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One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study n 1 × n 2 × n 3 third-order tensor completion based on transformed tensor singular value decomposition, and provide a bound on the number of required sample entries. Our approach is to make use of the multi-rank of the underlying tensor instead of its tubal rank in the bound. In numerical experiments on synthetic and imaging data sets, we demonstrate the effectiveness of our proposed bound for the number of sample entries. Moreover, our theoretical results are valid to any unitary transformation applied to n 3 -dimension under transformed tensor singular value decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we study the discovery of unknown dynamical systems with observed noisy data of the dynamics by neural networks. It is well-known that the performance of the neural network approach is degraded when observed data is noisy, even if the noise level is small. The main contribution of this paper is to propose a new network-based formulation for the dynamics discovery using numerical quadrature rules and to employ a self-supervision network to denoise observed data from the underlying dynamics. Our experimental results show that the performance of the proposed approach is better than that of existing dynamical discovery methods. • Design a neural network method for the discovery of unknown and noisy dynamical systems. • Propose a new quadrature rule. • Consider self-supervised denoising scheme in the network. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study a V-cycle multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. The coefficient matrices of the linear systems are structured such that their matrix-vector multiplications can be computed efficiently. The main advantage using the multigrid method is to handle the space-fractional diffusion equations on non-rectangular domains, and to solve the linear systems with non-constant coefficients more effectively. The main idea of the proposed multigrid method is to employ two banded splitting iteration schemes as pre-smoother and post-smoother. The pre-smoother and the post-smoother take banded splitting of the coefficient matrix under the x -dominant ordering and the y -dominant ordering, respectively. We prove the convergence of the proposed two banded splitting iteration schemes in the sense of infinity norm. Results of numerical experiments for time-dependent two-dimensional space-fractional diffusion equations on rectangular, L-shape and U-shape domains are reported to demonstrate that both computational time and iteration number required by the proposed method are significantly smaller than those of the other tested methods. [ABSTRACT FROM AUTHOR]

In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of samples are sufficiently taken. According to our established error bound, we also find that our physics-informed WGANs have higher requirement for the capacity of discriminators than that of generators. Numerical results on synthetic examples of partial differential equations are reported to validate our theoretical results and demonstrate how uncertainty quantification can be obtained for solutions of partial differential equations and the distributions of initial/boundary data. However, the quality or the accuracy of the uncertainty quantification theory in all the points in the interior is still the theoretical vacancy, and required for further research. • Wasserstein generative adversarial networks need little priors for uncertainty quantifications. • Physics-informed regularization term enforces the satisfaction of governing PDEs. • The proposed probabilistic model (WGAN-PINNs) is able to deal with non-Gaussian and spatially dependent noise. • WGAN-PINNs have good convergence and generalization properties. • Theoretically we have higher requirements on discriminators than generators. [ABSTRACT FROM AUTHOR]

In this paper, we study the block lower triangular Toeplitz-like with tri-diagonal blocks system which arises from the time-fractional partial differential equation. Existing fast numerical solver (e.g., fast approximate inversion method) cannot handle such linear system as the main diagonal blocks are different. The main contribution of this paper is to propose a fast direct method for solving this linear system, and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system. Our idea is based on the divide-and-conquer strategy and together with the fast Fourier transforms for calculating Toeplitz matrix–vector multiplication. The complexity needs O ( M N log 2 ⁡ M ) arithmetic operations, where M is the number of blocks (the number of time steps) in the system and N is the size (number of spatial grid points) of each block. Numerical examples from the finite difference discretization of time-fractional partial differential equations are also given to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

• A parallel-in-time preconditioning technique is proposed for the ill-conditioned and non-symmetric all-at-once system. • The condition number of the preconditioned matrix is proven to be uniformly bounded by a constant independent of matrix size. • A fast and exact implementation is proposed for the preconditioned matrix-vector product. In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. Such a problem has been intensively studied in recent years thanks to the numerous real world applications. However, due to the non-local property of the time evolution, solving the equation in PinT manner is difficult. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a square root of the constant Laplacian operator out of the constant-coefficient all-at-once matrix as a right preconditioner and take the remaining part as a left preconditioner, which constitutes our two-sided preconditioning. Exploiting the diagonalizability of the constant-Laplacian matrix and the triangular Toeplitz structure of the temporal discretization matrix, we obtain efficient representations of inverses of the right and the left preconditioners, because of which the iterative solution can be fast updated in a PinT manner. Theoretically, the condition number of the two-sided preconditioned matrix is proven to be uniformly bounded by a constant independent of the matrix size. To the best of our knowledge, for the non-local evolutionary equation with variable coefficients, this is the first attempt to develop a PinT preconditioning technique that has fast and exact implementation and that the corresponding preconditioned system has a uniformly bounded condition number. Numerical results are reported to confirm the efficiency of the proposed two-sided preconditioning technique. [ABSTRACT FROM AUTHOR]

In this paper, we propose a spatially variant total variational model to correct chromatic aberration (CA) that causes false color artifacts near edges in captured images. In general, it may be very difficult to determine suitably CA regions in captured images. Instead of using local image processing methods, our idea is to make use of spatially variant model to control the gradient and intensity matching between the red and blue color channels and the green color channel at the edges. The total variation regularization is also employed to constraint the change of the intensity of red and blue color channels during the gradient and intensity matching. We present both theoretical results and algorithms for the proposed model. Experimental results are given to illustrate the effectiveness of the proposed model and algorithm and show that their corrected images are visually better than those corrected by the other testing methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop an image pixel based histogram equalization model for image contrast enhancement. The approach is to propose a variational model containing an energy functional to adjust the pixel values of an input image directly so that the resulting histogram can be redistributed to be uniform. This idea is different from existing histogram equalization algorithms where a histogram based on the input image is constructed, a mapping is determined to output a uniform histogram and then the pixel values of the input image are adjusted based on the mapping. In the variational model, a mean brightness term is incorporated to preserve the brightness of the input image, and a geometry constraint can also be added to keep the geometry structure of the input image. Theoretically, the existence of the minimizer of the proposed model, and the convergence of the proposed algorithm are given. Experimental results are reported to demonstrate that the performance of the proposed model are competitive with the other testing histogram equalization methods for several testing images. [ABSTRACT FROM AUTHOR]

Recently, due to the advancement of acquisition techniques, visual tensor data have been accumulated in a great variety of engineering fields, e.g., biometrics, neuroscience, surveillance and remote sensing. How to analyze and learn with such tensor objects thus becomes an important and growing interest in machine learning community. In this paper, we propose a block linear discriminant analysis (BLDA) algorithm to extract features for visual tensor objects such as multichannel/hyperspectral face images or human gait videos. Taking the inherent characteristic of such tensor data into account, we unfold tensor objects according to their spatial information and frequency/time information, and represent them in a block matrix form. As a result, the block form between-class and within-class scatter matrices are constructed, and a related block eigen-decomposition is solved to extract features for classification. Comprehensive experiments have been carried out to test the effectiveness of the proposed method, and the results show that BLDA outperforms existing algorithms like DATER, 2DLDA, GTDA, UMLDA, STDA and MPCA for visual tensor object analysis. [ABSTRACT FROM AUTHOR]