In this paper, we introduce a class of stochastic Hopfield neural network with cross-diffusion (SHNNCD), and for the first time, investigate the existence of the ergodic stationary distribution for SHNNCD under time-varying modified Markov switching. In terms of the internal structure, we incorporate cross-diffusion to fully account for the coupling effects between different neuronal states. In terms of the external environment, compared with traditional Markov switching, we consider a class of Markov switching modified by time-varying functions, including periodic, decaying, and random types. Based on these innovations in both internal structure and external environment, we construct a global Lyapunov function for SHNNCD using graph theory under the consideration of cross-diffusion, and derive the criterion for the existence of the ergodic stationary distribution of SHNNCD. Finally, we verify the validity of our theoretical results through numerical simulations.

In this paper, we study the non-global local minimizers of the generalized trust-region subproblem (GTR), $\min\{x^\top A_0 x + 2b_0^\top x \mid x^\top A_1 x + 2b_1^\top x + c_1 \leq 0\}$, and its equality-constrained version (GTRE), which will be candidates of the global minimizers of the nonconvex quadratically constrained quadratic programming when the hard-case happens. Specifically, if there exists $\mu_1 \in \mathbb{R}$ such that $A_0 + \mu_1 A_1 \succ 0$, we prove for GTR and GTRE that, when $A_1 \succeq 0$ (or $A_1 \preceq 0$) there may exist at most one non-global local minimizer, and when $A_1$ is indefinite there may exist at most two non-global local minimizers. Moreover, if there exists $\mu_1 \in \mathbb{R}$ such that $A_0 + \mu_1 A_1 \prec 0$, we prove also that GTR and GTRE may have at most one non-global local minimizer. All the above three upper bounds are tight, i.e., none of them can be improved again. In summary, the famous Martínez's result is successfully generalized from $A_1 \succeq 0$ to the case that $\mu_0 A_0 + \mu_1 A_1 \succ 0$ for some $\mu_0, \mu_1 \in \mathbb{R}$. Finally, an algorithm is proposed either to find all the non-global local minimizers of GTR and GTRE or to confirm their nonexistence in a tolerance. Preliminary numerical results demonstrate the effectiveness of the algorithm.

In this paper, we consider a class of multi-term time-fractional nonlinear diffusion equations (MTFNDEs) with initial boundary value conditions. Due to the initial singularity of the solutions of MTFNDEs, many existing numerical methods suffer from order reduction. To overcome this challenge, we derive a new scheme with $\min\{(\delta + \alpha_m - \alpha_{m-1})/\beta, 2\}$-order accuracy in time for $0 < \alpha_{m-1} < \alpha_m \leq 1$ and $0 < \delta < 1$ by combining the technique of variable transformation $t = s^{1/\beta}$ ($0 < \beta \leq 1$) and the linear interpolation. Meanwhile, the unconditional stability and convergence of the scheme are proved through the Fourier analysis method. Finally, numerical experiments have been given to support the theoretical results and efficiency of our proposed scheme.

This work addresses the inverse problem of recovering sparse initial data in spatial fractional parabolic equations. The associated solution operator for initial-boundary value problem is continuous and compact, implying severe ill-posedness. To overcome this, an $ℓ_1$-regularized formulation is considered. The misfit functional is shown to be differentiable and strictly convex, and the well-posedness of the regularized problem is established, including existence, uniqueness, stability, and convergence. A numerical algorithm is proposed and implemented, incorporating Nesterov’s accelerated algorithm to enhance efficiency. Numerical experiments in both one and two spatial dimensions confirm the feasibility and accuracy of the proposed approach.

In this paper, we consider the sparse distributed control problem constrained by a random elliptic equation, which we reformulate as a nonsmooth stochastic optimization problem in Hilbert space. By incorporating the advantages of the stochastic approximation approach and the alternating direction method of multipliers (ADMM), we propose a stochastic ADMM algorithm. This method decouples the stochasticity arising from the random equation constraint from the nonsmoothness of the control objective, allowing them to be tackled separately within the iterations. We introduce stochastic gradients and develop a proximal linearization technique for the stochastic subproblem, allowing each subproblem to admit a closed-form solution. The convergence and a high-probability bound of the proposed method are analyzed for the model problem. Numerical results demonstrate the effectiveness and efficiency of our method.

The rate of convergence of the augmented Lagrangian method for solving nonlinear programming is studied under the Jacobian uniqueness conditions. It is demonstrated that, for a given multiplier vector $(\mu, \lambda)$, the rate of convergence of the augmented Lagrangian method is linear with respect to $\| (\mu, \lambda) - (\mu^{*}, \lambda^{*}) \|$ and the ratio constant is proportional to $1/c$ when the ratio $\| (\mu, \lambda)-(\mu^{*}, \lambda^{*}) \| /c$ is small enough, where $c$ is the penalty parameter that exceeds a threshold $c^{*} > 0$ and $(\mu^{*}, \lambda^{*})$ is the multiplier corresponding to a local minimum point. Importantly, the ratio constant of the Q-linear convergence of the sequence of multiplier vectors is estimated by the second-order derivative of the value function of the nonlinear optimization problem. This characterization gives an explicit expression for the rate constant of the Q-linear convergence of the sequence of multiplier vectors.

In this paper, we consider a class of three-composite nonconvex optimization problems, in which the nonsmooth function is further composed with a linear operator. This problem has many applications such as sparse signal recovery, image processing and machine learning. Based on the conjugate duality theory, we present an accelerated preconditioned primal-dual gradient algorithm for this problem. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function $h^∗$ which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. This may significantly reduce the computation load. We prove that the sequence generated by the proposed algorithm globally converges to a critical point when the function satisfies the Kurdyka- Lojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, numerical results on sparse signal recovery and image processing illustrate the efficiency and competitiveness of the proposed algorithm.

In this work, a subdiffusion equation with constant time delay $\tau$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t = 0^+$ and its second-order time derivative shows singularity at both $t = 0^+$ and $\tau^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Grönwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.

In this paper, we design novel high order probabilistic numerical algorithms for forward backward stochastic differential equations. Moreover, we derive the error estimates and prove the high order convergence rates of the proposed schemes. Because the proposed scheme involves conditional expectations, an estimator based on the multilevel Monte Carlo method is applied to approximate the conditional expectations. Furthermore, we theoretically demonstrate that the computational complexity of our numerical method is proportional to the square of prescribed accuracy. Numerical experiments are given to illustrate the theoretical results.

This paper is devoted to identifying the source term and initial value simultaneously in a time-fractional Black-Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of operator equations, and under certain source conditions, conditional stability is established. We propose a regularization method with two differential operators to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived, respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.