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Session III.5 - Information-Based Complexity

Monday, June 19, 14:30 ~ 15:00

On probabilistic stability for selected randomized schemes for ODEs

Tomasz Bochacik

AGH University of Science and Technology, Poland   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the stability of certain randomized algorithms approximating solutions of ordinary differential equations. We adapt notions of mean-square stability and asymptotic stability, considered in [4] in the context of numerical methods for SDEs, to the case of randomized schemes for ODEs. Moreover, we introduce the notion of stability in probability. We investigate relations between these three types of probabilistic stability, describe probabilistic stability regions, and compare them with absolute stability regions for deterministic schemes, cf. [1,2,3]. We focus on randomized Taylor and Euler schemes.

[1] T. Bochacik. A note on the probabilistic stability of randomized Taylor schemes. Electron. Trans. Numer. Anal. 58, 101-114, 2023.

[2] T. Bochacik, P. Przybyłowicz. On the randomized Euler schemes for ODEs under inexact information. Numer. Algorithms 91, 1205–1229, 2022.

[3] T. Bochacik, M. Goćwin, P. M. Morkisz, P. Przybyłowicz. Randomized Runge-Kutta method - Stability and convergence under inexact information. J. Complex. 65, 101554, 2021.

[4] D. J. Higham. Mean-square and asymptotic stability of the stochastic theta method. Siam J. Numer. Anal. 38, 753-769, 2000.

Joint work with Maciej Goćwin (AGH University of Science and Technology), Paweł M. Morkisz (AGH University of Science and Technology) and Paweł Przybyłowicz (AGH University of Science and Technology).

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