• 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.

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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.

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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.

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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.

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The phenomenon of distinct behaviors exhibited by neural networks under varying scales of initialization remains an enigma in deep learning research. In this paper, based on the earlier work [Luo et al., J. Mach. Learn. Res., 22:1–47, 2021], we present a phase diagram of initial condensation for two-layer neural networks. Condensation is a phenomenon wherein the weight vectors of neural networks concentrate on isolated orientations during the training process, and it is a feature in nonlinear learning process that enables neural networks to possess better generalization abilities. Our phase diagram serves to provide a comprehensive understanding of the dynamical regimes of neural networks and their dependence on the choice of hyperparameters related to initialization. Furthermore, we demonstrate in detail the underlying mechanisms by which small initialization leads to condensation at the initial training stage.

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Massive multiple-input multiple-output (MIMO) systems employ a large number of antennas to achieve gains in capacity, spectral efficiency, and energy efficiency. However, the large antenna array also incurs substantial storage and computational costs. This paper proposes a novel data compression framework for massive MIMO channel matrices based on tensor Tucker decomposition. To address the substantial storage and computational burdens of massive MIMO systems, we formulate the high-dimensional channel matrices as tensors and propose a novel groupwise Tucker decomposition model. This model efficiently compresses the tensorial channel representations while reducing SINR estimation overhead. We develop an alternating update algorithm and HOSVD-based initialization to compute the core tensors and factor matrices. Extensive simulations demonstrate significant channel storage savings with minimal SINR approximation errors. By exploiting tensor techniques, our approach balances channel compression against SINR computation complexity, providing an efficient means to simultaneously address the storage and computational challenges of massive MIMO.

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In this paper, we present a novel adaptive sampling strategy for enhancing the performance of physics-informed neural networks (PINNs) in addressing inverse problems with low regularity and high dimensionality. The framework is based on failure-informed PINNs, which was recently developed in [Gao et al., SIAM J. Sci. Comput., 45(4), 2023]. Specifically, we employ a truncated Gaussian mixture model to estimate the failure probability; this model additionally serves as an error indicator in our adaptive strategy. New samples for further computation are also produced using the truncated Gaussian mixture model. To describe the new framework, we consider two important classes of inverse problems: the inverse conductivity problem in electrical impedance tomography and the inverse source problem in a parabolic system. The effectiveness of our method is demonstrated through a series of numerical examples.

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