Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven, Jan S.; Rozza, Gianluigi; Stamm, Benjamin

doi:10.1007/978-3-319-22470-1

Cited by 805 publications

(1,235 citation statements)

References 94 publications

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“…Because of its flexibility and the fact that it does not require the evaluation of error bounds or indicators for the selection of basis functions, we rely on this latter, see e.g. [10,11,15] for a further understanding of (weak) greedy algorithms and POD. We start by computing n s high-fidelity solutions {u h (µ i )} ns i=1 (called snapshots) corresponding to the parameter values {µ i } ns i=1 .…”

Section: The Reduced Basis Methods For Parametrized Pdesmentioning

confidence: 99%

“…that is by a multiplicative combination of P −1 h (µ) and Q N k (µ), where P −1 h (µ) ∈ R N h ×N h is a nonsingular fine grid preconditioner and Q N k (µ) is an iteration-dependent coarse component which is a tailored for the error equation (11). When using a multiplicative combination of two preconditioners, the Richardson iterations can be rewritten by means of two half-steps, that is, if…”

Section: Msrb Preconditioners For the Richardson Methodmentioning

confidence: 99%

“…Reduced order modeling (ROM) techniques, and in particular the reduced basis (RB) method, represent an accurate, reliable and efficient tool to deal with parametrized problems, see e.g. [10,11]. Given µ ∈ D, the RB method seeks an approximation of the highfidelity solution u h (µ) in a reduced space that is spanned by a set of basis functions given by linear combinations of high-fidelity solutions corresponding to different parameter instances.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

Santo

Deparis

Manzoni

2017

Communications in Applied and Industrial Mathematics

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We analyze the numerical performance of a preconditioning technique recently proposed in [1] for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coefficients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.

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Section: The Reduced Basis Methods For Parametrized Pdesmentioning

confidence: 99%

Section: Msrb Preconditioners For the Richardson Methodmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

Santo

Deparis

Manzoni

2017

Communications in Applied and Industrial Mathematics

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“…For more details on the reduced basis theory we address the interested reader to e.g., Rozza et al [5], Hesthaven et al [27], and Quarteroni et al [28]. We already introduced U n+1 h that, at each time instant is the a high-fidelity approximation of the exact solution and is computed as a finite element solution with a sufficiently fine mesh.…”

Section: Numerical Reductionmentioning

confidence: 99%

“…For references to standard greedy algorithms applied to parametrized PDEs see e.g., Hesthaven et al [27] and Quarteroni et al [28].…”

Section: Greedy Enrichmentmentioning

confidence: 99%

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Colciago

Deparis

2018

Front. Appl. Math. Stat.

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This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

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Iga

2023

Iga

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Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Cited by 805 publications

References 94 publications

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Iga

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