Unconventional reservoirs, such as shale/tight oil and gas reservoirs, are typically multi-scale spaces with both nano- and centimetre-scale matrix porosity, micrometre-to-centimetre-scale natural fracture systems and metre-to-hundred-metre-scale man-made fractures resulting from large-scale fracturing. In this paper, we present a mathematical model of the coupled fluid flow over matrix, micro-fractures, macroscopic fractures, and large-scale fracture cavities in fractured porous media.

Component design (“digitalization”), performance optimization (“optimization”), and process simulation (“simulation”) are three critical modules in the 3D printing process. “Digitalization” refers to the transformation of design drawings or pre-processed physical objects into editable digital components through means such as images, videos, and scanning processes. “Optimization” involves the application of constraints from physical fields such as mechanics and thermodynamics to enhance the performance of digital components. “Simulation” entails the digital simulation and twin modeling of physicochemical changes during the manufacturing process, based on the optimized components, to emulate real-world physical conditions. This research aims to introduce integrated modeling and algorithmic studies in design, optimization, and simulation within the phase field framework. In the “digitalization” module, we will present three-dimensional reconstruction models, repair models, and lightweight support structure design models that correspond to common data types in computer-aided design. In the “optimization” module, we will introduce a series of multi-scale, multi-physical field, and multi-material coupled topology optimization problems and their corresponding solutions. In the “simulation” module, we will discuss macroscopic (phase transition) to microscopic (grain boundary) scale coupling theories and the physical field couplings such as laser-thermal-flow-solid, in addition to simulating methods for processes like Fused Deposition Modeling (FDM) and Selective Laser Melting (SLM), integrating 3D printing parameters and techniques. The integrated research approach to design, optimization, and simulation based on the phase field model framework aims to shorten the product development cycle, enhance design efficiency, and provide a theoretical basis and algorithmic support for quality traceability and root cause analysis of 3D printed components.

In the paper, for solving large sparse vertical linear complementarity problems, a twostep modulus-based matrix synchronous multisplitting parallel iteration method is constructed by using two-step multisplitting technique. The new method can be viewed as a generalization of the modulus-based matrix synchronous multisplitting iterative method and the two-step modulus-based matrix splitting iterative method in the existing literatures. Furthermore, under the assumption that the system matrices are H₊-matrices, the convergence analysis of the proposed method is given, and the convergence domain of the parameter matrix is obtained, which extends the convergence results of the existing methods. Finally, numerical experiments are carried out under the OpenMP framework for two numerical examples in the existing literatures. Numerical results show that the two-step multisplitting technique can improve the computational efficiency of the existing methods.

Compressed Sensing (CS) is an effective method for fast Magnetic Resonance Imaging (MRI), which aims to reconstruct MRI from a small amount of under-sampled data space and accelerate data acquisition in MRI. In order to improve the reconstruction accuracy and speed of current MRI, in this paper, we combine the traditional model-driven compressed sensing method and data-driven deep learning method to propose a new algorithm–ADMMCNN. This proposed algorithm solves a subproblem of the alternating direction multiplier method (ADMM) by convolutional neural network, and then optimizes the penalty parameter and the update rate of Lagrange multiplizer by constructing the loss function and using the gradient descent method to obtain the trained ADMM model, which can effectively recover the MRI signal from the compressed sensing measurement. Compared with the traditional reconstruction algorithms, the proposed algorithm has fewer regularization parameters and lower computational complexity, at the same time, we introduce a trained convolutional neural network directly to greatly simplify the training process and computation, also we use GPU to accelerate the training. Finally, we give some experimental results to show that ADMM-CNN has faster calculation speed and better reconstruction accuracy compared with traditional reconstruction algorithms and other deep learning methods.

In 2016, Karney proposed an exact sampling algorithm for the standard normal distribution. In this article, we present an exact sampling algorithm for the normal distribution of standard variance $\sqrt{1/(2\ln2)}$ and mean $0$, which can be called the binary Gaussian distribution, as its relative probability density function is given by $2^{-x^2}$ for $x\in\mathbb{R}$. Our proposed algorithm requires no floating-point arithmetic in practice, and can be regarded as the promotion of Karney's exact sampling technique. We give an estimate of the expected number of uniform deviates over the range $(0,1)$ used by this algorithm until outputting a sample value. Numerical experiments also demonstrate the effectiveness of the sampling algorithm. For any rational number $c$ greater than $1$ but less than Euler's number $e$, the idea of sampling exactly the binary Gaussian is generalized to a class of normal distributions of standard variance $\sqrt{1/(2\ln{c})}$ and mean $0$, called “$c$-ary Gaussian distributions”, and a similar complexity analysis is presented.

