3 research outputs found
NySALT: Nystr\"{o}m-type inference-based schemes adaptive to large time-stepping
Large time-stepping is important for efficient long-time simulations of
deterministic and stochastic Hamiltonian dynamical systems. Conventional
structure-preserving integrators, while being successful for generic systems,
have limited tolerance to time step size due to stability and accuracy
constraints. We propose to use data to innovate classical integrators so that
they can be adaptive to large time-stepping and are tailored to each specific
system. In particular, we introduce NySALT, Nystr\"{o}m-type inference-based
schemes adaptive to large time-stepping. The NySALT has optimal parameters for
each time step learnt from data by minimizing the one-step prediction error.
Thus, it is tailored for each time step size and the specific system to achieve
optimal performance and tolerate large time-stepping in an adaptive fashion. We
prove and numerically verify the convergence of the estimators as data size
increases. Furthermore, analysis and numerical tests on the deterministic and
stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal
admissible step size of linear stability, and quadruples the time step size of
the St\"{o}rmer--Verlet and the BAOAB when maintaining similar levels of
accuracy.Comment: 26 pages, 7 figure
Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds
A variational framework for accelerated optimization was recently introduced
on normed vector spaces and Riemannian manifolds in Wibisono et al. (2016) and
Duruisseaux and Leok (2021). It was observed that a careful combination of
timeadaptivity and symplecticity in the numerical integration can result in a
significant gain in computational efficiency. It is however well known that
symplectic integrators lose their near energy preservation properties when
variable time-steps are used. The most common approach to circumvent this
problem involves the Poincare transformation on the Hamiltonian side, and was
used in Duruisseaux et al. (2021) to construct efficient explicit algorithms
for symplectic accelerated optimization. However, the current formulations of
Hamiltonian variational integrators do not make intrinsic sense on more general
spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian
variational integrators are well-defined on manifolds, so we develop here a
framework for time-adaptivity in Lagrangian variational integrators and use the
resulting geometric integrators to solve optimization problems on normed vector
spaces and Lie groups.Comment: 30 pages, 4 figure