Acceleration of Newton-like methods for nonlinear systems by preflattening techniques

21 Mar 2024
Job Information

Organisation/Company

IFP Energies nouvelles (IFPEN)

Research Field

Mathematics » Computational mathematics

Researcher Profile

First Stage Researcher (R1)

Country

France

Application Deadline

31 Aug 2024 - 23:59 (Europe/Paris)

Type of Contract

Temporary

Job Status

Full-time

Hours Per Week

35

Offer Starting Date

4 Nov 2024

Is the job funded through the EU Research Framework Programme?

Not funded by an EU programme

Is the Job related to staff position within a Research Infrastructure?

No

Offer Description

The numerical discretization of physical models in many fields leads to very large nonlinear algebraic systems. The resolution of these systems, which is usually achieved by successive linearization like Newton's method, needs to be fast in order to increase software efficiency. However, one of the obstacles to this speed is precisely the degree of non-linearity of the system under consideration. Indeed, it is known that when the system is linear, Newton's method converges in a single iteration (at least in exact arithmetic). The best theoretical results available to date for Newton show that its rate of convergence depends on two problem-specific constants, one of which can be viewed as a measure of local nonlinearity.

Over the last two decades, preconditioning techniques have been introduced to speed up the resolution of nonlinear systems. They rely on solving an equivalent system that is judiciously built by analogy with the linear case. However, unlike the linear case, there is no guarantee that the new system will be more suitable for Newtonian solution than the original one, even though this is generally observed in numerical experiments. This is because, in the linear case, it is the conditioning of the matrix that governs the rate of convergence of the linear solver, and we can be sure that the new conditioning will be more favorable. In the nonlinear case, this is no longer the case. We do not know exactly which scalar quantity has decreased between the old and new systems. So, even if preconditioning works, its foundation remains heuristic.

Based on this observation, we propose to explore transformations of the original system into an equivalent system that is "less nonlinear" in a quantitative sense to be specified in relation to Newton's speed of convergence. To distinguish this approach (which acts only on the external non-linear level) from classical non-linear preconditioning (which acts simultaneously on both the linear and non-linear levels), we introduce the term pre-flattening. The conceptual clarity afforded by separating the two iteration levels would enable us to envisage targeted and relevant avenues of work.

Keywords: nonlinear systems, Newton’s method, rate of convergence, conditioning, preconditioning, nonlinearity measure, preflattening, reactive transport, porous media

Academic supervisors: Quang Huy TRAN (IFPEN) and Clément CANCÈS (INRIA Lille)

Other supervisor: Ibtihel BEN GHARBIA (IFPEN)

Doctoral School: ED STIC 580, University Paris-Saclay

Requirements

Research Field

Mathematics » Computational mathematics

Education Level

Master Degree or equivalent

Skills/Qualifications

Master’s degree in Numerical Analysis or Scientific Computing

Specific Requirements

Programming languages: C++, Python...

Languages

ENGLISH

Level

Excellent

Additional Information
Benefits

IFP Energies nouvelles is a French public-sector research, innovation and training center. Its mission is to develop efficient, economical, clean and sustainable technologies in the fields of energy, transport and the environment. For more information, see our WEB site .

IFPEN offers a stimulating research environment, with access to first in class laboratory infrastructures and computing facilities. IFPEN offers competitive salary and benefits packages. All PhD students have access to dedicated seminars and training sessions.

Eligibility criteria

This subject is part of a PEPR-type research program run by the French government. The admission of a candidate to the PhD position must be approved by a Defense Security Official of IFPEN's supervisory ministry, after an investigation that could induce some delay.

Work Location(s)

Number of offers available

1

Company/Institute

IFP Energies nouvelles

Country

France

City

Rueil-Malmaison

Postal Code

92852

Street

1 et 4 avenue de Bois-Préau

Geofield

Where to apply

E-mail

[email protected]

Contact

City

Rueil-Malmaison

Website

http://www.ifpenergiesnouvelles.com/

Street

4 avenue de Bois-Préau

Postal Code

92852

STATUS: EXPIRED