Optimal prediction methods estimate the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their effect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation, the orthogonal dynamics equation, needed to evaluate a non-Markovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. We present the constructions in detail together with several useful variants, provide simple examples, and point out the relation to the fluctuation-dissipation formulas of statistical physics.
Bibliographical noteFunding Information:
We would like to thank Prof. G.I. Barenblatt, Dr. E. Chorin, Mr. E. Ingerman, Dr. A. Kast, Mr. K. Lin, Mr. P. Okunev, Mr. B. Seibold, Mr. P. Stinis, Prof. J. Strain, and Prof. B. Turkington for helpful discussions and comments. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under Contract DE-AC03-76-SF00098, and in part by the National Science Foundation under Grant DMS98-14631. RK was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities and by the Alon Fellowship.
- Hamiltonian systems
- Hermite polynomials
- Langevin equations
- Optimal prediction
- Orghogonal dynamics