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For complex-valued or quaternionic neural networks, scholars and researchers usually decompose them into real-valued systems. The decomposed real-valued systems are equivalent to original systems. Then, the dynamical behaviors of real-valued systems obtained are investigated, including stability, synchronization, and chaos etc. In this paper, a class of quaternionic neural networks with time-varying delays is investigated. First, by designing a suitable PI controller, synchronization of the considered chaotic system is realized. By using a non-decomposition method and structuring a novel Lyapunov functional, sufficient conditions are derived to guarantee synchronization between the drive-response systems. It is worth mentioning that, unlike other methods, our approach does not require breaking down the quaternionic neural networks into four separate real-valued systems. Furthermore, we demonstrate the practical application of these chaotic quaternionic neural networks with time-varying delays in image encryption and decryption. Based on one sequence of chaotic signal from state trajectory of single quaternion-valued neuron and a new encryption algorithm, the application of chaotic system proposed, that is, image encryption, is researched. The process of image decryption is simply the reverse of the encryption process. Finally, numerical simulation examples are provided to validate the effectiveness of the designed PI controller and performance of image encryption and decryption.

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This paper deals with the numerical solutions of two-dimensional (2D) semi-linear reaction-diffusion equations (SLRDEs) with piecewise continuous argument (PCA) in reaction term. A high-order compact difference method called I-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy $\mathcal{O}(\tau^2+h^4_x+h^4_y),$ where $\tau,$ $h_x$ and $h_y$ are the calculation stepsizes of the method in $t$-, $x$- and $y$-direction, respectively. With the above method and Newton linearized technique, a ${\rm II}$-type basic scheme is also suggested. Based on the both basic schemes, the corresponding ${\rm I}$- and ${\rm II}$-type alternating direction implicit (ADI) schemes are derived. Finally, with a series of numerical experiments, the computational accuracy and efficiency of the four numerical schemes are further illustrated.

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Resonant tunneling diodes (RTDs) exhibit a distinctive characteristic known as negative resistance. Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices. This paper conducts a thorough investigation into the current-voltage (I-V) characteristics of RTDs utilizing various numerical methods. Through a series of numerical experiments, we verified that the transfer matrix method ensures robust convergence in I-V curves and proficiently determines the tunneling bias for energy potential functions with discontinuities. Our numerical analysis underscores the significant impact of variations in effective mass on I-V curves, emphasizing the need to consider this effect. Furthermore, we observe that increasing the doping concentration results in a reduction in tunneling bias and an enhancement in peak current. Leveraging the unique features of the I-V curve, we employ shallow neural networks to accurately fit the I-V curves, yielding satisfactory results with limited data.

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This report presents a series of implicit-explicit (IMEX) variable stepsize algorithms for natural convection equations. The presented method requires a minimally intrusive modification to an existing program, does not add to the computational complexity, and is conceptually simple. Here, IMEX means the nonlinear term is treated fully explicitly, while the remaining terms are treated implicitly. Due to the increasing demand for low memory solvers, the addition of time adaptive can improve the accuracy and efficiency of the algorithms. For the first-order algorithm, we prove the stability of the variable stepsize backward Euler scheme combined with Adams-Bashforth 2 (VSS BE-AB2) and analyze convergence. Then, the stability of Constant Timestep Filtered-BE-AB2 (BE-AB2+F) is proved. Moreover, we construct adaptive algorithms by extending the approach to variable stepsize. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

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Anderson acceleration is a kind of effective method for improving the convergence of the general fixed point iteration. In the linear case, Anderson acceleration can be used to improve the convergence rate of matrix splitting based iterative methods. In this paper, by using Anderson acceleration on general splitting iterative methods for linear systems, three classes of methods are given. The first one is obtained by directly applying Anderson acceleration on splitting iterative methods. For the second class of methods, Anderson acceleration is used periodically in the splitting iteration process. The third one is constructed by combining the Anderson acceleration and split iteration method in each iteration process. The key of this class of method is to determine a combination coefficient for Anderson acceleration and split iteration method. One optimal combination coefficient is given. Some theoretical results about the convergence of the considered three methods are established. Numerical experiments show that the proposed methods are effective.

