16,523 research outputs found
A reduced basis localized orthogonal decomposition
In this work we combine the framework of the Reduced Basis method (RB) with
the framework of the Localized Orthogonal Decomposition (LOD) in order to solve
parametrized elliptic multiscale problems. The idea of the LOD is to split a
high dimensional Finite Element space into a low dimensional space with
comparably good approximation properties and a remainder space with negligible
information. The low dimensional space is spanned by locally supported basis
functions associated with the node of a coarse mesh obtained by solving
decoupled local problems. However, for parameter dependent multiscale problems,
the local basis has to be computed repeatedly for each choice of the parameter.
To overcome this issue, we propose an RB approach to compute in an "offline"
stage LOD for suitable representative parameters. The online solution of the
multiscale problems can then be obtained in a coarse space (thanks to the LOD
decomposition) and for an arbitrary value of the parameters (thanks to a
suitable "interpolation" of the selected RB). The online RB-LOD has a basis
with local support and leads to sparse systems. Applications of the strategy to
both linear and nonlinear problems are given
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
- …