Fractional Klein-Kramers equation can well describe subdiffusion in phase space. In this paper, we develop the fully discrete scheme for time-fractional Klein-Kramers equation based on the backward Euler convolution quadrature and local discontinuous Galerkin methods. Thanks to the obtained sharp regularity estimates in temporal and spatial directions after overcoming the hypocoercivity of the operator, the complete error analyses of the fully discrete scheme are built. It is worth mentioning that the convergence of the provided scheme is independent of the temporal regularity of the exact solution. Finally, numerical results are proposed to verify the correctness of the theoretical results.

Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. For the nonlinear Schrödinger equation (NLS) with wave operator (NLSW) and weak nonlinearity controlled by a small value $ε ∈ (0, 1],$ an exponential wave integrator Fourier pseudo-spectral (EWIFP) discretization has been developed (Guo et al., 2021) and proved to be uniformly accurate about $ε$ up to the time at $\mathcal{O}(1/ε^2 ).$ However, the EWIFP method is not time symmetric and can not preserve the discrete energy. As we know, the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution. In this work, we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral (SEPEWIFP) method for the NLSW with periodic boundary conditions. Through rigorous error analysis, we establish uniform error bounds of the numerical solution at $\mathcal{O}(h^{m_0} + ε^{ 2−β}\tau^2)$ up to the time at $\mathcal{O}(1/ε^β)$ for $β ∈ [0, 2],$ where $h$ and $\tau$ are the mesh size and time step, respectively, and $m_0$ depends on the regularity conditions. The tools for error analysis mainly include cut-off technique and the standard energy method. We also extend the results on error bounds, energy-preservation and time symmetry to the oscillatory NLSW with wavelength at $\mathcal{O}(ε^2)$ in time which is equivalent to the NLSW with weak nonlinearity. Numerical experiments confirm that the theoretical results in this paper are correct. Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.

In this paper, we analyze the convergence properties of a stochastic augmented Lagrangian method for solving stochastic convex programming problems with inequality constraints. Approximation models for stochastic convex programming problems are constructed from stochastic observations of real objective and constraint functions. Based on relations between solutions of the primal problem and solutions of the dual problem, it is proved that the convergence of the algorithm from the perspective of the dual problem. Without assumptions on how these random models are generated, when estimates are merely sufficiently accurate to the real objective and constraint functions with high enough, but fixed, probability, the method converges globally to the optimal solution almost surely. In addition, sufficiently accurate random models are given under different noise assumptions. We also report numerical results that show the good performance of the algorithm for different convex programming problems with several random models.

In this paper, we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations, whose solutions have generally the initial singularity. To derive optimal order a posteriori error estimates, the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results, with the convergence of $α$ order for the nonsmooth case on a uniform mesh. To recover the optimal convergence order $2−α$ on a nonuniform mesh, we further develop a time adaptive algorithm by means of barrier function recently introduced. The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.

A time multipoint nonlocal problem for a Schrödinger equation driven by a cylindrical $Q$-Wiener process is presented. The initial value depends on a finite number of future values. Existence and uniqueness of a solution formulated as a mild solution is obtained. A single-step implicit Euler-Maruyama difference scheme, a Rothe-Maruyama scheme, is suggested as a numerical solution. Convergence rate for the solution of the difference scheme is established. The theoretical statements for the solution of this difference scheme is supported by a numerical example.

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous cavity. We shall develop a modified factorization method to reconstruct the shape and location of the interior interface of the inhomogeneous cavity by means of many internal measurements of the near-field data. Numerical examples are carried out to illustrate the practicability of the inversion algorithm.

In this paper, we consider the electromagnetic wave scattering problem from a periodic chiral structure. The scattering problem is simplified to a two-dimensional problem, and is discretized by a finite volume method combined with the perfectly matched layer (PML) technique. A residual-type a posteriori error estimate of the PML finite volume method is analyzed and the upper and lower bounds on the error are established in the $H^1$-norm. The crucial part of the a posteriori error analysis is to derive the error representation formula and use a $L^2$-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality. An adaptive PML finite volume method is proposed to solve the scattering problem. The PML parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimate. Finally, numerical experiments are presented to illustrate the efficiency of the proposed method.