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In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. This numerical scheme is easy to implement because it only need to modify the right side. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

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The main purpose of this paper is to give stability analysis and error estimates of the ultra-weak local discontinuous Galerkin (UWLDG) method coupled with a spectral deferred correction (SDC) temporal discretization method up to fourth order, for solving the fourth-order equation. The UWLDG method introduces fewer auxiliary variables than the local discontinuous Galerkin method and no internal penalty terms are required for stability, which is efficient for high order partial differential equations (PDEs). The SDC method we adopt in this paper is based on second-order time integration methods and the order of accuracy is increased by two for each additional iteration. With the energy techniques, we rigorously prove the fully discrete schemes are unconditionally stable. By the aid of special projections and initial conditions, the optimal error estimates of the fully discrete schemes are obtained. Furthermore, we generalize the analysis to PDEs with higher even-order derivatives. Numerical experiments are displayed to verify the theoretical results.

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An analysis of logistic stochastic differential equations (SDEs) with general power-law and driven by a Wiener process is conducted. We prove existence of unique, strong Markovian, continuous solutions. The solutions live (a.s.) on bounded domains $D = [0, K]$ required by applications to biology, ecology and physics with nonrandom threshold parameter $K>0$ (i.e. the maximum carrying constant). Moreover, we present and justify nonstandard numerical methods constructed by specified balanced implicit methods (BIMs). Their weak and $L^p$-convergence follows from the fact that these methods with local Lipschitz-continuous coefficients of logistic SDEs “produce” positive numerical approximations on bounded domain [0, K] (a.s.). As commonly known, standard numerical methods such as Taylor-type ones for SDEs fail to do that. Finally, asymptotic stability of nontrivial equilibria $x_∗=K$ is proven for both continuous time logistic SDEs and discrete time approximations by BIMs. We exploit the technique of positive, sufficiently smooth and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.

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The Rytov approximation has been commonly used to obtain reconstructed images for optical tomography. However, the method requires linearization of the nonlinear inverse problem. Here, we demonstrate nonlinear Rytov approximations by developing the inverse Rytov series for the time-dependent diffusion equation. The method is verified by a solid-phantom experiment.

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In this paper, we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations. The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed. The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error. The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle, quadrilateral and polygonal meshes with hanging nodes. Finally, numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.

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An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.

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In this work, we investigate the mechanism underlying loss spikes observed during neural network training. When the training enters a region with a lower-loss-as-sharper structure, the training becomes unstable, and the loss exponentially increases once the loss landscape is too sharp, resulting in the rapid ascent of the loss spike. The training stabilizes when it finds a flat region. From a frequency perspective, we explain the rapid descent in loss as being primarily influenced by low-frequency components. We observe a deviation in the first eigendirection, which can be reasonably explained by the frequency principle, as low-frequency information is captured rapidly, leading to the rapid descent. Inspired by our analysis of loss spikes, we revisit the link between the maximum eigenvalue of the loss Hessian ($λ_{{\rm max}}$), flatness and generalization. We suggest that $λ_{{\rm max}}$ is a good measure of sharpness but not a good measure for generalization. Furthermore, we experimentally observe that loss spikes can facilitate condensation, causing input weights to evolve towards the same direction. And our experiments show that there is a correlation (similar trend) between $λ_{{\rm max}}$ and condensation. This observation may provide valuable insights for further theoretical research on the relationship between loss spikes, $λ_{{\rm max}}$, and generalization.

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The h-version analysis technique developed in [Banjai et al., SIAM J. Numer. Anal., 55 (2017)] for Trefftz discontinuous Galerkin (DG) discretizations of the second order isotropic wave equation is extended to the time-dependent Maxwell equations in anisotropic media. While the discrete variational formulation and its stability and quasi-optimality are derived parallel to the acoustic wave case, the derivation of error estimates in a mesh-skeleton norm requires new transformation stabilities for the anisotropic case. The error estimates of the approximate solutions with respect to the condition number of the coefficient matrices are proved. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent nonhomogeneous Maxwell equations. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

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The weak convergence analysis plays an important role in error estimates for stochastic differential equations, which concerns with the approximation of the probability distribution of solutions. In this paper, we investigate the weak convergence order of a splitting-up method for stochastic differential equations. We first construct a splitting-up approximation, based on which we also set up a splitting-up numerical solution. We prove both of these two approximation methods are of first order of weak convergence with the help of Malliavin calculus. Finally, we present several numerical experiments to illustrate our theoretical analysis.

