A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations

In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system, and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and LANS-alpha models.Comment: v4: Minor corrections to bibliography, and final version that will apear in CPAM. v3: Minor corrections to the algebra in the last section. v2: Minor changes to introduction and refferences. 14 pages, 0 figure

Universal Nonlinear Filtering Using Feynman Path Integrals II: The Continuous-Continuous Model with Additive Noise

In this paper, the Feynman path integral formulation of the continuous-continuous filtering problem, a fundamental problem of applied science, is investigated for the case when the noise in the signal and measurement model is additive. It is shown that it leads to an independent and self-contained analysis and solution of the problem. A consequence of this analysis is Feynman path integral formula for the conditional probability density that manifests the underlying physics of the problem. A corollary of the path integral formula is the Yau algorithm that has been shown to be superior to all other known algorithms. The Feynman path integral formulation is shown to lead to practical and implementable algorithms. In particular, the solution of the Yau PDE is reduced to one of function computation and integration.Comment: Interdisciplinary, 41 pages, 5 figures, JHEP3 class; added more discussion and reference

Computing aspects of problems in non-linear prediction and filtering

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Gaining Insights into Denoising by Inpainting

The filling-in effect of diffusion processes is a powerful tool for various image analysis tasks such as inpainting-based compression and dense optic flow computation. For noisy data, an interesting side effect occurs: The interpolated data have higher confidence, since they average information from many noisy sources. This observation forms the basis of our denoising by inpainting (DbI) framework. It averages multiple inpainting results from different noisy subsets. Our goal is to obtain fundamental insights into key properties of DbI and its connections to existing methods. Like in inpainting-based image compression, we choose homogeneous diffusion as a very simple inpainting operator that performs well for highly optimized data. We propose several strategies to choose the location of the selected pixels. Moreover, to improve the global approximation quality further, we also allow to change the function values of the noisy pixels. In contrast to traditional denoising methods that adapt the operator to the data, our approach adapts the data to the operator. Experimentally we show that replacing homogeneous diffusion inpainting by biharmonic inpainting does not improve the reconstruction quality. This again emphasizes the importance of data adaptivity over operator adaptivity. On the foundational side, we establish deterministic and probabilistic theories with convergence estimates. In the non-adaptive 1-D case, we derive equivalence results between DbI on shifted regular grids and classical homogeneous diffusion filtering via an explicit relation between the density and the diffusion time

Review of CFD for wind-turbine wake aerodynamics

This article reviews the state of the art of the numerical calculation of wind-turbine wake aerodynamics. Different CFD techniques for modeling the rotor and the wake are discussed. Regarding rotor modeling, recent advances in the generalized actuator approach and the direct model are discussed, as far as it attributes to the wake description. For the wake, the focus is on the different turbulence models that are employed to study wake effects on downstream turbines

Review of computational fluid dynamics for wind turbine wake aerodynamics

This article reviews the state-of-the-art numerical calculation of wind turbine wake aerodynamics. Different computational fluid dynamics techniques for modeling the rotor and the wake are discussed. Regarding rotor modeling, recent advances in the generalized actuator approach and the direct model are discussed, as far as it attributes to the wake description. For the wake, the focus is on the different turbulence models that are employed to study wake effects on downstream turbines