50,937 research outputs found

    Effective descent for differential operators

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    A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator LL over a suitable differential field kk, which has an isotypical decomposition over the algebraic closure of kk, is a tensor product L=M⊗kNL=M\otimes_k N of an absolutely irreducible operator MM over kk and an irreducible operator NN over kk having a finite differential Galois group. Using the existence of the tensor decomposition L=M⊗NL=M\otimes N, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor FF of LL over a finite extension of kk. Here, an algorithmic approach to finding MM and NN is given, based on the knowledge of FF. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields kk which are C1C_1-fields.Comment: 21 page

    A machine learning framework for data driven acceleration of computations of differential equations

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    We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods

    Holonomic Gradient Descent and its Application to Fisher-Bingham Integral

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    We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere SnS^n, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.Comment: 23 pages, 1 figur

    Exponential asymptotics and Stokes lines in a partial differential equation

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    A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al. (Berk et al. 1982 J. Math. Phys.23, 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al. (Aoki et al. 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line
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