PhD Position F/M Fast solvers for studying light absorption by nanostructured imagers

The Inria centre at Université Côte d'Azur includes 37 research teams and 8 support services. The centre's staff (about 500 people) is made up of scientists of different nationalities, engineers, technicians and administrative staff. The teams are mainly located on the university campuses of Sophia Antipolis and Nice as well as Montpellier, in close collaboration with research and higher education laboratories and establishments (Université Côte d'Azur, CNRS, INRAE, INSERM ...), but also with the regiona economic players.

With a presence in the fields of computational neuroscience and biology, data science and modeling, software engineering and certification, as well as collaborative robotics, the Inria Centre at Université Côte d'Azur  is a major player in terms of scientific excellence through its results and collaborations at both European and international levels.

The exploitation of nanostructuring in order to improve the performance of CMOS imagers based on microlens grids is a very promising avenue. In this perspective, numerical modeling is a key component to accurately characterize and optimize the absorption properties of these complex imaging structures which are intrinsically multiscale (from the micrometer scale of the lenses to the nanometric characteristics of the nanostructured material layers). The present PhD project is proposed in the context of a collaboration between the Atlantis project-team of Inria research center at Université Côte d'Azur and STMicrolectronics (CMOS Imagers division of the Technology for Optical Sensors department) in Crolles. A Cifre funding will support this project.

The objectives of the project are to design (1) a fast electromagnetic simulation approach based on a model reduction technique, to characterize numerically the light trapping in digital imagers exploiting nanostructured pixels and, (2) a multi-objective optimization strategy of the geometrical characteristics of the nanostructuring in order to simultaneously maximize the light absorption in a pixel and to minimize the crosstalk phenomenon between neighboring pixels. For the first time, in addition to the rigorous methods for solving Maxwell's equations, we will be able to benefit from an alternative simulation approach based on model reduction. This new approach can be used in an optimization process and the expected gain in total computation time would be between 10 to 1000.

Atlantis is  a joint project-team  between Inria and  the Jean-Alexandre Dieudonné Mathematics Laboratory at  Université Côte d'Azur. The team  gathers applied mathematicians and  computational scientists who are collaboratively undertaking  research activities aiming at the design, analysis, development and  application of innovative numerical methods for systems of  partial differential equations (PDEs) modelling nanoscale light-matter interaction problems. In this context, the team is  developing  the   DIOGENeS  [https://diogenes.inria.fr/]  software suite,  which  implements  several Discontinuous  Galerkin  (DG)  type methods tailored to the systems  of time- and frequency-domain Maxwell equations  possibly coupled  to  differential  equations modeling  the behaviour of propagation  media at optical frequencies.  DIOGENeS is a unique  numerical   framework  leveraging   the  capabilities   of  DG techniques  for  the simulation  of  multiscale  problems relevant  tonanophotonics and nanoplasmonics.

The core of the present study is concerned with reduced-order modeling for time-domain electromagnetic wave propagation problems. In short, reduced-order models (ROMs) are simplifications of high fidelity, complex models. They capture the behavior of these source  models so that one can quickly study a system's dominant effects using minimal computational resources. The starting point will consist on some preliminary works and contributions that  the team has obtained during the recent years by considering a particular reduced-order modelling technique known as the proper orthogonal decomposition (POD) method.

In the POD approach, a reduced subspace with a significantly smaller dimension is constructed by a set of basis vectors extracted offline from snapshots that are extracted from simulations with a high order DGTD method. By doing so, a non-intrusive POD-based ROM has been developed for the solution of parameterized time-domain electromagnetic scattering problems. The considered parameters are the electric permittivity and the temporal variable. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Starting from these previous contributions, the present study will aim at two objectives: one one hand, the efficiency improvement, as well as the development of the proposed non-intrusive ROM methodology for three-dimensional (3d) parameterized time-domain electromagnetic scattering problems; on the other hand, the study of possible strategies for dealing with geometrical parameters.

Moreover, the recruited PhD student will also be exposed to alternative physics-based alternative numerical modeling in the framework of modal approaches developed at STMicroelectronics. These modal type methods are part of the reference simulation methods routinely in this industrial context.

• Bibliography study on data-driven ROM 
• Analysis of our previous works
• Getting started with existing Python codes for the POD-CSI method in the 2d case
• Specification and development (in Fortran 2003 and Python) of a POD-CSI software for 3d problems
• Adaptation of the 3d POD-CSI software to high performance computing systems
• Extension of the existing POD-CSI approach for handling geometrical parameters
• Detailed assessment of the novel ROM methdologies by considering practical scattering problems 
• Writing of scientific publications

Technical skills and level required :

• Master or engineering degree in numerical mathematics or scientific computing 
• Sound knowledge of numerical analysis for PDEs
• Basic knowledge of physiscs of electromagnetic wave propagation

Software development skills : Python and Fortran 2003, parallel programming with MPI and OpenMP

Relational skills : team worker (verbal communication, active listening, motivation and commitment)

Other valued appreciated : good level of spoken and written english

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Contribution to mutual insurance (subject to conditions)

Duration: 36 months

Location: Sophia Antipolis, France

Gross Salary per month: 2051€ brut per month (year 1 & 2) and 2158€ brut per month (year 3)