Complexity reduction methods are based on a wide variety of approaches, of which I will present a broad selection. Depending on the objectives - accuracy, speed, computational cost - one or other of these techniques will be favored, but the goal remains the same : to faithfully reproduce reality with controlled accuracy while minimizing algorithmic complexity and simulation time.
In concrete terms, after the " opening lecture ", this lecture will comprise :
- Four sessions dedicated to linear methods of complexity reduction.
- Four sessions dedicated to nonlinear approaches.
Each of these eight lessons will be followed by a guest seminar : an expert from industry or business will share his or her experience, illustrating the impact of these techniques, the gains achieved and the challenges still open in a wide variety of business fields. This interaction " research ↔ industry " aims to provide food for thought and inspire future research developments.
In the first lesson, the reduced basis method for parameterized partial differential equation models will be detailed :a priori and a posteriorierror estimation , greedy algorithm, complexity study, and introduction to EIM/GEIM methods for further cost reduction in the nonlinear case. The second lesson will cover the abstract notions of encoder and decoder, linear and non-linear, allowing complexity to be evaluated via Kolmogorov (linear and non-linear) and Gelfand thicknesses, and will present bounds depending on parametric dimension, illustrated by simple transformations on typical solutions. In the third lesson, the PBDW approach will be presented, which combines the input of an imperfect model and real measurements (possibly tainted by error) to i) reconstruct the state, ii) optimize the placement of measurement sensors and iii) propose error estimates. The fourth lesson will focus on preserving the structural properties of solutions (positivity, invariants).
The fifth lesson will explore nonlinear approximations, distinguishing between coefficients " degree of freedom " and " subject ", which can be represented by expressions derived by polynomial or advanced learning, while keeping computational complexity under control. In thefollowing two lessons, techniques for the numerical analysis of these schemes will be presented :a prioriestimates based on parametric regularity and on the optimality of greedy techniques, as well as a posteriori estimators for qualifying the accuracy of the approximation. The last lesson will propose heuristic approaches, notably via neural networks, which sometimes give solutions without recourse to the model, but which are also based on the complexity reduction developed earlier, with details of their implementation, resolution and the state of the associated numerical analysis.