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In science and engineering, there is often a need to repeatedly solve large-scale and high-resolution partial differential equations (PDEs). Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of PDEs. This paper introduces a novel Fourier neural operator with a multigrid architecture (MgFNO). The MgFNO combines the frequency principle of deep neural networks (DNNs) with the multigrid idea for solving linear systems. To speed up the training process of the FNO, a three-layer V-cycle multigrid architecture is used. This architecture involves training the model multiple times on a coarse grid and then transferring it to a fine grid to accelerate the training of the model. The DNN-based solver learns the solution from low to high frequency, while the multigrid method acquires the solution from high to low frequency. Note that the FNO is a resolution-invariant solution operator, therefore the corresponding calculations are greatly simplified. Finally, experiments are conducted on Burgers' equation, Darcy flow, and Navier-Stokes equation. The results demonstrate that the proposed MgFNO outperforms the traditional Fourier neural operator., Comment: 29 pages, 15 figures, 4 tables

In this paper, we propose an orthogonal block wise Kaczmarz (POBK) algorithm based on preprocessing techniques to solve large-scale sparse linear systems $Ax=f$. Firstly, the Reverse Cuthill McKee Algorithm (RCM) algorithm is used to preprocess the linear system, and then a new partitioning strategy is proposed to divide orthogonal blocks into one category, in order to accelerate the convergence rate of the Kaczmarz algorithm. The convergence of the POBK algorithm has been theoretically proven, and a theoretical analysis of its faster convergence is also provided. In addition, the experimental results confirm that this algorithm is far superior to GRBK, RBK(k), and GREBK(k) algorithms in both iteration steps (IT) and CPU time aspects., Comment: 22 pages, 11 figures, 3 tables

With the development of Big data technology, data analysis has become increasingly important. Traditional clustering algorithms such as K-means are highly sensitive to the initial centroid selection and perform poorly on non-convex datasets. In this paper, we address these problems by proposing a data-driven Bregman divergence parameter optimization clustering algorithm (DBGSA), which combines the Universal Gravitational Algorithm to bring similar points closer in the dataset. We construct a gravitational coefficient equation with a special property that gradually reduces the influence factor as the iteration progresses. Furthermore, we introduce the Bregman divergence generalized power mean information loss minimization to identify cluster centers and build a hyperparameter identification optimization model, which effectively solves the problems of manual adjustment and uncertainty in the improved dataset. Extensive experiments are conducted on four simulated datasets and six real datasets. The results demonstrate that DBGSA significantly improves the accuracy of various clustering algorithms by an average of 63.8\% compared to other similar approaches like enhanced clustering algorithms and improved datasets. Additionally, a three-dimensional grid search was established to compare the effects of different parameter values within threshold conditions, and it was discovered the parameter set provided by our model is optimal. This finding provides strong evidence of the high accuracy and robustness of the algorithm., Comment: 21 pages, 10 figures

In real-world tasks, there is usually a large amount of unlabeled data and labeled data. The task of combining the two to learn is known as semi-supervised learning. Experts can use logical rules to label unlabeled data, but this operation is costly. The combination of perception and reasoning has a good effect in processing such semi-supervised tasks with domain knowledge. However, acquiring domain knowledge and the correction, reduction and generation of rules remain complex problems to be solved. Rough set theory is an important method for solving knowledge processing in information systems. In this paper, we propose a rule general abductive learning by rough set (RS-ABL). By transforming the target concept and sub-concepts of rules into information tables, rough set theory is used to solve the acquisition of domain knowledge and the correction, reduction and generation of rules at a lower cost. This framework can also generate more extensive negative rules to enhance the breadth of the knowledge base. Compared with the traditional semi-supervised learning method, RS-ABL has higher accuracy in dealing with semi-supervised tasks.

