Extrapolation methods

[ChrisRackauckas]

http://www.columbia.edu/~mh2078/MonteCarlo/MCS_SDEs.pdf
https://arxiv.org/pdf/math/0612523.pdf
https://www.jstor.org/stable/2158354?seq=1#metadata_info_tab_contents
https://hal.inria.fr/inria-00075490/file/RR-1069.pdf (EXEM)

[frankschae]

If we define the functional of the expected value at a time point T as:

$$V_{11}^{dt} = E(f(Y(T)^{dt}))$$,

the extrapolation methods allow us to get a higher order convergence by simulating the time evolution with different step sizes dt, e.g., we can get a second-order weak convergence from a first-order one using:

$$ V_{12}^{dt} = 2 V_{11}^{dt} - V_{11}^{2dt}$$

We thus have to first apply f to the solution at each time step for both step sizes and then combine the results. Note that:

$$ V_{12}^{dt} != E(f( 2 Y(T)^{dt} - Y(T)^{2dt} ))$$,

such that we would have to store in each time step the solutions for all step sizes. This suggests that it might be computationally beneficial to handle extrapolation methods in DiffEqGPU (they should anyway be used in combination with EnsembleGPU/CPUArray) such that the extrapolation parts are handled by prob_func , output_func, and reduction.

http://www.columbia.edu/~mh2078/MonteCarlo/MCS_SDEs.pdf also mentions the use of MLMC approaches.