In this paper, we develop some numerical schemes to solve fractional nonlinear Schröd\-inger equation, which preserve one or more analytical properties of the fractional system. First, we apply BDF scheme, Crank-Nicolson scheme and relaxation scheme to discrete time derivative, and analytic conservation and dispersion error of the discrete schemes. Second, we use central difference scheme and compact difference scheme to discrete space fractional derivative of the fractional nonlinear Schrödinger equation with periodic boundary condition. We find that central difference scheme and compact difference scheme preserve mass and energy conservation laws very well for periodic boundary condition. Finally, the numerical experiments of some fractional nonlinear Schrödinger equations are given to verify the correctness of theoretical results.

Three explicit Runge-Kutta schemes are constructed from the viewpoint of Newton-Cotes integration, including a 3-stage $2^{nd}$ order Runge-Kutta scheme with parameters and two 4-stage $3^{rd}$ order Runge-Kutta schemes with parameters. The commonly used RK3 and RK4 can be obtained by taking special parameters from our schemes. The accuracy and the stability of these schemes are presented. Numerical examples are given to verify the stability, effectiveness and high accuracy of the constructed schemes. Results show that our explicit Runge-Kutta schemes have better stability than that of the classic explicit Runge-Kutta schemes.

Applying the conjugate gradient method and linear projection operator, an iterative algorithm is presented to solve the least squares solution of linear matrix equation $AXB=C$ under any linear subspace. It is proved that the least squares solution, the minimum-norm least squares solution and the optimal approximation of the matrix equation $AXB=C$ can be obtained in finite iteration steps by the method without considering rounding errors. The numerical examples verify the efficiency of the algorithm. The merit of our method is that it is easy to implement in any linear subspace.

The block Kaczmarz method is an iterative algorithm for solving large-scale linear systems. At each iteration, the current iteration points will be orthogonally projected to the solution space of the constraint subset. In the paper, according to the principle of the block Kaczmarz method, we propose the randomized block Kaczmarz method with the Gaussian mixture model (GMM-RBK(k)). The partition of blocks is carried out by the Gaussian mixture model. In order to avoid the computation of pseudo-inverse or the solution of the least-squares problem, we propose the randomized average block Kaczmarz method with the Gaussian mixture model (GMM-RABK(k)). We prove that the two algorithms converge when the linear system is consistent, and give the corresponding formula of convergence rate. The effectiveness of the GMM-RBK(k) method and GMM-RABK(k) method is also verified in the final numerical experiments, and avoiding the calculation of pseudo-inverse, the GMM-RABK(k) method is superior to the GMM-RBK(k) method.

This paper adopts the Newmark integration method based on the large-scale tearing finite element method to perform high-precision large-scale parallel solving of structural dynamic calculations. A multi-level load balancing strategy combining static and dynamic methods is designed for heterogeneous platforms. For inter-node computing, subdomain boundaries are partitioned based on the characteristics of the tearing finite element method, and a domain boundary balanced graph bipartition algorithm is used to balance the computation load of each subdomain. For intra-node computing, dynamic optimization of computation load is performed based on the performance differences of computing units on heterogeneous platforms. To improve the utilization rate of heterogeneous computing platforms, multi-stream parallel optimization is carried out for the core computing module's batch matrix-vector multiplication. The optimization in this paper has been integrated into the high-performance numerical simulation software for structural mechanics, HARSA-feti. The simulation performance is demonstrated using the flow-induced vibration simulation of a real reactor fuel component as an example. The results show that the simulation performance has increased by more than 71.3\%, and the high-precision simulation of a billion-grid-scale full-core fuel rod component has been achieved for the first time. Compared with 1, 000 GPUs, the strong and weak scalable parallel efficiency of 16, 000 GPUs reached 74.1\% and 81.1\%, respectively.

