T-IFISS: a toolbox for adaptive FEM computation

Bespalov, Alex; Rocchi, Leonardo; Silvester, David J.

doi:10.1016/j.camwa.2020.03.005

Cited by 15 publications

(17 citation statements)

References 48 publications

(75 reference statements)

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“…This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. We have implemented the computational codes on the basis of the MATLAB TT-Toolbox 58 and the T-IFISS 1.1 toolbox 59 and run on one core of Balena, an Intel E5-2650 v2 CPU.…”

Section: Numerical Resultsmentioning

confidence: 99%

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Benner

Dolgov

Onwunta

et al. 2020

Numerical Methods in Fluids

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Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

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Section: Numerical Resultsmentioning

confidence: 99%

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Benner

Dolgov

Onwunta

et al. 2020

Numerical Methods in Fluids

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“…We now consider two test problems: one where refractive index is a random field and the other close to a resonant frequency. In both problems, the discretisation over physical space uses a bilinear finite element approximation [6,15] and the implementation involves IFISS [13] and S-IFISS [4] packages.…”

Section: Numerical Resultsmentioning

confidence: 99%

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

Liao¹

2019

EAJAM

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The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

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“…To check the robustness of the SC error estimation strategy we can compare the pattern of refinement with the pattern that results when the same test problem is solved using the single-level stochastic Galerkin adaptive strategy in [7,6] that is built into T-IFISS [8] (with a slightly smaller accuracy tolerance). When we ran this test, the numerical solution generated by SG is visually identical to that reference solution in Fig.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

Bespalov¹,

Silvester²,

Feng³

2021

Preprint

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A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard & Nobile in 2018 (SIAM J. Numer. Anal., 56, 3121-3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available online.

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T-IFISS: a toolbox for adaptive FEM computation

Cited by 15 publications

References 48 publications

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

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