Optimal prediction for Hamiltonian partial differential equations. (English) Zbl 0960.65012
The paper deals with the problem of optimal prediction for the solution of an underresolved nonlinear problem. It is shown that perturbation theory provides a ready-made tool for applying the ideas of optimal prediction to problems where the invariant measure is non-Gaussian. Two ways to improve the prediction are under consideration: go to more sophisticated theory, or increase the number of collected variables. The paper is devoted to the second alternative.
A new derivation of the basic methodology is presented. It shows that field-theoretical perturbation theory provides a useful device to deal with quasi-linear problems, and provides a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels.
An examination of the formulas derived by perturbation theory shows that although it is a priori assumed that an invariant measure on the space of solutions is already known, all what is finally used in the process of derivation, presented in the paper, is just a set of moments. This opens space for new studies for applying optimal prediction methods in problems where not the entire invariant measure is known.
A new derivation of the basic methodology is presented. It shows that field-theoretical perturbation theory provides a useful device to deal with quasi-linear problems, and provides a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels.
An examination of the formulas derived by perturbation theory shows that although it is a priori assumed that an invariant measure on the space of solutions is already known, all what is finally used in the process of derivation, presented in the paper, is just a set of moments. This opens space for new studies for applying optimal prediction methods in problems where not the entire invariant measure is known.
Reviewer: T.Semerdjiev (Sofia)
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60G25 | Prediction theory (aspects of stochastic processes) |
| 62M20 | Inference from stochastic processes and prediction |
Keywords:
stochastic processes; inference from stochastic processes; optimal prediction; Hamiltonian partial differential equations; perturbation theory; invariant measure; quasi-linear problems; pseudo-spectral methodReferences:
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