## Section: New Results

### Model reduction / multiscale algorithms

#### Model Order Reduction

Participants : Mohamed Reda El Amri, Youssef Marzouk, Maëlle Nodet, Clémentine Prieur, Alessio Spantini, Olivier Zahm.

Another point developed in the team for sensitivity analysis is model reduction. To be more precise regarding model reduction, the aim is to reduce the number of unknown variables (to be computed by the model), using a well chosen basis. Instead of discretizing the model over a huge grid (with millions of points), the state vector of the model is projected on the subspace spanned by this basis (of a far lesser dimension). The choice of the basis is of course crucial and implies the success or failure of the reduced model. Various model reduction methods offer various choices of basis functions. A well-known method is called “proper orthogonal decomposition" or “principal component analysis". More recent and sophisticated methods also exist and may be studied, depending on the needs raised by the theoretical study. Model reduction is a natural way to overcome difficulties due to huge computational times due to discretizations on fine grids. In [68], the authors present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. The numerical experiments in the paper show significant computational savings, as well as efficiency of the error bound.

When a metamodel is used (for example reduced basis metamodel, but also kriging, regression, $...$) for estimating sensitivity indices by Monte Carlo type estimation, a twofold error appears: a sampling error and a metamodel error. Deriving confidence intervals taking into account these two sources of uncertainties is of great interest. We obtained results particularly well fitted for reduced basis metamodels [69]. In [66], the authors provide asymptotic confidence intervals in the double limit where the sample size goes to infinity and the metamodel converges to the true model. These results were also adapted to problems related to more general models such as Shallow-Water equations, in the context of the control of an open channel [70].

When considering parameter-dependent PDE, it happens that the quantity of interest is not the PDE's solution but a linear functional of it. In [67], we have proposed a probabilistic error bound for the reduced output of interest (goal-oriented error bound). By probabilistic we mean that this bound may be violated with small probability. The bound is efficiently and explicitly computable, and we show on different examples that this error bound is sharper than existing ones.

A collaboration has been started with Christophe Prieur (Gipsa-Lab) on the very challenging issue of sensitivity of a controlled system to its control parameters [70]. In [71], we propose a generalization of the probabilistic goal-oriented error estimation in [67] to parameter-dependent nonlinear problems. One aims at applying such results in the previous context of sensitivity of a controlled system.

More recently, in the context of the Inria associate team UNQUESTIONABLE, we have extended the focus of the axis on model order reduction. Our objectives are to understand the kinds of low-dimensional structure that may be present in important geophysical models; and to exploit this low-dimensional structure in order to extend Bayesian approaches to high-dimensional inverse problems, such as those encountered in geophysical applications. Our recent and future efforts are/will be concerned with parameter space dimension reduction techniques, low- rank structures in geophysical models and transport maps tools for probability measure approximation. At the moment, scientific progress has been achieved in different directions, as detailed below: A first paper [45] has been submitted on gradient- based dimension reduction of vector-valued functions. Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. In this work, we propose a gradient-based method for detecting these directions and using them to construct ridge approximations of such functions, in the case where the functions are vector-valued. The methodology consists of minimizing an upper bound on the approximation error, obtained by subspace Poincaré inequalities. We have provided a thorough mathematical analysis in the case where the parameter space is equipped with a Gaussian probability measure. A second work [46] has been submitted, which proposes a dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation noise. In this work, the likelihood function is approximated by a ridge function, i.e., a map which depends non-trivially only on a few linear combinations of the parameters. The ridge approximation is built by minimizing an upper bound on the Kullback-Leibler divergence between the posterior distribution and its approximation. This bound, obtained via logarithmic Sobolev inequalities, allows one to certify the error of the posterior approximation. A sample-based approximation of the upper bound is also proposed. In the framework of the PhD thesis of Reda El Amri, a work on data-driven stochastic inversion via functional quantization was submitted. In this paper [36], a new methodology is proposed for solving stochastic inversion problems through computer experiments, the stochasticity being driven by functional random variables. Main tools are a new greedy algorithm for functional quantization, and the adaptation of Stepwise Uncertainty Reduction techniques.