Currently, matrix decomposition is one of the most widely used collaborative filtering algorithms by using factor decomposition to effectively deal with large-scale rating matrix. It mainly uses the interaction records between users and items to predict ratings. Based on the characteristic attributes of items and users, this paper proposes a new UISVD++ model that fuses the type attributes of movies and the age attributes of users into SVD++ framework. By projecting the age attribute into the user's implicit space and the type attribute into the item's implicit space, the model enriches the side information of the users and items. At last, we conduct comparative experiments on two public data sets, Movielens-100K and Movielens-1M. Experiment results express that the prediction accuracy of this model is better than other baselines in the task of predicting scores. In addition, these results also show that UISVD++ can effectively alleviate the cold start situation., Comment: 11 pages, 8 figures, 9 tables

Although traditional symbolic reasoning methods are highly interpretable, their application in knowledge graph link prediction is limited due to their low computational efficiency. In this paper, we propose a new neural symbolic reasoning method: RNNCTPs, which improves computational efficiency by re-filtering the knowledge selection of Conditional Theorem Provers (CTPs), and is less sensitive to the embedding size parameter. RNNCTPs are divided into relation selectors and predictors. The relation selectors are trained efficiently and interpretably, so that the whole model can dynamically generate knowledge for the inference of the predictor. In all four datasets, the method shows competitive performance against traditional methods on the link prediction task, and can have higher applicability to the selection of datasets relative to CTPs., Comment: 16 pages, 8 figures,5 tables

Although traditional symbolic reasoning methods are highly interpretable, their application in knowledge graphs link prediction has been limited due to their computational inefficiency. A new RNNNTP method is proposed in this paper, using a generalized EM-based approach to continuously improve the computational efficiency of Neural Theorem Provers(NTPs). The RNNNTP is divided into relation generator and predictor. The relation generator is trained effectively and interpretably, so that the whole model can be carried out according to the development of the training, and the computational efficiency is also greatly improved. In all four data-sets, this method shows competitive performance on the link prediction task relative to traditional methods as well as one of the current strong competitive methods., Comment: 13 pages, 5 figures

The coordinate descent method is an effective iterative method for solving large linear least-squares problems. In this paper, for the highly coherent columns case, we construct an effective coordinate descent method which iteratively projects the estimate onto a solution space formed by two greedily selected hyperplanes via Gram-Schmidt orthogonalization. Our methods may be regarded as a simple block version of coordinate descent method which involves two active columns. The convergence analysis of this method is provided and numerical simulations also confirm the effectiveness for matrices with highly coherent columns., Comment: 12 pages,4 tables,8 figures

Recently, the Dodgson's determinant condensation algorithm was revisited in many papers [College Math. Journal 42(1)(2011): 43--54, College Math. Journal 38(2)(2007): 85--95, Math Horizons 14(2)(2006): 12--15},etc.]. This method is attractive, but there also exist some shortcomings. In this paper, a symbolic algorithm and the corresponding problems are discussed to overcome these shortcomings. Numerical experiments show that this symbolic modified algorithm has highly sensitivity on initial configuration of the matrix in condensation process., Comment: 10 pages, 3 figures

This paper clarifies the main research methods and ideas of the thesis [1,2,4]. The special calculation process is also realized by corresponding computer algorithm. Finally, we introduce zero rows sum case and give the corresponding algorithm, which greatly simplifies the relevant calculation of paper [1,4]., Comment: 10 pages, 26 figures

In this paper, we study the restarted Krylov subspace method, which is typically represented by the GMRES(m) method. Our work mainly focused on the amount of change in the iterative solution of GMRES(m) at each restart. We propose an extension of the GMRES(m) method based on the idea of projection. The algorithm is named as LGMRES. In addition, LLBGMRE method is also obtained by adding backtracking restart technology to LGMRES. Theoretical analysis and numerical experiments show that LGMRES and LLBGMRES have better convergence than traditional restart GMRES(m) method., Comment: 20 pages, 14 figures

