Abstract
We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where each input represents an angle of rotation. Our method is based on an anisotropic quasi-optimal estimate for the decay rate of the Fourier coeficients of the model; a least-squares fit to the coeficients of the interpolant is used to estimate the anisotropy. Thus, our adaptive approximation strategy begins with a coarse isotropic interpolant, which is gradually refined using the estimated anisotropic rates. The procedure takes several iterations where ever-more accurate interpolants are used to generate ever-improving anisotropy rates. We present several numerical examples of our algorithm where the adaptive procedure successfully recovers the theoretical \best" convergence rate, including an application to a periodic PES approximation. An open-source implementation of our algorithm resides in the Tasmanian UQ library developed at Oak Ridge National Laboratory.
| Original language | English |
|---|---|
| Pages (from-to) | A2436-A2460 |
| Journal | SIAM Journal on Scientific Computing |
| Volume | 42 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2020 |
Funding
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section August 26, 2019; accepted for publication (in revised form) May 4, 2020; published electronically August 24, 2020. https://doi.org/10.1137/19M1283483 Funding: The work of the first author was supported by an NSF Graduate Reearch Fellowship under grant DGE-1746939 and by an appointment to the Oak Ridge National Laboratory Advanced Short-Term Research Opportunity (ASTRO) Program, sponsored by the U.S. Department of Energy and administered by the Oak Ridge Institute for Science and Education. The work of the second author was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration; by the U.S. Defense Advanced Research Projects Agency, Defense Sciences Office under contract and award HR0011619523 and 1868-A017-15; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under 674 contract DE-AC05-00OR22725. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
| Funders | Funder number |
|---|---|
| Defense Sciences Office | 1868-A017-15, HR0011619523 |
| U.S. Department of Energy Office of Science | |
| National Science Foundation | DGE-1746939 |
| U.S. Department of Energy | |
| Directorate for Education and Human Resources | 1746939 |
| Defense Advanced Research Projects Agency | |
| National Nuclear Security Administration | |
| Oak Ridge Institute for Science and Education | 17-SC-20-SC |
| Laboratory Directed Research and Development | DE-AC05-00OR22725 |
Keywords
- Adaptive refinement
- Periodicitypreserving approximation
- Sparse interpolation
- Trigonometric interpolation