Error behavior and optimal discretization of chaotic differential equations

In this thesis, the simulation of chaotic systems is considered. For many chaotic systems, we desire to make estimates of mean values of quantities of interest, and in this case, the effect of chaos is to introduce behavior that naturally lends itself to statistical, rather than deterministic, description. When simulating chaotic systems using discrete versions of governing differential equations, then, chaos introduces statistical errors alongside discretization errors. These statistical errors are generally one of two types: transient spin-up error before the system reaches the attractor (i.e. the stationary distribution of long-run states) and sampling error due to finite-time averaging of trajectories on the attractor. In this work, we first propose an error model to describe the expected absolute errors on the attractor of a chaotic ordinary differential equation system. This model for the error implies optimal choices of timestep and sampling time to minimize the error in the simulation- including discretization error and sampling error- given some computational budget. Adding a model for the spin-up error, this allows the description of the optimal choice of timestep, sampling, and spin-up times. Next, we develop a small-sample Bayesian approach that allows the estimation of the discretization and the sampling error using only a small number of simulation results with distinct timesteps and sampling times on the attractor. We then extend the approach for spatiotemporally chaotic partial differential equation systems, which introduces error due to spatial discretization in addition to the temporal discretization errors and statistical errors. Finally, we augment the small-sample approach with corrections for non-negligible spin-up transient behavior, then embed the resulting small-sample method in a naive explore-exploit algorithm. Using this algorithm, we demonstrate that given a fixed total computational budget such an approach can allow chaotic simulations that achieve near-optimal estimates without strong prior knowledge of the behavior of the system. In addition to this near-optimal discretization, the method allows an a posteriori estimate of the simulation error in the final result after the exploitation stage.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Aeronautics and Astronautics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2023

Author dc:creator

• Frontin, Cory

Advisor dc:contributor.advisor

• Darmofal, David

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright MIT

Licence dc:rights.uri

• http://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/150280

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/150280