The system of generalized absolute value equations (GAVE) has attracted more and more attention in the optimization community. In this paper, by introducing a smoothing function, we develop a smoothing Newton algorithm with non-monotone line search to solve the GAVE. We show that the non-monotone algorithm is globally and locally quadratically convergent under a weaker assumption than those given in most existing algorithms for solving the GAVE. Numerical results are given to demonstrate the viability and efficiency of the approach.

In this paper, we establish the oracle inequalities of highly corrupted linear observations $b = Ax_0 + f_0 + e ∈ \mathbb{R}^m.$ Here the vector $x_0 ∈ \mathbb{R}^n$ with $n ≫ m$ is a (approximately) sparse signal and $f_0 ∈ \mathbb{R}^m$ is a sparse error vector with nonzero entries that can be possible infinitely large, $e ∼\mathcal{ N} (0, σ^2 I_m)$ represents the Gaussian random noise vector. We extend the oracle inequality $||\hat{x}-x_0||^2_2\lesssim \sum_i{\rm min}\{|x_0(i)|^2,\sigma^2\}$ for Dantzig selector and Lasso models in [E.J. Candès and T. Tao, Ann. Statist., 35 (2007), 2313–2351] and [T.T. Cai, L. Wang, and G. Xu, IEEE Trans. Inf. Theory, 56 (2010), 3516–3522] to $||\hat{x}-x_0||^2_2+||\hat{f}-f_0||^2_2\lesssim \sum_i{\rm min}\{|x_0(i)|^2,\sigma^2\}+\sum_j{\rm min}\{|\lambda f_0(j)|^2,\sigma^2\}$ for the extended Dantzig selector and Lasso models. Here $(\hat{x}, \hat{f})$ is the solution of the extended model, and $λ > 0$ is the balance parameter between $||x||_1$ and $||f||_1$ i.e. $||x||_1+\lambda||f||_1.$

A fundamental problem in some applications including group testing and communications is to acquire the support of a $K$-sparse signal $x,$ whose nonzero elements are 1, from an underdetermined noisy linear model. This paper first designs an algorithm called binary least squares (BLS) to reconstruct $x$ and analyzes its complexity. Then, we establish two sufficient conditions for the exact reconstruction of $x$’s support with $K$ iterations of BLS based on the mutual coherence and restricted isometry property of the measurement matrix, respectively. Finally, extensive numerical tests are performed to compare the efficiency and effectiveness of BLS with those of batch orthogonal matching pursuit (Batch-OMP) which to our best knowledge is the fastest implementation of OMP, orthogonal least squares (OLS), compressive sampling matching pursuit (CoSaMP), hard thresholding pursuit (HTP), Newton-step-based iterative hard thresholding (NSIHT), Newton-step-based hard thresholding pursuit (NSHTP), binary matching pursuit (BMP) and $ℓ_1$-regularized least squares. Test results show that: (1) BLS can be 10-200 times more efficient than Batch-OMP, OLS, CoSaMP, HTP, NSIHT and NSHTP with higher probability of support reconstruction, and the improvement can be 20%-80%; (2) BLS has more than 25% improvement on the support reconstruction probability than the explicit BMP algorithm with a little higher computational complexity; (3) BLS is around 100 times faster than $ℓ_1$-regularized least squares with lower support reconstruction probability for small $K$ and higher support reconstruction probability for large $K.$ Numerical tests on the generalized space shift keying (GSSK) detection indicate that although BLS is a little slower than BMP, it is more efficient than the other seven tested sparse recovery algorithms, and although it is less effective than $ℓ_1$-regularized least squares, it is more effective than the other seven algorithms.