Hopf Bifurcations of Moore-Greitzer PDE Model with Additive Noise

The Moore-Greitzer partial differential equation (PDE) is a commonly used mathematical model for capturing flow and pressure changes in axial-flow jet engine compressors. Determined by compressor geometry, the deterministic model is characterized by three types of Hopf bifurcations as the throttle coefficient decreases, namely surge (mean flow oscillations), stall (inlet flow disturbances) or a combination of both. Instabilities place fundamental limits on jet-engine operating range and thus limit the design space. In contrast to the deterministic PDEs, the Hopf bifurcation in stochastic PDEs is not well understood. The goal of this particular work is to rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic stall bifurcation points in the presence of additive noise acting on the fast modes. We also show that the reduced-dimensional approximations (SDEs) contain multiplicative noise. Instability margins in the presence of uncertainties can be thus approximated, which will eventually lead to lighter and more efficient jet engine design.

C∞− regularization of ODEs perturbed by noise

We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space–time regularity properties of local times, and solutions are constructed using the approach of Catellier–Gubinelli based on nonlinear Young integrals.

The infinitesimal generator of the stochastic Burgers equation

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.

Particle filtering in high-dimensional chaotic systems

We present an efficient particle filtering algorithm for multiscale systems, which is adapted for simple atmospheric dynamics models that are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. In this paper, we propose a reduced-order particle filtering algorithm based on the homogenized multiscale filtering framework developed in Imkeller et al. "Dimensional reduction in nonlinear filtering: A homogenization approach," Ann. Appl. Probab. (to be published). In order to adapt the proposed algorithm to chaotic signals, importance sampling and control theoretic methods are employed for the construction of the proposal density for the particle filter. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 [E. N. Lorenz, "Predictability: A problem partly solved," in Predictability of Weather and Climate, ECMWF, 2006 (ECMWF, 2006), pp. 40-58] atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.