Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions

A Non-Intrusive Data-Driven Reduced Order Model for Parametrized CFD-DEM Numerical Simulations

The investigation of fluid-solid systems is very important in a lot of industrial processes. From a computational point of view, the simulation of such systems is very expensive, especially when a huge number of parametric configurations needs to be studied. In this context, we develop a non-intrusive data-driven reduced order model (ROM) built using the proper orthogonal decomposition with interpolation (PODI) method for Computational Fluid Dynamics (CFD) -- Discrete Element Method (DEM) simulations. The main novelties of the proposed approach rely in (i) the combination of ROM and FV methods, (ii) a numerical sensitivity analysis of the ROM accuracy with respect to the number of POD modes and to the cardinality of the training set and (iii) a parametric study with respect to the Stokes number. We test our ROM on the fluidized bed benchmark problem. The accuracy of the ROM is assessed against results obtained with the FOM both for Eulerian (the fluid volume fraction) and Lagrangian (position and velocity of the particles) quantities. We also discuss the efficiency of our ROM approach

Nonintrusive approximation of parametrized limits of matrix power algorithms -- application to matrix inverses and log-determinants

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones