3 research outputs found
A Dynamical Sparse Grid Collocation Method for Differential Equations Driven by White Noise
We propose a sparse grid stochastic collocation method for long-time
simulations of stochastic differential equations (SDEs) driven by white noise.
The method uses pre-determined sparse quadrature rules for the forcing term and
constructs evolving set of sparse quadrature rules for the solution variables
in time. We carry out a restarting scheme to keep the dimension of random
variables for the forcing term, therefore also the number of quadrature points,
independent of time. At each restart, a sparse quadrature rule for the current
solution variables is constructed based on the knowledge of moments and the
previous quadrature rules via a minimization procedure. In this way, the method
allows us to capture the long-time solutions accurately using small degrees of
freedom. We apply the algorithm to low-dimensional nonlinear SDEs and
demonstrate its capability in long-time simulations numerically
Variance-based sensitivity analysis for time-dependent processes
The global sensitivity analysis of time-dependent processes requires
history-aware approaches. We develop for that purpose a variance-based method
that leverages the correlation structure of the problems under study and
employs surrogate models to accelerate the computations. The errors resulting
from fixing unimportant uncertain parameters to their nominal values are
analyzed through a priori estimates. We illustrate our approach on a harmonic
oscillator example and on a nonlinear dynamic cholera model.Comment: 28 Pages; revised version; accepted for publication in Reliability
Engineering & System Safet
Stochastic dynamical low-rank approximation method
In this paper, we extend the dynamical low-rank approximation method to the
space of finite signed measures. Under this framework, we derive stochastic
low-rank dynamics for stochastic differential equations (SDEs) coming from
classical stochastic dynamics or unraveling of Lindblad quantum master
equations. We justify the proposed method by error analysis and also numerical
examples for applications in solving high-dimensional SDE, stochastic Burgers'
equation, and high-dimensional Lindblad equation.Comment: 27 pages, 8 figure