Abstract
In this work, we present a generalized methodology for analyzing the convergence of quasi-optimal Taylor and Legendre approximations, applicable to a wide class of parameterized elliptic PDEs with finite-dimensional deterministic and stochastic inputs. Such methods construct an optimal index set that corresponds to the sharp estimates of the polynomial coefficients. Our analysis, furthermore, represents a novel approach for estimating best M-term approximation errors by means of coefficient bounds, without the use of the standard Stechkin inequality. In particular, the framework we propose for analyzing asymptotic truncation errors is based on an extension of the underlying multi-index set into a continuous domain, and then an approximation of the cardinality (number of integer multi-indices) by its Lebesgue measure. Several types of isotropic and anisotropic (weighted) multi-index sets are explored, and rigorous proofs reveal sharp asymptotic error estimates in which we achieve sub-exponential convergence rates [of the form Mexp (- (κM) 1 / N) , with κ a constant depending on the shape and size of multi-index sets] with respect to the total number of degrees of freedom. Through several theoretical examples, we explicitly derive the constant κ and use the resulting sharp bounds to illustrate the effectiveness of Legendre over Taylor approximations, as well as compare our rates of convergence with current published results. Computational evidence complements the theory and shows the advantage of our generalized framework compared to previously developed estimates.
Original language | English |
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Pages (from-to) | 451-493 |
Number of pages | 43 |
Journal | Numerische Mathematik |
Volume | 137 |
Issue number | 2 |
DOIs | |
State | Published - Oct 1 2017 |
Funding
The authors wish to graciously thank Prof. Ron DeVore for his interest in our work, his patience in discussing the analysis of “best M-term” approximations, and his tremendously helpful insights into the theoretical developments we pursued in this paper. This material is based upon work supported in part by the US Air Force of Scientific Research under Grant Number 1854-V521-12 and by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract and Award Numbers ERKJ314, ERKJ259, and ERKJE45; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the US Department of Energy under Contract DE-AC05-00OR22725.
Funders | Funder number |
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US Air Force of Scientific Research | 1854-V521-12 |
U.S. Department of Energy | |
Office of Science | |
Advanced Scientific Computing Research | ERKJ259, ERKJ314, ERKJE45 |
Laboratory Directed Research and Development | DE-AC05-00OR22725 |
Keywords
- 05A16
- 41A10
- 65N12