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In this paper, we propose and analyze a multiphysics finite element method for a nonlinear poroelasticity model. To more effectively capture the deformation and diffusion processes, we reformulate the original nonlinear fluid-solid coupling problem into a fluid-fluid coupling problem using a multiphysics approach. We then establish the growth, coercivity, and monotonicity properties of the nonlinear stress-strain relation, derive energy estimates, and use Schaefer’s fixed point theorem to prove the existence and uniqueness of the weak solution. Furthermore, we design a fully discrete time-stepping scheme – multiphysics finite element method with $P_2−P_1−P_1$ elements for spatial variables and the backward Euler method for time variable. To handle nonlinearity, we employ the Newton iterative method, establish discrete energy laws and give the optimal convergence order error estimates. Finally, we show some numerical examples to verify the rationality of theoretical analysis and the proposed method has no “locking phenomenon”.

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Numerical simulation of time-periodic problems is a special area of research, since the time periodicity modifies the problem structure, and then it is desirable to use parallel methods to solve such problems. The classical parareal algorithm for time-periodic problems, which is parallel in time, solving an initial value coarse problem, called the periodic parareal algorithm with initial value coarse problem (PP-IC), usually converges very slowly, and even diverges for wave propagation problems. In this paper, we first present a new PP-IC algorithm based on a diagonalization technique proposed recently. In this new algorithm, we approximate the coarse propagator $G$ in the classical PP-IC algorithm with a head-tail coupled condition such that $G$ can be parallelized using diagonalization in time. We analyze the convergence factors of the diagonalization-based PP-IC algorithm for both the linear and nonlinear cases. Then, we further design and analyze a new parallel-in-time algorithm for time-periodic problems by combining the Krylov subspace method with the diagonalization-based PP-IC algorithm to accelerate the convergence. Finally, we also determine an appropriate choice of the parameter $α$ in the head-tail coupling condition, and illustrate our theoretical results with several numerical experiments, both for model problems and the realistic application of Maxwell’s equations.

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The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study. Different from the standard virtual element method, the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation. Consequently, if the interpolation operators are utilized to approximate the nonlinear terms, the implementation of Newton’s iteration algorithm can be simplified. Nonhomogeneous Dirichlet boundary conditions are considered in this paper. The strategy of approximating curved domains with polygonal domains is taken into consideration, and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly, Nitsche-based projection method is employed to impose the boundary conditions weakly. For time discretization, Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted. Based on the concrete formulation of Nitsche-based projection method, a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed. Building upon these approximation properties, error estimates are derived for the fully discrete scheme. Additionally, the extension of the fully discrete scheme to 3D case is also included. Finally, we present two numerical experiments to corroborate the theoretical findings.

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In this paper, we present a generalized Jacobi spectral Galerkin method for fractional Volterra integro-differential equations (FVIDEs). The basis functions of the proposed method are generalized Jacobi functions, which serve as natural basis functions for appropriately designed spectral methods for FVIDEs. We establish a convergence analysis of the generalized Jacobi spectral Galerkin method under reasonable assumptions. Numerical experiments are provided to demonstrate the effectiveness of the proposed method.

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In this research article, we present convenient analytical-approximate solutions for fluid flow models known as multi-dimensional Navier-Stokes equations containing time-fractional order by using a relatively new analytical method called modified generalized Mittag-Leffler function method. The Caputo fractional derivative is used to describe fractional mathematical formalism. The approximate solutions for five problems are implemented to demonstrate the validity and accuracy of the proposed method. It is also demonstrated that the solutions obtained from our method when $α = 1$ coincide with the exact solutions, this is displayed by using some 2D and 3D plots for each problem. Moreover, the comparison between our outcomes with given exact solutions and results obtained by other methods in the literature besides absolute error is provided in some tables. Additionally, we offer some plots when $α$ has different values to present the effect of fractional order on the solution of each suggested problem. The numerical simulation presented in this work indicates that the proposed method is efficient, reliable, accurate and easy which has less computational ability to give analytical-approximate solution form. So, this method can be extended to implement on different related problems arising in various areas of innovation and research.

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Achieving linear complexity is crucial for demonstrating optimal convergence rates in adaptive refinement. It has been shown that the existing linear complexity local refinement algorithm for ${\rm T}$-splines generally produces more degrees of freedom than the existing greedy refinement, which lacks linear complexity. This paper introduces a novel greedy local refinement algorithm for analysis-suitable ${\rm T}$-splines, which achieves linear complexity and requires fewer control points than existing algorithms with linear complexity. Our approach is based on the observation that confining refinements around each ${\rm T}$-junction to a preestablished feasible region ensures the algorithm’s linear complexity. Building on this constraint, we propose a greedy optimization local refinement algorithm that upholds linear complexity while significantly reducing the degrees of freedom relative to previous linear complexity local refinement methods.

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