We develop and analyze an algorithm for the generalized Darcy–Forchheimer equations in porous media, combining a two-level mixed finite element discretization with a backtracking correction step. The governing operator is G(u) =μ/ρK^-1u+β/ρ|u|^m-1u with exponent m ∈ (1,2] , which recovers the classical quadratic Forchheimer law at m = 2 and captures sub-quadratic inertial corrections for m < 2 . The algorithm proceeds in three steps: a nonlinear solve on a coarse mesh 𝒯_H , a linearized solve on a fine mesh 𝒯_h , and a backtracking correction on 𝒯_H that refines the coarse-grid solution using second-order information from the fine-grid residual. We establish well-posedness of all three discrete problems and derive a priori error estimates for the velocity in L^m(Ω ) and the pressure gradient in L^(m+1)/m(Ω ) . Under the mesh-size and regularization balancing conditions h = O(H^2) and ε = O(h^m/2(m-1)) , the corrected two-level approximation achieves the same first-order accuracy as the fine-grid solution. Explicit stability and convergence bounds are derived for both the velocity and pressure corrections, showing that the correction step introduces no degradation in the overall convergence rate. Numerical experiments for m = 2 , 3/2 , and 6/5 confirm the theoretical rates, extend the two-level analysis to the deep sub-quadratic Darcy–Forchheimer regime for the first time, and demonstrate that the backtracking correction reduces errors consistently at negligible additional computational cost.

We develop and analyse a PINN-assisted two-level mixed finite element method for the Darcy–Forchheimer equation. The coarse nonlinear solve of the classical two-level scheme is replaced by a physics-informed neural network (PINN) surrogate; the network output is projected onto the coarse finite element space and used as the linearization point for the fine-grid correction. Numerical results demonstrate that the proposed approach achieves accuracy comparable to the classical method, with nearly first-order convergence for both velocity and pressure. In multi-query settings where the trained network is reused across repeated solves with fixed physical parameters, the one-time training cost is offset by savings on repeated coarse solves.

The premise of blind image deblurring revolves around the restoration of a clear image from a blurred one without prior knowledge of the specific blur kernel employed. Within this realm, various image priors have been extensively investigated and applied to address this inherently challenging problem. Throughout the image deblurring process, ensuring the resulting image intensities remain strictly non-negative is often imperative. However, prevalent numerical methodologies utilized to solve this issue have shown instances where the outcomes are not consistently favorable, leading to undesirable negative intensities that contribute to significant areas of darkness in the restored images. This study introduces a novel model designed to tackle the blind image deblurring problem by leveraging mean curvature. The proposed model not only assures positive outcomes but also confines the upper limit of image intensity values, thereby maintaining them within a predefined range. Additionally, new numerical algorithms are introduced, which not only restore the image but also estimate the blur kernel. Comparative analyses between these proposed algorithms and existing numerical techniques have been conducted to showcase the effectiveness and feasibility of our suggested approach.

In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.

Most of the time, while deblurring an image, we need the restored image’s intensities to be precisely non-negative. However, it has been noted that using current numerical techniques to solve the problem could produce outcomes that are not always favorable. In restored images, these negative intensities produce a sizable amount of dark space. In this study, we provide a model for the non-blind, mean curvature-based image deblurring problem. The suggested model limits the top boundary of the image intensity values, keeping them within a specified range, in addition to guaranteeing a strictly positive result. Removing of negative intensities or keeping them in prescribed range also help in increasing the quality of deblurred images. The main idea of this paper is to convert the mean curvature based constrained image deblurring problem into an unconstrained one by introducing a Lagrange multiplier. New numerical algorithms are described and compared with the currently used numerical techniques in order to solve the resulting non-linear partial differential equations. Furthermore, a new circulant preconditioned matrix is introduced to overcome the problem of slow convergence when employing a generalized minimal residual method inside of the augmented Lagrangian method. The viability of our suggested approach is demonstrated on test problems.

The numerical solution of the variable-order time fractional mobile/immobile (VO-TFMIE) equation is computationally challenging due to the nonlocal property of fractional operators. In this paper, an efficient numerical scheme, that is explicit decoupled group (EDG) method is proposed for solving the VO-TFMIE in two space dimensions. In this method, the variable-order temporal derivative is approximated by means of the L1 scheme, while the spatial derivatives are discretized by means of a skewed finite difference scheme. A Crank-Nicolson (C-N) difference scheme is also developed for comparison purposes. Stability and convergence results of the suggested methods are discussed. Several numerical simulations are conducted to solve the problem under consideration. The results validate the accuracy of the suggested methods, and the computational superiority of the EDG method over the C-N difference scheme. To enhance the computational efficiency furthermore, parallel implementation of the EDG method is also considered in this work. Lastly, the effectiveness and efficiency of the parallel EDG (PEDG) method in solving the considered problem is confirmed through numerical experiments. The obtained numerical results show the pivotal role of parallel computing in diminishing the computational limitations associated with the simulation of multi-dimensional variable-order fractional differential equations.

