Timeline for Conditions for convergence of Euler's method
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 4, 2023 at 11:28 | comment | added | Nawaf Bou-Rabee | It is a sufficient condition. | |
| Jul 4, 2023 at 11:22 | comment | added | ViktorStein | But $(\star)$ is an extra condition, it does not follow from the Lipschitz continuity, right? | |
| Jul 4, 2023 at 9:09 | comment | added | Nawaf Bou-Rabee | Under Assumption $(\star)$, the result certainly holds on a Banach space. | |
| Jul 4, 2023 at 8:23 | comment | added | ViktorStein | Thank you for your response. But the argument about the Rademacher theorem only holds in finite dimensions, right? I am under the impression that for the existence of this finite constant $M$ you need that $f$ almost everywhere differentiable with respect to $t$, which does not follow from the Lipschitz continuity if $X$ is merely a Banach space. | |
| Jul 3, 2023 at 15:40 | comment | added | Nawaf Bou-Rabee | (1) Yes. (2) $\epsilon_k := \| \tilde{y}_k - y(t_k) \|$ where $\tilde{y}_k$ is the forward Euler solution at $t=k h$. | |
| Jun 29, 2023 at 13:14 | comment | added | ViktorStein | Is $f \colon [0, T] \times X \to X$, where $X$ is a Banach space? And would you mind defining $\epsilon_k$ exactly? | |
| May 4, 2023 at 15:46 | comment | added | ViktorStein | Ok, I will look into it in more detail. I am interested in a gradient flow-type autonomous differential inclusion $u'(t) \in \nabla_- F(u(t))$, where $\nabla_- F$ are the steepest descent directions of the proper, lower semicontinuous functional $F \colon \mathbb R^d \to (- \infty, \infty]$. | |
| May 4, 2023 at 14:17 | comment | added | Nawaf Bou-Rabee | The answer to which part of the book depends on the model (conservative, hamiltonian, dissipating or gradient type). The book is well written and easy to read. “Foster” (named after the statistician Gordon Foster) is for the general stochastic ode case en.wikipedia.org/wiki/Foster%27s_theorem | |
| May 4, 2023 at 14:10 | comment | added | ViktorStein |
Thanks for the reply. In what part of the book can I find these arguments? (The word foster is not mentioned in the book as far as I can tell).
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| May 4, 2023 at 13:16 | comment | added | Nawaf Bou-Rabee | No, Grönwall’s inequality en.wikipedia.org/wiki/Grönwall%27s_inequality degenerates as $T$ increases. One would need to replace this inequality with something like a Foster-Lyapunov argument as in books.google.com/books/about/…. Hope that clarifies. | |
| S May 4, 2023 at 13:12 | history | suggested | ViktorStein | CC BY-SA 4.0 |
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| May 4, 2023 at 12:18 | comment | added | ViktorStein | Can the analysis be preformed similarly if $T = \infty$, that is, we are searching for a solution $\gamma \colon [0, \infty) \to \mathbb R^d$? | |
| May 4, 2023 at 11:22 | review | Suggested edits | |||
| S May 4, 2023 at 13:12 | |||||
| Mar 19, 2015 at 18:38 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 5:09 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 5:01 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 4:51 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 4:42 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 4:27 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Mar 19, 2015 at 4:22 | history | answered | Nawaf Bou-Rabee | CC BY-SA 3.0 |