Identification of space-dependent permeability in nonlinear diffusion equation from interior measurements using wavelet multiscale method

Abstract

Numerical identification of the space-dependent permeability function in a nonlinear diffusion equation is considered. This problem plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the identification is formulated as a minimization problem with regularization terms. To overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this inverse problem. This method works by decomposing the inverse problem into multiple scales using wavelet transform so that the original inverse problem is reformulated to be a sequence of subinverse problems relying on scale variables, and successively solving these subinverse problems according to the size of scale from the longest to the shortest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical simulations show the wide convergence, computational efficiency, anti-noise and de-noising abilities of the proposed algorithm.

Parameter Estimation for Nonlinear Diffusion Problems by the Constrained Homotopy Method

“…where α 1 , α 2 are the regularization parameters, B 1 , B 2 are, respectively, the second-order smooth matrices in the x− and y−direction (see [37]), and Υ 0 is an initial estimate.…”

“…The concrete expression for ∇ • (υ i,j N k i,j ∇u k i,j ) is not the focus of this article, so we do not describe it here. For interested readers, see [37].…”

Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation

“…is the discrete form of the nonlinear diffusion term, which can be found in reference [50]. Subsequently, Equation (7) can construct the nonlinear operator Equation ( 5), and…”

A wavelet multi-scale method for the inverse problem of diffuse optical tomography

“…A relatively new method has emerged in the field of inversion, namely the wavelet multi-scale method [18][19][20][21][22][23][24][25]. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one.…”

“…This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Successful applications of this method include the inversion of the Maxwell equations [19], the identification of space-dependent porosity in fluid-saturated porous media [20,21], the inversion of the two-dimensional acoustic wave equation [22], the reconstruction of permittivity distribution by the inversion of the electrical capacitance tomography model [23], the parameter estimation of elliptical partial differential equations [18,24] and the identification of space-dependent permeability in a nonlinear diffusion equation [25]. It is shown in these papers that the wavelet multi-scale method can enhance stability of inversion, accelerate convergence and cope with the presence of local minima to reach the global minimum.…”