The mean curvature-based image deblurring model is widely used in image restoration to preserve edges and remove staircase effect in the resulting images. However, the Euler–Lagrange equations of the mean curvature model lead to the challenging problem of solving a nonlinear fourth order integro-differential equation. Furthermore the discretization of the Euler–Lagrange equations produces a nonlinear ill-conditioned system which affects the convergence of the numerical algorithms such as Krylov subspace methods (GMRES, etc.) In this paper, we have treated the high order nonlinearity by converting the nonlinear fourth order integro-differential equation into a system of first order equations. To speed up convergence by GMRES method, we have introduced a new circulant preconditioned matrix. Fast convergence is assured by the proved analytical property of our proposed new preconditioner. The first order error estimates are also established for the finite difference discretization. The effectiveness of our algorithm can be observed through fast convergence rates in numerical examples.

The mean curvature (MC)-based image denoising and image deblurring models are used to enhance the quality of the denoised images and deblurred images respectively. These models are very efficient in removing staircase effect, preserving edges and other nice properties. However, high order derivatives appear in the Euler–Lagrange equations of the MC-based models which create problems in developing an efficient numerical algorithm. To overcome this difficulty, we present a robust and efficient Two-Level method for MC-based image denoising and image deblurring models. The Two-Level method consists of solving one small problem and one large problem. The small problem is a nonlinear system, having high order derivative, on Level I (image having small number of pixels). The large problem is one less expensive system, having low order derivative, on Level II (image having large number of pixels). The derivation of the optimal regularization parameter of Level II is studied and formula is presented. Numerical experiments on digital images are presented to exhibit the performance of the Two-Level method.

Image deblurring models with a mean curvature functional has been widely used to preserve edges and remove the staircase effect in the resulting images. However, the Euler-Lagrange equations of a mean curvature model can be used to solve fourth-order non-linear integro-differential equations. Furthermore, the discretization of fourth-order non-linear integro-differential equations produces an ill-conditioned system so that the numerical schemes like Krylov subspace methods (conjugate gradient etc.) have slow convergence. In this paper, we propose an augmented Lagrangian method for a mean curvature-based primal form of the image deblurring problem. A new circulant preconditioned matrix is introduced to overcome the problem of slow convergence when employing a conjugate gradient method inside of the augmented Lagrangian method. By using the proposed new preconditioner fast convergence has been observed in the numerical results. Moreover, a comparison with the existing numerical methods further reveal the effectiveness of the preconditioned augmented Lagrangian method.

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler–Lagrange equations produces a nonlinear ill-conditioned system which affects the convergence of the numerical algorithms such as Krylov subspace methods (generalized minimal residual etc.) To overcome this difficulty, in this paper, we present three new circulant preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem, which combines a fixed point iteration with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new circulant preconditioners.

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

A two-level mixed finite element method is developed and analyzed to solve the Darcy--Forchheimer equation modeling non-Darcy flows in porous media. Instead of solving a large nonlinear system of equations on a fine mesh, the two-level method provides the flexibility of solving a small nonlinear system of equations on a coarse mesh with mesh size H followed by solving one linear system of equations on a fine mesh with mesh size h. In constructing the two-level algorithm, we introduce a small positive constant $\epsilon$ to the original nonlinear term of the Darcy--Forchheimer equation to avoid difficulties associated with nondifferentiability of the Euclidean norm at zero. The priori error estimates for the velocity and pressure gradient are obtained in $L^2$ and $L^{3/2}$ discrete norms, respectively. These estimates state that if piecewise constant velocities and piecewise continuous, linear pressures are used, then the coarse and fine meshes are related by $h = O(H^2)$. Numerical examples are provided to show the efficiency and accuracy of the method. Numerical results show that the two-level Galerkin mixed finite element method is computationally more cost-effective than the standard Galerkin mixed finite element and the rate of convergence also agrees with the theoretical results.

