Optimal Prediction for Hamiltonian Partial Differential Equations

Alexandre J. Chorin*, Raz Kupferman, Doron Levy

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show that field-theoretical perturbation theory provides a useful device for dealing with quasi-linear problems, and provide a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well.

Original languageAmerican English
Pages (from-to)267-297
Number of pages31
JournalJournal of Computational Physics
Volume162
Issue number1
DOIs
StatePublished - 20 Jul 2000

Bibliographical note

Funding Information:
1This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract DE-AC03-76-SF00098, and in part by the National Science Foundation under Grant DMS94-14631.

Keywords

  • Nonlinear Schrödinger
  • Optimal prediction
  • Perturbation methods
  • Pseudo-spectral methods
  • Regression
  • Underresolution

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