Computer methods for ordinary differential equations and differential-algebraic equations

By Uri M. Ascher, Linda R. Petzold. 1998

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Learning Deep Transformer Models for Machine Translation (2019, methodology): “• Inspired by the linear multi-step method in numerical analysis (Ascher and Petzold, 1998), we propose an approach based on dynamic linear combination of layers (D LCL ) to memorizing the features extracted from all preceding layers.”
Elucidating the Design Space of Diffusion-Based Generative Models (2022, methodology): “Through extensive tests, we have found Heun’s 2 nd order method [2] (a.k.a. improved Euler, trapezoidal rule) — previously explored in the context of diffusion models by Jolicoeur-Martineau et al. [24] — to provide an excellent tradeoff between truncation error and NFE.”
Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations (2017, methodology): “More importantly, we demonstrate that the connection between deep networks and numerical ODEs enables us to design new and more effective deep networks.”
Numerical collapse simulation of large‐scale structural systems using an optimization‐based algorithm (2009, methodology): “in the algorithm developed here, the analysis increment can be chosen purely based on truncation error control (see, for example, [29]).”
Option Hedging with Smooth Market Impact (2016, methodology): “The action was similar to a forward Euler discretization of an ordinary differential equation [Ascher and Petzold, 1998] which can introduce instability into a stable problem. As an additional example, on Oct. 15, 2014, the US Treasury market underwent the largest intraday move since 2009 [Almgren, ”
Diffusion Models: A Comprehensive Survey of Methods and Applications (2022, background): “In the following, we will discuss them in detail.”
Consolidation of unorganized point clouds for surface reconstruction (2009, methodology): “We then estimate particle normals via an iterative predictor-corrector scheme (see [Ascher and Petzold 1998] for origins and analogies for this term).”
Stable architectures for deep neural networks (2017, methodology): “, [2, 4] for the proof of stability criteria for the forward Euler method.”
Discrete mechanics and variational integrators (2001, methodology): “Constrained mechanical systems are particular examples of differential algebraic systems, and many of the techniques for the numerical integration of such systems can also be applied in the mechanical setting (see Hairer and Wanner (1996) and Ascher and Petzold (1998)).”
Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach (2004, methodology): “Common methods for very stiff equations are based on backward differentiation formulas (BDF) or on collocation at Radau points (Hairer and Wanner, 1991; Ascher and Petzold, 1998; Heling, 1998; Bastian, 1999; Turek, 1999).”
Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation (2010, unknown): “i) There is a close analogy between hyperbolic systems with stiﬀ relaxation and diﬀerential algebraic equations (DAE) [3].”
High-order multi-implicit spectral deferred correction methods for problems of reactive flow (2003, unknown): “13 displays the L1 errors of the solutions computed using the MISDC(4; 4; 2; 6), MISDC(5; 5; 2; 5), ARK4(3)6L[2]SA (hereafter ARK4), and ARK5(4)8LSA (hereafter ARK5) methods for N 1⁄4 32, 64, 128, 256, and 512.”
Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution (2002, methodology): “Asymptotic numerical stability, the stability for a fixed stepsize h and n → ∞, has been thoroughly studied for ODE and DAE systems [18, 2, 1, 16, 17].”