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We present convergence estimates of two types of greedy algorithms in terms of the entropy numbers of underlying compact sets. In the first part, we measure the error of a standard greedy reduced basis method for parametric PDEs by the entropy numbers of the solution manifold in Banach spaces. This contrasts with the classical analysis based on the Kolmogorov [Formula: see text]-widths and enables us to obtain direct comparisons between the algorithm error and the entropy numbers, where the multiplicative constants are explicit and simple. The entropy-based convergence estimate is sharp and improves upon the classical width-based analysis of reduced basis methods for elliptic model problems. In the second part, we derive a novel and simple convergence analysis of the classical orthogonal greedy algorithm for nonlinear dictionary approximation using the entropy numbers of the symmetric convex hull of the dictionary. This also improves upon existing results by giving a direct comparison between the algorithm error and the entropy numbers.
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Accelerating many-nucleon basis generation for high performance computing enabled ab initio nuclear structure studies
We present the problem of generating a many-nucleon basis in [Formula: see text]-scheme for ab initio nuclear structure calculations in a symmetry-adapted no-core shell model framework. We first discuss and analyze the basis construction algorithm whose baseline implementation quickly becomes a significant bottleneck for large model spaces and heavier nuclei. The outcomes of this analysis are utilized to propose a new scalable version of the algorithm. Its performance is consequently studied empirically using the Blue Waters supercomputer. The measurements show significant acceleration achieved with over two orders of magnitude speedups realized for larger model spaces.
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- PAR ID:
- 10547640
- Publisher / Repository:
- SAGE Publications
- Date Published:
- Journal Name:
- The International Journal of High Performance Computing Applications
- Volume:
- 33
- Issue:
- 3
- ISSN:
- 1094-3420
- Format(s):
- Medium: X Size: p. 522-533
- Size(s):
- p. 522-533
- Sponsoring Org:
- National Science Foundation
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