Structure-Preserving Algorithms for Oscillatory Differential Equations


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The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations from [Huisken ] for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. The stability analysis is based on the matrix-vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. We will also present various numerical experiments to illustrate and complement the theoretical results.

Furthermore, we will give an outlook towards problems coupling mean curvature forced by a surface PDE. The c onnections b etween d iscrete g eometric m echanics, i nformation g eometry and m achine l earning. Melvin Leok. Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry.

A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions.

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A bounded numerical solution with a small mesh size implies the existence of a smooth solution of the Navier—Stokes equations. Buyang Li. Hong Kong Polytechnic University, China. Lorentz c ovariant s ymplectic a lgorithms for d ynamics of c harged p articles. Jian Liu. University of Science and Technology of China, China. Lorentz covariance is a key property of physical systems according to the theory of relativity. The Lorentz covariance of algorithms is introduced in this talk. Under Lorentz transformation, both the form and performance of a Lorentz covariant algorithm are invariant.

To acquire the advantages of symplectic algorithms and Lorentz covariance, a general procedure for constructing Lorentz covariant canonical symplectic algorithms LCCSAs is provided, based on which an explicit LCCSA for dynamics of relativistic charged particles is built. LCCSA possesses Lorentz invariance as well as long-term numerical accuracy and stability, due to the preservation of a discrete symplectic structure and the Lorentz symmetry of the system.

Neural Differential Equations

Variationally evolving Gaussians revisited. Christian Lubich. This work is joint with Caroline Lasser. Quantifying discretisation errors by isotonic regression and its application to estimating ODE models. Yuto Miyatake. Osaka University, Japan. However, such a method does not account for the discretisation error in numerical solutions and has limited estimation accuracy. In this talk, we present an estimation method with discretisation error quantification.

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Note that it is hoped that the discretisation error is quantified as efficiently as possible and when a structure-preserving method is employed its structure is effectively used. These are achieved by modelling the discretisation error as random variables and estimating their variances by an isotonic regression algorithm. This talk is based on joint work with Takeru Matsuda.


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Low-regularity time integrators. Alexander Ostermann. In this talk, we introduce low-regularity Fourier integrators as an alternative. They are obtained from Duhamel's formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. Similar improvements can also be obtained for other dispersive problems. Adaptive Patankar-Runge-Kutta methods.


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  5. Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387).
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Helmut Podhaisky. Martin Luther University, Germany. Patankar-Runge-Kutta methods are numerical methods for solving positive and conservative production - destruction system. They are linearly implicit and unconditionally positive.

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In this talk, we will discuss error estimation and step size control for methods of order 3 which have been constructed recently by Stefan Kopecz and Andreas Meister. The talk is based on a joint work with Stefan Kopecz. Word-series high-order averaging of highly oscillatory differential equations with delay. Jesus Maria Sanz-Serna. We show that, for appropriate combinations of the values of the delay and the forcing frequency, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay.

Our approach is based on the use of word-series techniques to obtain high-order averaged equations for differential equations without delay.

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This work is joint with Beibei Zhu. Nonlinear Fourier integrators for dispersive equations. Katharina Schratz. Heriot-Watt University, United Kingdom. In many situations these classical schemes allow a precise and efficient approximation. Nevertheless, non-smooth phenomena play a fundamental role in modern physical modeling e. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization- linking the finite dimensional discretization to powerful existence results of nonlinear dispersive PDEs in low regularity spaces.

Stable parallel-in-time computation of multiscale hyperbolic problems.

Yen-Hsi Richard Tsai. However, it is not originally intended for multiscale problems. It is well-known that typical parareal schemes do not perform well for hyperbolic problems.


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  • ISBN 13: 9783642051579.
  • Our idea is to generalize the parareal schemes by defining new effective propagators using the computed data that are collected during the iterations. Multirevolution integrators for differential equations with fast stochastic oscillations. Gilles Vilmart. We introduce new integrators for highly oscillatory stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise.

    The approach is based on the framework of multi-revolution composition methods previously introduced for deterministic and stochastic problems and it inherits its geometric features, in particular to design integrators preserving exactly quadratic first integrals. Long-term analysis of energy-preserving methods.

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    Bin Wang. Qufu Normal University, China. We study the long-time numerical oscillatory energy conservation of EP methods for highly oscillatory Hamiltonian ODEs. Moreover, this talk analyses the long-time actions and momentum conservations of EP methods when applied to nonlinear wave equations. All the results are proved by using the mathematical tool of modulated Fourier expansion.

    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations
    Structure-Preserving Algorithms for Oscillatory Differential Equations Structure-Preserving Algorithms for Oscillatory Differential Equations

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