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A theoretical analysis of deep neural networks and parametric PDEs
We derive upper bounds on the complexity of ReLU neural networks approximating the
solution maps of parametric partial differential equations. In particular, without any …
solution maps of parametric partial differential equations. In particular, without any …
Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
We estimate the expressive power of certain deep neural networks (DNNs for short) on a
class of countably-parametric, holomorphic maps u: U→ ℝ on the parameter domain U=[− 1 …
class of countably-parametric, holomorphic maps u: U→ ℝ on the parameter domain U=[− 1 …
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients∗
Page 1 ESAIM: M2AN 51 (2017) 341–363 ESAIM: Mathematical Modelling and Numerical …
Page 1 ESAIM: M2AN 51 (2017) 341–363 ESAIM: Mathematical Modelling and Numerical …
Convergence rates of high dimensional Smolyak quadrature
We analyse convergence rates of Smolyak integration for parametric maps u: U→ X taking
values in a Banach space X, defined on the parameter domain U=[− 1, 1] N. For parametric …
values in a Banach space X, defined on the parameter domain U=[− 1, 1] N. For parametric …
Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)
We give a convergence proof for the approximation by sparse collocation of Hilbert-space-
valued functions depending on countably many Gaussian random variables. Such functions …
valued functions depending on countably many Gaussian random variables. Such functions …
Reduced basis greedy selection using random training sets
Reduced bases have been introduced for the approximation of parametrized PDEs in
applications where many online queries are required. Their numerical efficiency for such …
applications where many online queries are required. Their numerical efficiency for such …
Fully discrete approximation of parametric and stochastic elliptic PDEs
It has recently been demonstrated that locality of spatial supports in the parametrization of
coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial …
coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial …