This paper delves into a rapid and accurate numerical solution for the inverse problem of the nonlinear diffusion equation in the context of multiphase porous media flow. For the realization of this, the combination of the multigrid method with constraint data is utilized and investigated. Additionally, to address the ill-posedness of the inverse problem, the Tikhonov regularization is incorporated. Numerical results demonstrate the computational performance of this method. The proposed combination strategy displays remarkable capabilities in reducing noise, avoiding local minima, and accelerating convergence. Moreover, this combination method performs better than any one method used alone.

Poisson regression is a statistical method specifically designed for analyzing count data. Considering the case where the functional and vector-valued covariates exhibit a linear relationship with the log-transformed Poisson mean, while the covariates in complex domains act as nonlinear random effects, an intrinsic functional partially linear Poisson regression model is proposed. This model flexibly integrates predictors from different spaces, including functional covariates, vector-valued covariates, and other non-Euclidean covariates taking values in complex domains. A truncation scheme is applied to approximate the functional covariates, and the random effects related to non-Euclidean covariates are modeled based on the reproducing kernel method. A quasi-Newton iterative algorithm is employed to optimize the parameters of the proposed model. Furthermore, to capture the intrinsic geometric structure of the covariates in complex domains, the heat kernel is employed as the kernel function, estimated via Brownian motion simulations. Both simulation studies and real data analysis demonstrate that the proposed method offers significant advantages over the classical Poisson regression model.

The matrix sign function has a key role in several applications in numerical linear algebra. This paper presents a novel iterative approach with a sixth order of convergence to efficiently compute this function. The scheme is constructed via the employment of a nonlinear equations solver for simple roots. Then, the convergence of the extended matrix procedure is investigated to demonstrate the sixth rate of convergence. Basins of attractions for the proposed solver are given to show its global convergence behavior as well. Finally, the numerical experiments demonstrate the effectiveness of our approach compared to classical methods.