A Wiener Chaos Based Approach to Stability Analysis of Stochastic Shear Flows

As the aviation industry expands, consuming oil reserves, generating carbon dioxide gas and adding to environmental concerns, there is an increasing need for drag reduction technology. The ability to maintain a laminar flow promises significant reductions in drag, with economic and environmental benefits. Whilst development of flow control technology has gained interest, few studies investigate the impacts that uncertainty, in flow properties, can have on flow stability. Inclusion of uncertainty, inherent in all physical systems, facilitates a more realistic analysis, and is therefore central to this research. To this end, we study the stability of stochastic shear flows, and adopt a framework based upon the Wiener Chaos expansion for efficient numerical computations. We explore the stability of stochastic Poiseuille, Couette and Blasius boundary layer type base flows, presenting stochastic results for both the modal and non modal problem, contrasting with the deterministic case and identifying the responsible flow characteristics. From a numerical perspective we show that the Wiener Chaos expansion offers a highly efficient framework for the study of relatively low dimensional stochastic flow problems, whilst Monte Carlo methods remain superior in higher dimensions. Further, we demonstrate that a Gaussian auto-covariance provides a suitable model for the stochasticity present in typical wind tunnel tests, at least in the case of a Blasius boundary layer. From a physical perspective we demonstrate that it is neither the number of inflection points in a defect, nor the input variance attributed to a defect, that influences the variance in stability characteristics for Poiseuille flow, but the shape/symmetry of the defect. Conversely, we show the symmetry of defects to be less important in the case of the Blasius boundary layer, where we find that defects which increase curvature in the vicinity of the critical point generally reduce stability. In addition, we show that defects which enhance gradients in the outer regions of a boundary layer can excite centre modes with the potential to significantly impact neutral curves. Such effects can lead to the development of an additional lobe at lower wave-numbers, can be related to jet flows, and can significantly reduce the critical Reynolds number.EPSR

Inverse problems for discrete heat equations and random walks

We study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk.Comment: 31 pages, 3 figures. arXiv admin note: text overlap with arXiv:2101.1002

Reflected Diffusion Models

Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift with thresholding, which projects to the natural data domain (such as pixel space for images) after each diffusion step, but this leads to a mismatch between the training and generative processes. To incorporate data constraints in a principled manner, we present Reflected Diffusion Models, which instead reverse a reflected stochastic differential equation evolving on the support of the data. Our approach learns the perturbed score function through a generalized score matching loss and extends key components of standard diffusion models including diffusion guidance, likelihood-based training, and ODE sampling. We also bridge the theoretical gap with thresholding: such schemes are just discretizations of reflected SDEs. On standard image benchmarks, our method is competitive with or surpasses the state of the art and, for classifier-free guidance, our approach enables fast exact sampling with ODEs and produces more faithful samples under high guidance weight.Comment: Preprint. 27 Pages, 21 Figure