyond the discrete set covered by the standard C libm but where
the full power of multiple-precision libraries like MPFR or FLINT may be
overkill.

Finally, we intend to explore the theoretical and practical complexity of
evaluating holonomic functions. It is now classical that holonomic functions
can be evaluated fast at very high precisions. However, most
complexity estimates use the target precision as the only parameter. It would
be interesting to better understand the evaluation cost as a function of the
evaluation point and the complexity of the differential equation to be solved.
This includes a finer study of low and medium precisions,
which can be backed by concrete timings, ideally speaking.
In a similar vein, we wish to better understand the relation between
complexity, stability, and singularities.

Your Work Environment

- The PhD will be co-supervised by Marc Mezzarobba.
- The supervisors plan to have regular meetings with the student.
- The student will have office space, a workstation or laptop, within a standard scientific working environment.

Compensation and benefits

Compensation

2300 € gross monthly

Annual leave and RTT

44 jours

Remote Working practice and compensation

Pratique et indemnisation du TT

Transport

Prise en charge à 75% du coût et forfait mobilité durable jusqu’à 300€

About the offer

About the CNRS

The CNRS is a major player in fundamental research on a global scale. The CNRS is the only French organization active in all scientific fields. Its unique position as a multi-specialist allows it to bring together different disciplines to address the most important challenges of the contemporary world, in connection with the actors of change.

CNRS

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