Symplectic methods for Hamiltonian systems with additive noise

Stochastic systems, phase flows of which have integral invariants, are considered. Hamiltonian systems with additive noise being a wide class of such systems possess the property of preserving symplectic structure. For them, numerical methods preserving the symplectic structure are constructed. A special attention is paid to systems with separable Hamiltonians, to second order differential equations with additive noise, and to Hamiltonian systems with small additive noise

Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches

Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators

We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge-Kutta methods. We show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained when inserting the training data in the MIRK formulae, unlocking symmetric and high-order integrators that would otherwise be implicit for initial value problems. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with experiments using data from several (chaotic) Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, supporting the favorable performance of MII.Comment: 23 pages, 10 figure

Comparison of geometric integrator methods for Hamilton systems

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2009Includes bibliographical references (leaves: 79)Text in English; Abstract: Turkish and Englishxii, 115leavesGeometric numerical integration is relatively new area of numerical analysis The aim of a series numerical methods is to preserve some geometric properties of the flow of a differential equation such as symplecticity or reversibility In this thesis, we illustrate the effectiveness of geometric integration methods. For this purpose symplectic Euler method, adjoint of symplectic Euler method, midpoint rule, Störmer-Verlet method and higher order methods obtained by composition of midpoint or Störmer-Verlet method are considered as geometric integration methods. Whereas explicit Euler, implicit Euler, trapezoidal rule, classic Runge-Kutta methods are chosen as non-geometric integration methods. Both geometric and non-geometric integration methods are applied to the Kepler problem which has three conserved quantities: energy, angular momentum and the Runge-Lenz vector, in order to determine which those quantities are preserved better by these methods