Numerical Solution of Stochastic Differential Equations

By Taye Samuel Faniran. 2015

2,840 citations recorded by the source corpus.

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Citations in the corpus

DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps (2022, methodology): “When discretizing SDEs, the step size is limited by the randomness of the Wiener process [27, Chap.”
Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speech (2021, methodology): “In this case, generation boils down to sampling random noise from N (0 , I ) and then just solving the SDE describing dynamics of the reverse diffusion with any numerical solver (usually a simple ﬁrst-order Euler-Maruyama scheme (Kloeden & Platen, 1992) is used).”
Stochastic Modified Equations and Adaptive Stochastic Gradient Algorithms (2015, methodology): “The deﬁnition above is standard in numerical analysis of SDEs Milstein (1995); Kloeden & Platen (2011).”
Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient (2017, methodology): “Following the standard approach, introduced first in the SDE setting, see [46] and the monographs [31] and [36], the weak error (33) is decomposed as follows: Eφ ( X(T ) ) − Eφ ( XN ) = E [ u(T, x)− u(0, XN) ]”
Sequential Monte Carlo with Highly Informative Observations (2014, background): “In such cases higher-order schemes, such as the Milstein and semi-implicit schemes [25], or those of the Runge–Kutta family, are to be preferred, and indeed may be essential.”
Mathematical framework for large-scale brain network modeling in The Virtual Brain (2015, unknown): “One of the simplest forms for colored noise is exponentially correlated Gaussian noise (Klöden and Platen, 1995).”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “The crude approach of sampling an Euler discretisation (Kloeden and Platen (1992)) of (1) and us-ing it as an approximate sample from π was introduced in the applied literature (Ermak (1975); Doll and Dion (1976)).”
On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients (2015, background): “Both the strong and weak rates of convergence of X(n) to X are known when b and σ satisfy some Lipschitz continuous condition (see [18, 3]).”
Lagrangian-Eulerian methods for multiphase flows (2013, unknown): “cesses [137].”
Stochastic Normalizing Flows (2020, methodology): “Likewise, the Davie scheme (6) corresponds to the Milstein integrator for SDEs (Kloeden & Platen, 2013, §10.3).”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, methodology): “The crude approach of sampling an Euler discretisation (Kloeden and Platen 1992) of (1) and using it as an approximate sample from π was introduced in the applied literature (Ermak 1975; Doll and Dion 1976).”
Diffusion-Based Voice Conversion with Fast Maximum Likelihood Sampling Scheme (2021, methodology): “This solver differs from general-purpose Euler-Maruyama SDE solver (Kloeden & Platen, 1992) by inﬁnitesimally small values which can however become signiﬁcant when we sample from diffusion model using a few iterations.”
Asymptotically exact inference in differentiable generative models (2016, methodology): “A simulator for these SDEs can be formed by using an Euler–Maruyama [47] integration scheme to generate simulated realisations of the stochastic process at discrete time points.”
A mild Itô formula for SPDEs (2010, methodology): “The main difference to the finite dimensional setting in Kloeden & Platen [59] is that the linear operators L (0) t , t ∈ (0, T ], and L(1)t , t ∈ (0, T ], in (55) and (57) here depend explicitly on the time variable t ∈ (0, T ] too (compare (55) and (57) here with (1.13) and (1.14) in Chapter 5 in ”
Comparative Analysis of the Efficiency of Application of Legendre Polynomials and Trigonometric Functions to the Numerical Integration of Ito Stochastic Differential Equations (2018, unknown): “In this paper we use the definition of the Stratonovich stochastic integral from [49], [50].”