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This paper proposes a novel deep learning (DL) based source reconstruction method (SRM). The proposed DL-based SRM employs the deep convolutional asymmetric encoder–decoder structure (DCAEDS), which only demands one-time single-frequency far-field measurement on EM scattered field as its input and further predicts the equivalent source on target scatterers. During the offline training, an EM scattering simulator is designed to compute EM scattered field originated from the predicted equivalent source on target scatterers. The DCAEDS is combined with the EM scattering simulator to optimize the loss function, including two parts: (1) Data-induced loss, which directly evaluates the difference between the prediction of the proposed DCAEDS and the true-labelled equivalent source of target scatterers; (2) Physics-induced loss, which evaluates the difference between received EM scattered field and the computed EM scattered field originated from the prediction of DCAEDS. Moreover, the proposed DL-based SRM can overcome the limitation of conventional methods, involving high computation cost and strong ill-conditions. Consequently, the proposed DL-based SRM can realize the reconstruction of the equivalent current source with higher accuracy and lower computation complexity. Numerical examples illustrate the feasibility of the proposed DL-based SRM, which opens the new path for DL-based EM computation approaches. [ABSTRACT FROM AUTHOR]

We study a class of spectral learning methods with dependent observations, including popular ridge regression, Landweber iteration, spectral cut-off and so on. We derive an explicit risk bound in terms of the correlation of the observations, regularity of the regression function, and effective dimension of the reproducing kernel Hilbert space. By appropriately choosing regularization parameter according to the sample size, the risk bound yields a nearly optimal learning rate with a logarithmic term for strongly mixing sequences. We thus extend the applicable range of spectral algorithm to non-i.i.d. sampling process. Particularly, it is shown that the learning rates for i.i.d. samples in the literature refer to our special case, i.e., the mixing condition parameter tends to zero. [ABSTRACT FROM AUTHOR]

Abstract: We consider the iterative solution of unconstrained minimization problems arising from nonlinear image restoration. Our approach is based on a novel generalized BFGS method for such large-scale image restoration minimization problems. The complexity per step of the method is of operations and only memory allocations are required, where is the number of image pixels. Based on the results given in [Carmine Di Fiore, Stefano Fanelli, Filomena Lepore, Paolo Zellini, Matrix algebras in quasi-Newton methods for unconstrained minimization, Numer. Math. 94 (2003) 479–500], we show that the method is globally convergent for our nonlinear image restoration problems. Experimental results are presented to illustrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

We study efficient iterative methods for Toeplitz systems based on the circulant and skew-circulant splitting (CSCS) of the Toeplitz matrix. Theoretical analysis show that if the circulant and the skew-circulant splitting matrices are positive definite, then the CSCS method converges to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew-circulant matrices involved. Numerical examples are presented to demonstrate the method. [Copyright &y& Elsevier]

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods. [Copyright &y& Elsevier]

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a multi-rank under the Discrete Cosine Transform-related transform tensor-tensor product. With suitable incoherence conditions on the underlying low multi-rank rank tensor, we show that the proposed Riemannian optimization method is guaranteed to converge to the underlying low multi-rank tensor with a high probability. Number of sampling entries required for convergence are also derived. Numerical examples of synthetic data and real data sets are reported to demonstrate that the performance of the proposed method is better than that of tensor-based method using the Tucker-rank model in terms of computational time, and that of the matrix-based completion method in terms of number of sampling entries. [ABSTRACT FROM AUTHOR]

In this paper, we study and analyze absolute and relative condition numbers of the spectral projections for regular periodic eigenproblems. The main contribution is to derive explicit expressions of the condition numbers of the j-th left and right spectral projections. Numerical examples are given to illustrate the proposed condition numbers for the spectral projections associated with periodic eigenproblems. [ABSTRACT FROM AUTHOR]

Abstract: Given a Probabilistic Boolean Network (PBN), an important problem is to study its steady-state probability distribution for network analysis. In this paper, we present a new perturbation bound of the steady-state probability distribution of PBNs with gene perturbation. The main contribution of our results is that this new bound is established without additional condition required by the existing method. The other contribution of this paper is to propose a fast algorithm based on the special structure of a transition probability matrix of PBNs with gene perturbation to compute its steady-state probability distribution. Experimental results are given to demonstrate the effectiveness of the new bound, and the efficiency of the proposed method. [Copyright &y& Elsevier]