Timeline for Solving IVP backward in time via python
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 19, 2023 at 7:07 | comment | added | JesseJC | @LutzLehmann, thank you for the detailed explanations!! | |
| Jul 18, 2023 at 10:13 | history | edited | Lutz Lehmann | CC BY-SA 4.0 |
additional interpretation
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| Jul 14, 2023 at 10:47 | comment | added | Lutz Lehmann | You will always get this kind of problem when you start close to a stationary point (unstable or saddle point in the current forward direction). A relatively small area, such as given by the absolute tolerance, will split in exponentially diverging solutions. | |
| Jul 9, 2023 at 7:22 | comment | added | JesseJC | Or perhaps by any chance we could check priorly (by its drift? eigenvalues?) whether an ode can be solved feasibly and numerically via Python package…? | |
| Jul 9, 2023 at 7:01 | comment | added | JesseJC | @Lutz Lehmann, thank you for the explanation. Ummm… I wonder if there’s any workaround to solve an ODE backward… | |
| Jul 8, 2023 at 16:15 | comment | added | Lutz Lehmann |
@JesseJC : The contrast between large and small remains, but care about the y scale. When setting tol=1e-11, the divergence backwards has only size 7e-6, which is the expected scale for the above magnitude napkin calculation. So to overcome this you would have to use tol=1e-20 or smaller. But that is not feasible with 64bit double numbers
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| Jul 8, 2023 at 15:53 | comment | added | JesseJC | Lutz Lehmann, I'm sorry that my reputation isn't enough so I couldn't thumb-up your answer. | |
| Jul 8, 2023 at 15:52 | comment | added | JesseJC | Thank you very much for both of your replies. So it seems like even the forward DE is easily solvable. But solving it backward is sensitive and generally not workable? Is there a principle way to deal with this issue...? | |
| Jul 8, 2023 at 13:22 | comment | added | whpowell96 | The forward problem is stable, but the backwards problem is unstable. Numerically integrating unstable systems with large eigenvalues requires either very tight tolerances or more sophisticated integration algorithms | |
| Jul 8, 2023 at 12:50 | comment | added | JesseJC | Thanks for answering. However, I'm not fully understanding it (sorry that I wasn't familiar with numerical methods). Wasn't the forward integration and backward integration computed independently...? Did you suggest to choose different and smaller scales of atol, or rtol? I tested it but the backward solution still explodes... | |
| Jul 8, 2023 at 12:12 | history | answered | Lutz Lehmann | CC BY-SA 4.0 |