High-dimensional data set arises in many fields, which means the feature size is greater than or far greater than the sample size. To deal with these, there have been a lot of researches on regularized models, whose formulations are an objective function composed by the loss function and regularization term. These two terms are combined by the tuning parameter. It is well known that tuning parameter selection is very important. Theoretically, this parameter characterizes properties of the model solution and determines the model effect. Practically, the calculation cost and computational effect are different under different tuning parameters. As far as we known, there are three main methods to select the optimal tuning parameter, which are cross validation, information criterion and bilevel programming. For cross validation and information criterion, they all require big computational costs, causing by that fact that they need to calculate solutions under different tuning parameters. Besides that, how to appropriately choose the sequence of possible tuning parameters is an essential problem. For the purpose of reducing the computational cost of cross validation and information criterion, screening rules are proposed to eliminate inactive features in data sets and speed up the tuning parameter selection procedure. Comparing to these popular ways, transforming the tuning parameter selection problem to a bilevel programming is a more direct way. But this usually lead to a nonconvex optimization problem and still need to be explored. This paper will review existing works from tuning parameter selection methods and acceleration perspectives, respectively. Based on these, we propose the future works.

The physical quantities in the electronic structure calculation of periodic systems involve the band integral over the Brillouin zone. For metals, the integrand is discontinuous when the Fermi surface crosses through the energy band, which makes it difficult to improve the calculation accuracy of general numerical integration methods. Based on smearing methods, we propose two numerical formats with higher order precision about the broadening parameters by first and second extrapolation respectively. The numerical method based on this kind of extrapolation scheme gives more precise integral approximation over the Brillouin zone with the premise of the same discrete $\mathbf{k}$-point in Brillouin region, and significantly improves the computational efficiency. The efficiency of the extrapolation methods is verified by numerical tests on two typical systems.

Reconstructing clear images from blurred and noisy ones is a typical ill-posed problem. When the blurring kernel is unknown, reconstructing both the blurring kernel and the image is required. Such blind denoising and deblurring problems have attracted widespread attention in academia. Using variational methods, we established a partial differential equation model for the blind denoising and deblurring problem of remote sensing images. Then, combining alternating direction method and finite difference method, we constructed a fully discrete numerical format for solving unknown kernel functions and clear images. Numerical experiments were conducted to analyze the effect of parameters on image processing performance and to determine reasonable parameters. Finally, numerical experiments were conducted on several remote sensing images, and the results demonstrated the effectiveness of the model.

In high-dimensional data analysis, penalized quantile regression is an effective tool for variable selection and parameter estimation. In many real applications, variables are structured into groups. In order to achieve the desired effect of sparsity within and between groups, we study the sparse group lasso penalized quantile regression model that combines lasso and group Lasso. To solve computational challenges caused by non-smoothness of object function, we approximate the quantile loss function using quantile Huber function, and the quantile Huber regression with sparse group Lasso penalty (SGLQHR) is obtained. We introduce Groupwise Majorization Descent (GMD) algorithm for computing the proposed model. Numerical examples and real data analysis demonstrate the competitive performance of our algorithm.

One-dimensional unsteady seepage model is the basic model in well-test research, existing solution includes two categories such as analytic solution and numerical solution. Analytic solution is depend on Laplace transform and numerical Inversion, and Numerical solution is depend on mesh generation and equation discretization. Iterative solution of non-linear equation is present which is direct solution for One-dimensional unsteady seepage equation. Based on steady state sequential replacement method and mass conservation equation, the unsteady seepage equation is solved by transformed as steady seepage equation, the nonlinear equation of investigation radius under different outer boundary were build, well-bore pressure and Investigation radius at each moment were obtained by Newton method. The error between iterative solution and analytic solution is small, and bigger error appeared in early stage. Compared with analytic solution and numerical solution, the iterative solution can be spread used in other One-dimensional unsteady seepage problems because of its simple algorithm, acceptable accuracy and fewer time-consuming.

Multi-dimensional hypersingular integrals are widely used in many engineering fields such as elasticity and scattering of electromagnetic fields. In order to improve the calculation accuracy, we construct the formula of two-dimensional and three-dimensional hypersingular integrals. In this paper, the composite rectangle quadrature formula is used to approximate the part without singularity in the divided $N$ subinterval and the remaining part is solved by the analytic expression of the hypersingular integral. Based on the extrapolation, the modified composite rectangle quadrature formula of one-dimensional hypersingular integral is constructed. Finally, the modified rectangle quadrature formula is extended to the numerical quadrature of two-dimensional and three-dimensional surface hypersingular integrals. Numerical examples at the end of the paper verify the feasibility of the proposed method.

