Stabilization of unstable procedures: the recursive projection method
SIAM Journal on Numerical Analysis
ACM Transactions on Mathematical Software (TOMS)
An Adaptive Newton--Picard Algorithm with Subspace Iteration for Computing Periodic Solutions
SIAM Journal on Scientific Computing
Numerical Initial Value Problems in Ordinary Differential Equations
Numerical Initial Value Problems in Ordinary Differential Equations
Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum
SIAM Journal on Scientific Computing
An Equation-Free, Multiscale Approach to Uncertainty Quantification
Computing in Science and Engineering
Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation
Journal of Scientific Computing
Numerical stability analysis of an acceleration scheme for step size constrained time integrators
Journal of Computational and Applied Mathematics
Projective and coarse projective integration for problems with continuous symmetries
Journal of Computational Physics
Variance reduction for particle filters of systems with time scale separation
IEEE Transactions on Signal Processing
Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit
SIAM Journal on Scientific Computing
Hi-index | 31.46 |
In "equation-free" multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of "coarse limit cycles" of the macroscopic behavior, and the approximation of their stability through estimates of the leading "coarse Floquet multipliers".