Image deblurring with total fractional-order variation model is used to improve the quality of the deblurred images. This model is very efficient in preserving edges and removing staircase effect. However, the regularization matrix associated with the total fractional-order model is dense which complicate developing an efficient numerical algorithm. In this research work, we present an efficient and robust Two-Level method to overcome the dense problem. The Two-Level method started by reducing the problem to one small non-linear system with dense regularization matrix (Level-I) and one less expensive large linear system with sparse regularization matrix (Level-II). The derivation of the optimal regularization parameter of Level-II is studied and formula is presented. Numerical experiments on several images are also provided to demonstrate the efficiency of the Two-Level method.

Under necessary compatibility condition, and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and stability of the solution of generalized Dary-Forchheimer model.

We considered mixed finite element discretization of generalized Darcy–Forchheimer model in a two or three dimensional porous domain, using a piecewise constant and continuous piecewise linear finite elements. The existence, uniqueness, stability and convergence of the discrete solution are established. The error estimates for the velocity and pressure in L m and L m + 1 m respectively are obtained for any m ∈ ( 1 , 2 ]. Numerical examples are also presented to confirm the rate of convergence.

Even though using the total fractional-order variation (TFOV) model reduces or eliminates the stair casing effect in image de-blurring problems, but in the same time it generates a dense and ill-conditioned linear system of equations. These properties lead to slow convergence of any iterative method such as Krylov subspace methods. One treatment of the slowness property is to apply the preconditioning technique. In this research work, we propose two block diagonal preconditioners to construct an image from a given blurred image using circulant matrices. These matrices allow us to use the fast Fourier transform (FFT) and the convolution theorem. Using FFT and the convolution theorem reduces the cost of the computations from O(n(2)) into O(n log n) operations in each iteration and also reduces the storages. Our proposed preconditioners are of Murphy, Golub, and Wathen type. Numerical examples are given to illustrate the efficiency of our preconditioners.

The Brinkman’s equation simplifies the numerical modeling of karst aquifers by allowing the use of a single transport equation to model the flow of fluids in both the free-flow and porous regions, in effect reducing the error arising from improper modeling of the interface between the two regions. Most equations available to model flow within karst aquifers deal with steady flow conditions. This may not be accurate in aquifers where unsteady conditions exist. We considered the effects of unsteady flow conditions in karst aquifers by assessing the addition of an unsteady flow term to the Brinkman’s equation. We solved the coupled mass conservation-transport equations that models unsteady fluid transport in karst aquifers and studied the effects of unsteady flow conditions on tracer transport in two different sample aquifers and compared to the results obtained from the steady flow Brinkman’s equation. The solution method adopted is sequential and it involves solving the unsteady Brinkman’s model first, followed by the advection-diffusion-adsorption equation using the cell-centered finite volume approach. The first example presented here is a simple aquifer model consisting of a single conduit surrounded by porous regions. The second example is a complicated structure consisting of complex geometrical caves embedded in a highly heterogeneous porous media. The results show that, inside the caves, the unsteady Brinkman’s model yielded lower tracer concentrations at early times when compared to the steady flow model. At longer times, both models produced almost similar results. In particular, the results obtained from the simplified example case (Example 1) indicate that the velocity profiles for unsteady flow within open conduits do not instantly yield a parabolic shape expected from the Brinkman’s equation, but gradually develops into one starting from a linear profile. Results obtained also show that the addition of unsteady flow term to the Brinkman’s model does not affect the flow of tracer within porous media in any significantly observable manner.

In this research work, we consider an indefinite and ill-conditioned linear system of equations resulted from discretizing the non-linear integro-differential equations associated with image deblurring problem. The suitable iterative method for this system is the minimal residuals (MINRES) method. However, this method is very slow with ill-conditioned linear system of equations. To overcome this slowness, we present two block diagonal preconditioners to speed up the convergence of this method. Since, the (2,2) block of the coefficient matrix of this system has the block Toepelitz with Toepelitz block (BTTB) structure, the first preconditioner is based on approximating the BTTB matrix by the Strang circulant while the second preconditioner is based on the optimal circulant approximation for the BTTB matrices. We study the eigenvalue bounds of the preconditioned matrices and investigate the efficiency of these two preconditioners by several numerical computations in term of CPU-time, iteration numbers and the quality of the reconstruction images.

SummaryIn this paper, we consider the solution of a large linear system of equations, which is obtained from discretizing the Euler–Lagrange equations associated with the image deblurring problem. The coefficient matrix of this system is of the generalized saddle point form with high condition number. One of the blocks of this matrix has the block Toeplitz with Toeplitz block structure. This system can be efficiently solved using the minimal residual iteration method with preconditioners based on the fast Fourier transform. Eigenvalue bounds for the preconditioner matrix are obtained. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.