1,720,981 research outputs found
Regularity and sparse approximation of the recursive first moment equations for the lognormal Darcy problem
Full text linkWe study the Darcy boundary value problem with lognormal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Hölder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived
Perturbation analysis for the stochastic Darcy problem
No full textWe study the single-phase flow in a saturated, bounded randomly heterogeneous porous medium. We model the permeability tensor as a lognormal random field. We perform a perturbation analysis, expanding the solution in Taylor series. We give a theoretical upper bound for the norm of the residual of the Taylor series which predicts the divergence of the series as the polynomial degree goes to infinity. We then numerically determine the optimal order for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach
Shape optimization for a noise reduction problem by non-intrusive parametric reduced modeling
No full textWe study a PDE-constrained optimization problem, where the shape and liner material of the nacelle of an aircraft engine are optimized in order to minimize the noise radiated by the engine. More precisely, the acoustic problem is modeled by the Helmholtz equation with varying wavenumber k on an exterior domain. A model reduction strategy is employed to alleviate the cost of the design optimization: the minimal rational interpolation technique is used to construct a surrogate (w.r.t. k) for the quantity of interest at fixed shape/material parameter values, and a parametric model order reduction approach is employed to combine surrogates at different shape/material designs, resulting in a nonintrusive methodology. Numerical experiments for shape and shape/material optimization are provided, to showcase the effectiveness of the presented methodology
Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots
Full text linkWe introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives in a different discrete space that resolves the local singularities of the analytical solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding a function-valued version of the the standard rational interpolation method (V-SRI) and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive V-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the V-SRI method seems to be the best-performing one
Finite element differential forms on curvilinear cubic meshes and their approximation properties.
No full textMoment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term
Full text linkWe study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (n≥ 1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m≥ 1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces
Convergence analysis of Padé approximations for Helmholtz frequency response problems
Full text linkA Tensor-Product Finite Element Cochain Complex with Arbitrary Continuity
Full text linkWe develop tensor product finite element cochain complexes of arbitrary
smoothness on Cartesian meshes of arbitrary dimension. The first step is the
construction of a one-dimensional -conforming finite element cochain
complex based on a modified Hermite interpolation operator, which is proved to
commute with the exterior derivative by means of a general commutation lemma.
Adhering to a strict tensor product construction we then derive finite element
complexes in higher dimensions
Perturbation analysis for the Darcy problem with log-normal permeability
Full text linkWe study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theoretical upper bound for the norm of the residual of the Taylor expansion which predicts the divergence of the series as the polynomial degree goes to infinity. We provide a formula to compute the optimal degree for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach. Our theoretical findings are confirmed by numerical experiments in the simple case where the permeability field is described using one random variableCSQ
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