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Results for “LIPSCHITZ CONDITION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2026

A stiff order condition theory for Runge–Kutta methods applied to semilinear ODEs

Classical convergence theory of Runge–Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss–Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Here, our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term.

Mathematics and Computing↗

DOE OSTI2025

Direction of impact for explainable risk assessment modeling

Abstract Several graphical indicators have been recently introduced to help analysts visualize the marginal effects of inputs in complex models. The insights derived from such tools may help decision‐makers and risk analysts in designing interventions. However, we know little about the adequacy and consistency of different indicators. This work investigates popular marginal effect indicators to understand whether they yield indications consistent with the properties of the quantitative model under inspection. Specifically, we examine the notions of monotonicity, Lipschitz, and concavity consistency. Surprisingly, only PD functions satisfy all these notions of consistency. However, when selecting the indicators, in addition to consistency, analysts need to consider the risk of model extrapolation. For situations where such risk is under control, we utilize individual conditional expectations together with PD plots. Two applications, on a NASA space risk assessment model and a susceptible exposed infected recovered (SEIR) model for the COVID‐19 pandemic illustrate the insights obtained from these indicators.

Mathematical Methods In Social Sciences↗

DOE OSTI2024

Asymptotic consistency of the WSINDy algorithm in the limit of continuum data

In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold σ c . We provide explicit bounds on σ c in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.

asymptotic consistency↗