Scientific Training V

Inverse problems: theory, algorithms and applications

PhD course introducing the basics of meta-modelling techniques for efficient inverse problems. Introduction to inverse problems regularised gradient-based and gradient-free optimisers, and probabilistic approaches such as Bayesian inversion. Lectures on the application of inverse problems in the context of image processing and data to simulation transfers.


Professor Pierre Kerfriden,

Learning objectives





  • Dr. P. Kerfriden, Centre des Matériaux Mines ParisTech, PSL University, France / Cardiff University, School of Engineering
  • Prof. L. Champion, École Normale Supérieure de Paris-Saclay, France 
  • Dr. M. Genet, Mechanics Department & Solid Mechanics Laboratory (M3DISIM team) École Polytechnique, Palaiseau, France
  • Sami Hilal

Syllabus and systems/software prerequisites

See specifications under each day programme below.


Day 1: 24 February 2020 - Inverse problems: general concepts and approaches

A. Deterministic approach
- regularisation
- optimisation algorithms

B. Probabilistic approach
- Bayesian formulation
- sampling techniques for posterior distributions
C. Model approximations
- Meta-modelling (polynomial Chaos, Gaussian processes)

Necessary pre-installations:
Exercises using FEniCS (python package dolfin), Scipy and Numpy (algebra and optimisatin packages for python) Sklearn (machine learning) and and/or OpenTurns (uncertainty quantification). Students are advised to install these packages on their laptops prior to the course (preferably using anaconda).

Day 2: 25 February 2020 - Data assimilation and Reduced Order Modelling

A. Inverse problems in complex nonlinear mechanics
- duality-based mCRE approach
- goal-oriented version
- model selection & enrichment

B. Model reduction
- Generalities and applications in inverse problems
- Focus on the PGD method
- PGD-based inverse analysis with applications

C. Sequential data assimilation
- Kalman filtering and regularization through the physics
- full Bayesian formulation
- real-time assimilation using PGD (DDDAS concept)

Day 3: 26 February 2020 - Image processing and parameter identification in biomechanics

A. Motion quantification/Image registration/Image correlation
- Finite element method
- Image similarity metrics
- Mechanical regularization

B. Model parameter identification
- Cost functions
- Optimization algorithms

C. Unloaded configuration identification
- Inverse elastostatic problem
- Including residual stresses

D. On the fly registration+identification/Integrated correlation
- Cost functions
- Optimization algorithms

Necessary pre-installations:
Exercises will use FEniCS and VTK, within the dolfin_dic library. Students with Linux  machines will need to install these libraries; others will need to install Docker on their machines (a container will be provided).