Reconstruction of a permeability field with the wavelet multiscale–homotopy method for a nonlinear convection–diffusion equation

Estimation of a Permeability Field within the Two‐Phase Porous Media Flow Using Nonlinear Multigrid Method

“…We perform the simulations on a 2.6 GHz PC with 4 GB RAM in MATLAB 2012b environment under Windows 7. In our simulations, the permeability field is estimated both with the nonlinear multigrid method (NMGM) and with the fixed grid method (FGM, i.e., the Levenberg-Marquardt method [31]). For the estimation, we use the parameters as follows: In the first two examples, the spatial and temporal step sizes Δ 1 , Δ 2 , and Δ are, respectively, chosen to be 1/8, 1/8, and 0.0025 s, and the initial permeability field 0 is chosen to be constant 2.…”

“…Wavelet multiscale method is a specific form of multiscale techniques, which has recently been used to solve various parameter estimation problems. Successful applications of this method include the Bayesian tomography [25,26], the Bayesian formulations of emission tomography [27], the thermal wave tomography [28], the diffuse optical tomography [29], the velocity estimation problems of a two-dimensional wave equation [30], the permeability estimation problems of a nonlinear convection-diffusion equation [31], and the parameter estimation problems of partial differential equations [32,33]. It is shown in these papers that the performance of iterative parameter identification methods is much better with the help of the multiscale techniques.…”

Coiflet Wavelet-Homotopy Solution of Channel Flow due to Orthogonally Moving Porous Walls Governed by the Navier–Stokes Equations

“…Here, ⊗ denotes the operator of the Kronecker tensor product, A 1 , A 2 , and B are (2 j + 1) × (2 j + 1) matrices, and the generalized connection coefficients Γ and Γ are defined as equation (23).…”

“…At the same time, many homotopybased methods have also been proposed. Liu studied a parameter identification problem of two-phase porous media fractional flow equations [22] and then solved a nonlinear convection-diffusion equation [23] by a wavelet multiscale-homotopy method. In addition, based on the homotopy method, a multigrid-homotopy method was used to solve a nonlinear inverse problem governed by a convection-diffusion equation [24], and the numerical identification of diffusion parameters in a nonlinear convectiondiffusion equation was investigated by an adaptive homotopy perturbation method [25].…”

A Recommender System for Healthy Food Choices Based on Integer Programming

“…However, the recipe recommendation system can help narrow the richness of online information and consider our personal needs. Liu et al focused on the study of the homotopy method [3,4], and the combination of homotopy and multigrid [5,6] or wavelet [7,8]. Based on certain assumptions, this study introduces the design, implementation and evaluation of an intelligent recipe recommendation model based on the balanced diet and economic level of patients with diabetes mellitus.…”