169 research outputs found
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 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
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
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
Projection-based reduced order modeling of an iterative coupling scheme for thermo-poroelasticity
This paper explores an iterative coupling approach to solve
thermo-poroelasticity problems, with its application as a high-fidelity
discretization utilizing finite elements during the training of
projection-based reduced order models. One of the main challenges in addressing
coupled multi-physics problems is the complexity and computational expenses
involved. In this study, we introduce a decoupled iterative solution approach,
integrated with reduced order modeling, aimed at augmenting the efficiency of
the computational algorithm. The iterative coupling technique we employ builds
upon the established fixed-stress splitting scheme that has been extensively
investigated for Biot's poroelasticity. By leveraging solutions derived from
this coupled iterative scheme, the reduced order model employs an additional
Galerkin projection onto a reduced basis space formed by a small number of
modes obtained through proper orthogonal decomposition. The effectiveness of
the proposed algorithm is demonstrated through numerical experiments,
showcasing its computational prowess
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
Weighted reduced order methods for parametrized partial differential equations with random inputs
In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown
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