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-Maryuama 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 article presents an image space branch-reduction-bound algorithm for globally solving the sum of affine ratios problem. The algorithm works by solving its equivalent problem, and by using convex hull and concave hull approximation of bilinear function, we can construct the affine relaxation problem of the equivalent problem, which can be used to compute the lower bounds during the branch-and-bound search. By subsequently refining the initial image space rectangle and solving a series of affine relaxation problems, the proposed algorithm is convergent to the global optima of the primal problem. For improving the convergence speed, an image space region reducing method is adopted for compressing the investigated image space rectangle. In addition, the global convergence of the algorithm is proved, and its computational complexity is analyzed. Finally, comparing with some existing methods, numerical results indicate that the algorithm has better computational performance.

In this paper, a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin (SUPG) virtual element method (VEM) discretization of convection dominated diffusion equation. A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm, where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors. Based on the Dörfler's marking strategy, adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes. Numerical experiments show the robustness of the a posteriori error estimates.

In this article, we propose a new finite element space $Λ_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space $Λ_h$ is designed in such a way that the strong requirement $V_h ⊂ Λ_h$ in [9] is weakened to $\{v_h ∈ V_h; {\rm div} v_h = 0\} ⊂ Λ_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new $Λ_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, $d = 2, 3$ and for triangular partitions in $\mathbb{R}^2.$ Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.

In this paper, we study a stochastic Newton method for nonlinear equations, whose exact function information is difficult to obtain while only stochastic approximations are available. At each iteration of the proposed algorithm, an inexact Newton step is first computed based on stochastic zeroth-and first-order oracles. To encourage the possible reduction of the optimality error, we then take the unit step size if it is acceptable by an inexact Armijo line search condition. Otherwise, a small step size will be taken to help induce desired good properties. Then we investigate convergence properties of the proposed algorithm and obtain the almost sure global convergence under certain conditions. We also explore the computational complexities to find an approximate solution in terms of calls to stochastic zeroth-and first-order oracles, when the proposed algorithm returns a randomly chosen output. Furthermore, we analyze the local convergence properties of the algorithm and establish the local convergence rate in high probability. At last we present preliminary numerical tests and the results demonstrate the promising performances of the proposed algorithm.

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y.$ The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrdinger equations. Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument. The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions. Numerical experiments are done to verify the theoretical results.

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.