In this paper, we consider the convergence analysis of two accelerated iterative algorithms for solving a nonsymmetric algebraic Riccati equation arising in transport theory. This equation has two parameters $\alpha\in[0,1), c\in(0,1]$. We prove two accelerated iterative algorithms have the same convergence rates, and show that two algorithms converge linearly in $(\alpha,c)\neq(0,1)$ and sublinearly in $(\alpha,c)=(0,1)$.

Gegenbauer’s addition theorem is an important theory of Bessel functions. This paper studies the truncation error of Gegenbauer’s addition theorem. Based on the limiting forms of Bessel and Neumann functions as the argument tend to zero, an explicit bound of the truncation error and its convergence order are derived. Numerical experiments show that the estimates are valid and sharp.

In this paper, we use the concatenating method to construct a local momentum-preserving algorithm for the Zakharov system based on the idea that the numerical algorithm should keep the intrinsic properties of the original problem as much as possible. The advantage of the proposed algorithm is that it preserves the local momentum conservation law and the local mass conservation law in any time-space regions, i.e., independence on any boundary conditions. Certainly, if the boundary condition is suitable, such as homogeneous boundary or periodic boundary, the proposed algorithm also preserves the global momentum conservation law and the global mass conservation law. Numerical experiments do not only show the second-order convergence, but also verify the stability in long term calculations and the conservation of invariants for the proposed algorithm.

In this paper, based on the active set identification, we propose a Proximal Newton method for $\ell_1$ problems. One advantage of this method is that it uses the good support identification property of the ISTA algorithm to determine the free variables and active set variables, and another advantage is that it uses the information of the partial Hessian matrix to update the free variables. Under appropriate conditions, we demonstrate the global convergence of the algorithm with a nonmonotonic line search strategy. The results of numerical experiments show that our proposed algorithm is effective.

This paper proposes a partial linear regression model based on Fourier basis by combining nonparametric regression model and quantile regression model. We use the Fourier approximation of function model of the nonlinear function, and combined with quantile regression model estimation method of the model are given, under some basic assumptions The consistency of parameter vector and nonlinear function estimation is proved.The effectiveness of this method is demonstrated by simulation studies.At the end of the paper, the meteorological data of Beijing Capital International Airport are empirically analyzed with this model, and a new method will be proposed to accurately predict the diffusion of PM2.5 by this model.

The two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm are effective methods for solving the coherent linear system. By adding the soft shrinkage in the two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm, this paper introduces the sparse two-subspace projection algorithm and the sparse multi-step inertial randomized Kaczmarz algorithm. The linear convergence rate in expectation of the proposed algorithms in noise and noiseless cases is presented. Finally, we give some numerical experiments to illustrate the effectiveness and advantage of our algorithms.

It designs a combined high-order compact method for KdV equation in this work. This method simultaneously and compactly calculates the first-order and third-order spatial derivatives which overcomes many shortcomings of classic high-order compact methods. The KdV equation is discretized by the combined high-order compact method in space, and is approximated by the Crank-Nicolson scheme combined with extrapolation method in time. In addition, projection method is used to pull the numerical solution back to the energy-preserving manifold. Finally, some numerical experiments are conducted to verify the numerical accuracy, computational efficiency, and the property of energy-preserving.

In this paper, we first provide a new convergence theorem for the greedy Kaczmarz (GK) method in reference [14]. Secondly, in order to improve the efficiency of solving consistent linear equations, a new greedy block Kaczmarz (RDBK) method based on the greedy strategy of the GK method is proposed, and the convergence theorem of the RDBK method is provided. Finally, the numerical results demonstrate that the RDBK method is significantly superior to the GK method in terms of iteration steps and computation time.

In this present work, we propose two local energy-preserving schemes for the CamassaHolm equation, which can preserve both the local energy conservation law and the local mass conservation law, that is to say, these two schemes can accurately preserve energy and mass in any time and space regions. The local energy-preserving scheme is an extension of the global energy-preserving scheme, which eliminates the dependence on boundary conditions of the latter. Moreover, under suitable boundary conditions, such as periodic or homogeneous boundary condition, the local energy conservation law and local mass conservation law can be transformed into the corresponding global conservation laws. Finally, numerical experiments verify the splendid effect of the proposed schemes.