The applicability of stochastic programming models and methods to PDE constrained stochastic optimization problem is discussed. The problem concerning mathematical model involves a PDE-type constaint and an uncertain parameter related to the exernal load. A computational scheme for this type of problems is proposed, including discretization methods for random elements (scenario based two-stage stochastic programming) and the PDE constraint (finite difference method). Several deterministic reformulations are presented and compared using numerical and graphical results. and Obsahuje seznam literatury
The two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finitedifference method for the transport of solutes through a homogenous two-dimensional domain that is finite and porous. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the firstorder decay and zero-order production parameters proportional to the seepage velocity, and periodic boundary conditions at the origin and at the end of the domain. Results agree well with analytical solutions that were reported in the literature for special cases. It is shown that the solute concentration profile is influenced strongly by periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in finite media, which is especially important when arbitrary initial and boundary conditions are required.
This study addresses a particular phenomenon in open channel flows for which the basic assumption of hydrostatic pressure distribution is essentially invalid, and expands previous suggestions to flows where streamline curvature is significant. The proposed model incorporates the effects of the vertical curvature of the streamline and steep slope, in making the pressure distribution non-hydrostatic, and overcomes the accuracy problem of the Saint-Venant equations when simulating curvilinear free surface flow problems. Furthermore, the model is demonstrated to be a higher-order one-dimensional model that includes terms accounting for wave-like variations of the free surface on a constant slope channel. Test results of predicted flow surface and pressure profiles for flow in a channel transition from mild to steep slopes, transcritical flow over a short-crested weir and flow with dual free surfaces are compared with experimental data and previous numerical results. A good agreement is attained between the experimental and computed results. The overall simulation results reveal the satisfactory performance of the proposed model in simulating rapidly varied gravity-driven flows with predominant non-hydrostatic pressure distribution effects. This study suggests that a higher-order pressure equation should be used for modelling the pressure distribution of a curvilinear flow in a steeply sloping channel.
The paper deals with a filter design for nonlinear continuous stochastic systems with discrete-time measurements. The general recursive solution is given by the Fokker-Planck equation (FPE) and by the Bayesian rule. The stress is laid on the computation of the predictive conditional probability density function from the FPE. The solution of the FPE and its integration into the estimation algorithm is the cornerstone for the whole recursive computation. A new usable numerical scheme for the FPE is designed. In the scheme, the separation technique based on the upwind volume method and the finite difference method for hyperbolic and parabolic part of the FPE is used. It is supposed that separation of the FPE and choice of a suitable numerical method for each part can achieve better estimation quality comparing to application of a single numerical method to the unseparated FPE. The approach is illustrated in some numerical examples.
The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton-Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.