A Reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations

In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours \cite{Ali2018}. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.Comment: arXiv admin note: text overlap with arXiv:2001.0082

Space-time POD-Galerkin approach for parametric flow control

In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control can be used in order to fill the gap between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by viscous Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not losing accuracy

A monolithic and a partitioned Reduced Basis Method for Fluid-Structure Interaction problems

The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid-Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek-Hron benchmark test case, with a fluid Reynolds number Re = 100

A Weighted POD Method for Elliptic PDEs with Random Inputs

In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems. \ua9 2018, Springer Science+Business Media, LLC, part of Springer Nature

ATHENA: Advanced Techniques for High Dimensional Parameter Spaces to Enhance Numerical Analysis

ATHENA is an open source Python package for reduction in parameter space. It implements several advanced numerical analysis techniques such as Active Subspaces (AS), Kernel-based Active Subspaces (KAS), and Nonlinear Level-set Learning (NLL) method. It is intended as a tool for regression, sensitivity analysis, and in general to enhance existing numerical simulations' pipelines tackling the curse of dimensionality. Source code, documentation, and several tutorials are available on GitHub at https://github.com/mathLab/ATHENA under the MIT license

Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems

In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.Comment: 26 pages, 11 figure

Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting

In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases