Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations

By T. H. Gronwall. 1919

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Implicit Regularization of Discrete Gradient Dynamics in Deep Linear Neural Networks (2019, methodology): “Thus, using that ‖S‖ = O( ), we have ‖W1(t)−W 01 (t)‖ ≤ O( ) +O(1) ∫ t 0 s(‖W2(s)−W 02 (s)‖+ ‖W1(s)−W1(s)0‖)ds The same way we can prove that, ‖W2(t)−W 02 (t)‖ ≤ O( ) +O(1) ∫ t 0 s(‖W2(t)−W 02 (t)‖+ ‖W1(t)−W 01 (t)‖)dt And consequently we can sum these two inequalities and apply Grönwall’s inequali”
Discrete non-linear inequalities and applications to boundary value problems (2006, background): “Introduction The celebrated Gronwall–Bellman inequality [3,9] states that if u and f are non-negative continuous functions on an interval [a, b] satisfying u(t) c + ∫ t”
Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning (2020, unknown): “Lemma 4 (Grönwall’s Lemma [64]) .”
Existence and Uniqueness Theorem on Uncertain Dierential Equations with Local Lipschitz Condition (2012, unknown): “Setting A = |X0(γ)| + τ(1 + K(γ))Gτ , by Gronwell inequality [5], we have |Xt(γ)| ≤ A · exp[(1 + K(γ))Gτ t] ≤ A · exp[(1 + K(γ))Gττ ] = H1 < +∞, ∀t ∈ [0, τ), that is, Xt(γ) is bounded on [0, τ).”
The Magnus expansion and some of its applications (2008, unknown): “0 A(s)Y ds, one gets ‖Y ‖ ≤ 1 + |ε| ∫ t 0 ‖A(s)‖ ‖Y ‖ds, and application of Gronwall’s lemma [97] leads to ‖Y (t; ε)‖ ≤ exp ( |ε| ∫ t”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “The derivation will involve an application of the logarithmic norm [24] and the integral form of Gronwall’s lemma [40, 13].”