• Abstract

In this paper, we propose a new multiphysics finite element method for a Biot model with secondary consolidation in soil dynamics. To better describe the multiphysical processes underlying in the original model and propose stable numerical methods to overcome “locking phenomenon” of pressure and displacement, we reformulate the swelling clay model with secondary consolidation by a new multiphysics approach, which transforms the fluid-solid coupling problem to a fluid coupled problem. Then, we give the energy law and prior error estimate of the weak solution. Also, we design a fully discrete time-stepping scheme to use multiphysics finite element method with $P_2−P_1−P_1$ element pairs for the space variables and backward Euler method for the time variable, and we derive the discrete energy laws and the optimal convergence order error estimates. Also, we show some numerical examples to verify the theoretical results and there is no “locking phenomenon”. Finally, we draw conclusions to summarize the main results of this paper.

• General Summary

• Abstract

Scattered data interpolation aims to reconstruct a continuous (smooth) function that approximates the underlying function by fitting (meshless) data points. There are extensive applications of scattered data interpolation in computer graphics, fluid dynamics, inverse kinematics, machine learning, etc. In this paper, we consider a novel generalized Mercel kernel in the reproducing kernel Banach space for scattered data interpolation. The system of interpolation equations is formulated as a multilinear system with a structural tensor, which is an absolutely and uniformly convergent infinite series of symmetric rank-one tensors. Then we design a fast numerical method for computing the product of the structural tensor and any vector in arbitrary precision. Whereafter, a scalable optimization approach equipped with limited-memory BFGS and Wolfe line-search techniques is customized for solving these multilinear systems. Using the Łojasiewicz inequality, we prove that the proposed scalable optimization approach is a globally convergent algorithm and possesses a linear or sublinear convergence rate. Numerical experiments illustrate that the proposed scalable optimization approach can improve the accuracy of interpolation fitting and computational efficiency.

• General Summary

• Abstract

In this paper, we develop total variations for hue, saturation, and value of a color image, and we propose a novel hue-saturation-value total variation model for color image restoration. We first refine the hue formulation of a color image to make it mathematically and applicationally meaningful by assigning different hue values to different colors. We then develop the proposed hue-saturation-value total variation based on the conception of hue/saturation/value gradient. We investigate the dual formulation and the properties of the proposed hue-saturation-value total variation, and we finally propose a color image restoration model which is formulated by combing the proposed hue-saturation-value total variation regularization with the data-fitting term between the objective color image and the observed color image. We develop an efficient alternating iterative algorithm to solve the proposed optimization model in practice, and we give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed hue-saturation-value total variation and the proposed color restoration model is better than that of other testing methods in terms of visual quality, peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and S-CIELAB color error.

• General Summary

• Abstract

Computed tomography (CT) reconstruction with sparse-view projections is a challenging problem in medical imaging. The learning-based methods lack generalization ability and mathematical interpretability. Since the model-based iterative reconstruction (IR) methods need inner gradient-based iterations to deal with the CT system matrix, the algorithms may not be efficient enough, and IR methods with deep networks have no convergence guarantees. In this paper, we propose an efficient deep semi-proximal iterative method (DeepSPIM) to reconstruct CT images from sparseview projections. Unlike the existing IR methods, a carefully designed semi-proximal term is introduced to make the system matrix-related subproblem solvable. Theoretically, we give some useful mathematical analysis, including the existence of the solutions to the reconstruction model with an implicit image prior, the global convergence of the proposed method under gradient step denoiser assumption. Experimental results show that DeepSPIM is efficient and outperforms the closely related state-of-the-art methods regarding quantitative image quality values, details preservation, and structure recovery.

• General Summary