Title: On derivatives of smooth functions represented in multiwavelet bases

Abstract

We construct high-order derivative operators for smooth functions represented via discontinuous multiwavelet bases. The need for such operators arises in order to avoid artifacts when computing functionals involving high-order derivatives of solutions of integral equations. Previously high-order derivatives had to be formed by repeated application of a first-derivative operator that, while uniquely defined, has a spectral norm that grows quadratically with polynomial order and, hence, greatly amplifies numerical noise (truncation error) in the multiwavelet computation. The new constructions proceed via least-squares projection onto smooth bases and provide substantially improved numerical properties as well as permitting direct construction of high-order derivatives. We employ either b-splines or bandlimited exponentials as the intermediate smooth basis, with the former maintaining the concept of approximation order while the latter preserves the pure imaginary spectrum of the first-derivative operator and provides more direct control over the bandlimit and accuracy of computation. We demonstrate the properties of these new operators via several numerical tests as well as application to a problem in nuclear physics.

Authors:

Anderson, Joel; Harrison, Robert J. ;

• Search DOE PAGES for author "Harrison, Robert J."
• Search DOE PAGES for ORCID "0000-0002-8777-7466"

Sekino, Hideo; Sundahl, Bryan; Beylkin, Gregory ;

• Search DOE PAGES for author "Beylkin, Gregory"
• Search DOE PAGES for ORCID "0000-0003-3447-1460"

Fann, George I.; Jensen, Stig R.; Sagert, Irina

Publication Date:

Sun Sep 01 00:00:00 EDT 2019

Research Org.:

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:

USDOE Office of Science (SC), Nuclear Physics (NP); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:

1546869

Alternate Identifier(s):

OSTI ID: 1557781; OSTI ID: 1559577

Report Number(s):

LA-UR-18-31060
Journal ID: ISSN 2590-0552; S2590055219300496; 100033; PII: S2590055219300496

Grant/Contract Number:  

AC05-00OR22725; 89233218CNA000001; FG02-87ER40365; SC0018083

Resource Type:

Published Article

Journal Name:

Journal of Computational Physics: X

Additional Journal Information:

Journal Name: Journal of Computational Physics: X Journal Volume: 4 Journal Issue: C; Journal ID: ISSN 2590-0552

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Multiwavelets; Multiresolution; Derivatives; Numerical; Discontinuous; Bandlimited exponentials