The object of this paper is to introduce a new and fascinating method of solving large linear equations, based on Cramer's rule or Gaussian elimination but employing Sylvester's determinant identity in its computation process. In addition, a scheme suitable for parallel computing is presented for this kind of generalized Chi\`{o}'s determinant condensation processes, which makes this new method have a property of natural parallelism. Finally, some numerical experiments also confirm our theoretical analysis., Comment: 15 pages, 4 figures

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iterationsolver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods., Comment: 22 pages, 2 figures. This manuscript has been accepted (by J. Appl. Anal. Comput., in press) for publication (2017/02). arXiv admin note: text overlap with arXiv:1403.7013 by other authors

If $A$ and $B$ are nonnegative matrices, a sharp upper bound on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices is given, and the minimum eigenvalue $\tau(A\star B)$ of the Fan product of two $M$-matrices $A$ and $B$ is discussed. In addition, we also give a sharp lower bound on $\tau(A\circ B^{-1})$ for the Hadamard product of $A$ and $B^{-1}$. Several examples, illustrating that the given bound is stronger than the existing bounds, are also given., Comment: 15 pages, Submitted to LAA on 2012.10.14

Many problems in science and engineering fields require the solution of shifted linear systems. To solve such systems efficiently, the recycling BiCG (RBiCG) algorithm in [SIAM J. SCI. COMPUT, 34 (2012) 1925-1949] is extended in this paper. However, the shift-invariant property could no longer hold over the augmented Krylov subspace due to adding the recycling spaces. To remedy this situation, a strategy to enforce the collinearity condition on the shifted system is adopted and then a short term recurrence for the solution update of the shifted system is derived when the seed system is solving. The new method not only improves the convergence but also has a potential to simultaneously compute approximate solutions for shifted linear systems using only as many matrix-vector multiplications as the solution of a single system requires. In addition, some numerical experiments also confirm the efficiency of our method., Comment: This paper has been withdrawn by the author due to a crucial sign error in article

The finite difference scheme with the shifted Gr\"{u}nwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed for solving these linear systems. One of the main results is that if an $A_{\nu_1,\nu_2}$-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I+L, where I is the identity matrix and the rank of L is at most $2m(\nu_1+\nu_2)$. It means that when the GMRES method is applied to solve the preconditioned linear systems, the method will converge in at most $2m(\nu_1+\nu_2)+1$ iterations.Finally, extensive numerical experiments are reported to illustrate the effectiveness of our methods for solving the fractional diffusion equations., Comment: 19 pages,4 figures

As we all known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing and signal processing etc. In this paper, an algorithm for nonnegative matrix approximation is proposed. This method mainly bases on the active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. Our method improves the recent results of those methods in [Pattern Recognition, 45(2012)3557-3565; SIAM J. Sci. Comput., 33(6)(2011)3261-3281; Neural Computation, 19(10)(2007)2756-2779, etc.]. Moreover, the object function decreases faster than many other NMF methods. In addition, some numerical experiments are presented in the synthetic data, imaging processing and text clustering. By comparing with the other six nonnegative matrix approximation methods, our experiments confirm to our analysis., Comment: 19 pages, 13 figures, Submitted to PP on Feb. 5, 2013

This study is mainly focused on iterative solutions to shifted linear systems arising from a Quantum Chromodynamics (QCD) problem. To solve such system efficiently, we explore a kind of shifted QMRCGstab (SQMRCGstab) methods, which is derived by extending the quasi-minimum residual to the shifted BiCGstab. The shifted QMRCGstab method takes advantage of the shifted structure, so that the number of matrix-vector products and the number of inner products are the same as a single linear system. Moreover, the SQMRCGstab achieves a smoothing of the residual compared to the shifted BiCGstab, and is more competitive than the MS-QMRIDR(s) and the shifted BiCGstab on the QCD problem. Numerical examples show also the efficiency of the method when one applies it to the real problems., Comment: 19 pages,14 figures, Submitted to CPC on Jan 7, 2012

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(\tau^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis., Comment: 30 pages,8 figures

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture., Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 2012