Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics

The simulation of dynamical systems involving random coefficients by means of stochastic spectral methods (Polynomial Chaos or other types of orthogonal stochastic expansions) is faced with well known computational difficulties, arising in particular due to the broadening of the solution spectrum as time evolves. The simulation of such systems thus requires increasing the basis dimension and computational resources for long time integration. This paper deals with systems having almost surely a stable limit cycles. It is proposed to reformulate the problem at hand in a rescaled time framework such that the spectrum of the rescaled solution remains narrow-banded. Two variants of this approach are considered and evaluated. The first relies on an explicit expression of a time-dependent, uncertain, time scale related to some distance between the corresponding solution and a reference deterministic system. The time scale is adjusted at each time step so that the distance from the reference system solution remains small, mimicking "in phase'' behavior. The second variant achieves the same objective by borrowing concepts from optimal control theory, and yields more precise time-scale estimates at the price of a higher CPU cost. It is thus more appropriate for uncertain systems exhibiting a stiff behavior and complex limit cycles. The method is applied to the case of a linear oscillator with uncertain properties, and to a stiff nonlinear chemical system involving uncertain reaction constants. The tests demonstrate the effectiveness of the proposed approaches, at least in situations where the topology of the limit cycle does not change when the uncertain system parameters vary.