Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Markus Bachmayr,Albert Cohen +1 more
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In this paper, the decay of the n-widths can be controlled by that of the error achieved by best n-term approximations using polynomials in the parametric variable.Abstract:
Kolmogorov n-widths and low-rank approximations are studied for families of ellip-tic diffusion PDEs parametrized by the diffusion coefficients. The decay of the n-widths can be controlled by that of the error achieved by best n-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the n-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the n-widths.read more
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Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
Kookjin Lee,Kevin Carlberg +1 more
TL;DR: The ability of the method to significantly outperform even the optimal linear-subspace ROM on benchmark advection-dominated problems is demonstrated, thereby demonstrating the method's ability to overcome the intrinsic $n$-width limitations of linear subspaces.
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Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
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A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
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A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
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Practical error bounds for a non-intrusive bi-fidelity approach to parametric/stochastic model reduction
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References
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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
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Convergence Rates for Greedy Algorithms in Reduced Basis Methods
Peter Binev,Albert Cohen,Wolfgang Dahmen,Ronald A. DeVore,Guergana Petrova,Przemysław Wojtaszczyk +5 more
TL;DR: The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs by determining a “good” n-dimensional space to be used in approximating the elements of a compact set $\mathcal{F}$ in a Hilbert space $\ mathscal{H}$.
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Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde's
TL;DR: In this article, the authors considered a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters and showed that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth.
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Convergence Rates of Best N -term Galerkin Approximations for a Class of Elliptic sPDEs
TL;DR: New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error.
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Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
TL;DR: Partial differential equations with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension to derive representation of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series.