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Reduced basis methods with adaptive snapshot computations
Springer Science and Business Media LLC - Tập 43 - Trang 257-294 - 2016
We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the co......
Blossoms and Optimal Bases
Springer Science and Business Media LLC - - 2004
It is now classical to define blossoms by means of intersections of osculating flats. We consider here the most general context of spline spaces with sections in arbitrary extended Chebyshev spaces and with connections defined by arbitrary lower triangular matrices with positive diagonal elements. We show how the existence of blossoms in such spaces automatically leads to optimal bases in the sens......
Classification of Bivariate Configurations with Simple Lagrange Interpolation Formulae
Springer Science and Business Media LLC - Tập 20 - Trang 5-16 - 2004
In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes than the degree is used for classifying ......
Adaptive energy preserving methods for partial differential equations
Springer Science and Business Media LLC - - 2018
Approximation of functions from Korobov spaces by deep convolutional neural networksAbstract The efficiency of deep convolutional neural networks (DCNNs) has been demonstrated empirically in many practical applications. In this paper, we establish a theory for approximating functions from Korobov spaces by DCNNs. It verifies rigorously the efficiency of DCNNs in approximating functions of many variables with some variable structures and their abili... ...
Springer Science and Business Media LLC - - 2022
Efficient construction of 2-chains representing a basis of $H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})$
Springer Science and Business Media LLC - Tập 44 - Trang 1411-1440 - 2018
We present an efficient algorithm for the construction of a basis of
$H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})$
via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of
......
Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions
Springer Science and Business Media LLC - - 2019
Optimal recovery of 3D X-ray tomographic data via shearlet decomposition
Springer Science and Business Media LLC - Tập 39 - Trang 227-255 - 2012
This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-r......
The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions
Springer Science and Business Media LLC - Tập 29 - Trang 179-206 - 2007
In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approac......
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