Reduced order modeling of delayed PEEC circuits

We propose a novel model order reduction technique that is able to accurately reduce electrically large systems with delay elements, which can be described by means of neutral delayed differential equations. It is based on an adaptive multipoint expansion and model order reduction of equivalent first order systems. The neutral delayed differential formulation is preserved in the reduced model. Pertinent numerical results validate the proposed model order reduction approach

Inverse Reduced-Order Modeling

International audienceWe propose a general probabilistic formulation of reduced-order modeling in the case the system state is hidden and characterized by some uncertainty. The objective is to integrate noisy and incomplete observations in the process of building a reduced-order model. We call this problematic inverse reduced-order modeling. This problematic arises in many scientific domains where there exists a need of accurate low-order descriptions of highly-complex phenomena, which can not be directly and/or deterministically observed. Among others, it concerns geophysical studies dealing with image data, which are important for the characterization of global warming or the prediction of natural disasters

Reduced-Order Modeling based on Approximated Lax Pairs

A reduced-order model algorithm, based on approximations of Lax pairs, is proposed to solve nonlinear evolution partial differential equations. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the space where the solution is searched for evolves according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive waves or front propagation. Numerical examples are shown for the KdV and FKPP (nonlinear reaction diffusion) equations, in one and two dimensions

Probabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations

We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the coefficients (e.g. material parameters) exhibit random, fine scale variability. The direct solution to such problems requires grids that are small enough to resolve this fine scale variability which unavoidably requires the repeated solution of very large systems of algebraic equations. We establish a physically inspired, data-driven coarse-grained model which learns a low- dimensional set of microstructural features that are predictive of the fine-grained model (FG) response. Once learned, those features provide a sharp distribution over the coarse scale effec- tive coefficients of the PDE that are most suitable for prediction of the fine scale model output. This ultimately allows to replace the computationally expensive FG by a generative proba- bilistic model based on evaluating the much cheaper CG several times. Sparsity enforcing pri- ors further increase predictive efficiency and reveal microstructural features that are important in predicting the FG response. Moreover, the model yields probabilistic rather than single-point predictions, which enables the quantification of the unavoidable epistemic uncertainty that is present due to the information loss that occurs during the coarse-graining process