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Tomography Reconstructions With Stochastic Level-Set Methods
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
Bruno Sixou, Lin Wang, Françoise Peyrin
We have studied some aspects of the nonlinear inverse problem associated with the reconstruction of the real and imaginary parts of the refractive index in phase contrast tomography and with the binary tomographic reconstruction problem. Both are regularized with level-set functions and with Total-Sobolev penalty term. The deterministic optimization of the regularization functional leads to local minima with large reconstruction errors. The reconstruction results are improved with a stochastic perturbation of the shape of the reconstructed regions with a stochastic level-set evolution. The evolution is based on a stochastic partial differential equation with the Stratonovich formulation. The stochastic algorithm leads to a decreased reconstruction errors localized on the boundaries for different noise levels. The method gives better reconstruction results than the classical simulated annealing method.
Unsaturated Water Movement
Published in Shingo Iwata, Toshio Tabuchi, Benno P. Warkentin, Soil-Water Interactions, 2020
Shingo Iwata, Toshio Tabuchi, Benno P. Warkentin
3) Substituting Eqs. 6.12, 6.13 and 6.14 into Eq. 6.11 gives a stochastic partial differential equation which depends on the random fields, In Kj and a. In other words, the dependent variable ψ(x,t) is a random field which can only be described in terms of its statistical properties. This also applies to a variable such as θ(x,t) directly related to ψ. A small perturbation approach is used to derive approximate expressions for the mean and variance of ψ. Their mean tension satisfies a deterministic differential equation having the same form as the Richards equation.
Stochastic Optimal Control for Estuarine Management
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
Stochastic optimal control of distributed-parameter systems has been extensively discussed by Omatu and Seinfeld (1989). The distributed-parameter system herein is stochastic partial differential equation which may be coupled with measurement equations. They used dynamic programming to derive the Hamilton–Jacobi differential equation for a stochastic, optimal control problem of a linear, quadratic, distributed-parameter system with measurement equations. Optimal sensor and actuator locations problems have also been discussed by Omatu and Seinfeld (1989).
Large deviation principles for a 2D liquid crystal model with jump noise
Published in Applicable Analysis, 2022
Stochastic partial differential equations (SPDE) are used to model physical systems subjected to influence of internal, external or environmental noises or to describe systems that are too complex to be described deterministically, e.g. a flow of a chemical substance in a river subjected by wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, a laser beam subjected to turbulent movement of the atmosphere, spread of an epidemic in some regions and the spatial spread of infectious diseases. SPDEs are also used in the physical sciences (e.g. in plasmas turbulence, physics of growth phenomena such as molecular beam epitaxy and fluid flow in porous media with applications to the production of semiconductors and to the oil industry) and biology (e.g. bacteria growth and DNA structure). Models related to the so called passive scalar equations have potential applications to the understanding of waste (e.g. nuclear) convection under the earth's surface, [16, 17].
On the weak solutions to a stochastic two-phase flow model
Published in Applicable Analysis, 2022
Stochastic partial differential equations (SPDE) are used to model physical systems subjected to influence of internal, external or environmental noises or to describe systems that are too complex to be described deterministically, e.g. flow of a chemical substance in a river subjected by wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, a laser beam subjected to turbulent movement of the atmosphere, spread of an epidemic in some regions and the spatial spread of infectious diseases. SPDEs are also used in the physical sciences (e.g. in plasmas turbulence, physics of growth phenomena such as molecular beam epitaxy and fluid flow in porous media with applications to the production of semiconductors and to the oil industry) and biology (e.g. bacteria growth and DNA structure). Models related to the so-called passive scalar equations have potential applications to the understanding of waste (e.g. nuclear) convection under the earth's surface [1–3]. We recall that the presence of noise can lead to new and important phenomena. For example, the two-dimensional Navier–Stokes equations with sufficiently degenerate noise have a unique invariant measure and hence exhibit ergodic behavior in the sense that the time average of a solution is equal to the average overall possible initial data. Despite continuous efforts in the last 30 years, such property has so far not been found for the deterministic counterpart of these equations. This property could lead to a profound understanding of the nature of turbulence.
On inverse initial value problems for the stochastic strongly damped wave equation
Published in Applicable Analysis, 2022
Tran Bao Ngoc, Tran Ngoc Thach, Donal O'Regan, Nguyen Huy Tuan
The terminal value problem for strongly damped wave equation in the deterministic case (see [8,9]) is well known to be severely ill-posed in the sense that a solution corresponding to the data does not always exist, and in the case of existence, it does not depend continuously on the given data. In fact, from small noise in ξ, the corresponding solution will have large errors. Hence, one has to resort to regularization in order to restore stability. Stochastic partial differential equations (SPDEs) played an increasingly important role in modeling the phenomena in physics, chemistry, economics, social sciences, finance and engineering. For the SPDEs driven by white noise , we refer the reader to [10–14]. For more general SPDEs where white noise is replaced by fractional noise , we refer the reader to [15–19]. It should be noted that if , then the fractional Brownian motion becomes the Wiener process W. Furthermore, the fBm forms a subclass of Gaussian processes, which are positively correlated (res. negatively correlated) for (res. ).