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In this paper, we investigate solving the elliptical multiple eigenvalue (EME) problems using a Feedforward Neural Network. Firstly, we propose a general formulation for computing EME based on penalized variational forms of elliptical eigenvalue problems. Next, we solve the penalized variational form using the Deep Ritz Method. We establish an upper bound on the error between the estimated eigenvalues and true ones in terms of the depth D , width W of the neural network, and training sample size n. By exploring the regularity of the EME and selecting an appropriate depth D and width W , we demonstrate that the desired bound enjoys a convergence rate of O (1 / n 16) , which circumvents the curse of dimensionality. We also present several high-dimensional simulation results to illustrate the effectiveness of our proposed method and support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider recovering n - dimensional signals from m binary measurements corrupted by noises and sign flips under the assumption that the target signals have low generative intrinsic dimension, i.e., the target signals can be approximately generated via an L-Lipschitz generator G : R k → R n , k ≪ n . Although the binary measurements model is highly nonlinear, we propose a least square decoder and prove that, up to a constant c, with high probability, the least square decoder achieves a sharp estimation error C k log (L n) m as long as m ≥ C (k log (L n)) . Extensive numerical simulations and comparisons with state-of-the-art methods demonstrated the least square decoder is robust to noise and sign flips, as indicated by our theory. By constructing a ReLU network with properly chosen depth and width, we verify the (approximately) deep generative prior, which is of independent interest. [ABSTRACT FROM AUTHOR]

We propose a fast Newton algorithm for ℓ 0 regularized high-dimensional generalized linear models based on support detection and root finding. We refer to the proposed method as GSDAR. GSDAR is developed based on the KKT conditions for ℓ 0 -penalized maximum likelihood estimators and generates a sequence of solutions of the KKT system iteratively. We show that GSDAR can be equivalently formulated as a generalized Newton algorithm. Under a restricted invertibility condition on the likelihood function and a sparsity condition on the regression coefficient, we establish an explicit upper bound on the estimation errors of the solution sequence generated by GSDAR in supremum norm and show that it achieves the optimal order in finite iterations with high probability. Moreover, we show that the oracle estimator can be recovered with high probability if the target signal is above the detectable level. These results directly concern the solution sequence generated from the GSDAR algorithm, instead of a theoretically defined global solution. We conduct simulations and real data analysis to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A natural image u is often sparse under a given transformation W, one can use L0 norm of Wu as a regularisation term in image reconstructions. Since minimizing the L0 norm is a NP hard problem, the L1 norm is widely used as an replacement. However, recent studies show that nonconvex penalties, e.g., MCP, enjoy better performance for sparse signal recovery. In this paper, we propose a nonconvex model for image restoration with a minimax concave penalty (MCP). First we establish the existence of a global minimizer for the nonconvex model. Then we solve this model by using the alternating direction method of multipliers algorithm. The convergence of the proposed algorithm is analysed with properly chosen parameters. Numerical experiments show that the MCP model outperforms TV model in terms of efficiency and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a novel numerical algorithm is presented to compute the optimal time of a time optimal control problem where the governing system is a linear ordinary differential equation. By the equivalence between time optimal control problem and norm optimal control problem, computation of the optimal time can be obtained by solving a sequence of norm optimal control problems, which are transferred into their Lagrangian dual problems. The nonsmooth structure of the dual problem is approximated by the iteratively reweighted least square strategy. Several numerical tests are given to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated. The convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained. A monotonic decreasing algorithm is presented to solve the discretized problem and numerical examples are given to demonstrate the error estimation as well as the efficiency of the method. [ABSTRACT FROM AUTHOR]

We investigate an inverse problem of identifying a Robin coefficient with a sparse structure in the Laplace equation from noisy boundary measurements. The sparse structure of the Robin coefficient γ is understood as a small perturbation of a reference profile γ in the sense that their difference γ− γ has a small support. This problem is formulated as an optimal control problem with an L-regularization term. An iteratively reweighted least-squares algorithm with an inner semismooth Newton iteration is employed to solve the resulting optimization problem, and the convergence of the iteratively weighted least-squares algorithm is established. Numerical results for two-dimensional problems are presented to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Optimal control for an elliptic system with pointwise Euclidean norm constraints on the control variables is investigated. First order optimality conditions are derived in a manner that is amenable for numerical realisation. An efficient semismooth Newton algorithm is proposed based on this optimality system. Numerical examples are given to validate the superlinear convergence of the semismooth Newton algorithm. [ABSTRACT FROM AUTHOR]