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The main purpose of this paper is to design a local discontinuous Galerkin (LDG) method for the Benjamin-Ono equation. We analyze the stability and error estimates for the semi-discrete LDG scheme. We prove that the scheme is $L^2$-stable and it converges at a rate $\mathcal{O}(h^{k+1/2})$ for general nonlinear flux. Furthermore, we develop a fully discrete LDG scheme using the four-stage fourth order Runge-Kutta method and ensure the devised scheme is strongly stable in case of linear flux using two-step and three-step stability approach under an appropriate time step constraint. Numerical examples are provided to validate the efficiency and accuracy of the method.

Oscillatory second order linear ordinary differential equations arise in many scientific calculations. Because the running times of standard solvers increase linearly with frequency when they are applied to such problems, a variety of specialized methods, most of them quite complicated, have been proposed. Here, we point out that one of the simplest approaches not only works, but yields a scheme for solving oscillatory second order linear ordinary differential equations which is significantly faster than current state-of-the-art techniques. Our method, which operates by constructing a slowly varying phase function representing a basis of solutions of the differential equation, runs in time independent of the frequency and can be applied to second order equations whose solutions are oscillatory in some regions and slowly varying in others. In the high-frequency regime, our algorithm discretizes the nonlinear Riccati equation satisfied by the derivative of the phase function via a Chebyshev spectral collocation method and applies the Newton-Kantorovich method to the resulting system of nonlinear algebraic equations. We prove that the iterates converge quadratically to a nonoscillatory solution of the Riccati equation. The quadratic convergence of the Newton-Kantorovich method and the simple form of the linearized equations ensure that this procedure is extremely efficient. Our algorithm then extends the slowly varying phase function calculated in the high-frequency regime throughout the solution domain by solving a certain third order linear ordinary differential equation related to the Riccati equation. We describe the results of numerical experiments showing that our algorithm is orders of magnitude faster than existing schemes, including the modified Magnus method [18], the current state-of-the-art approach [7] and the recently introduced ARDC method [1].

We consider non-linear Bayesian inverse problems of determining the parameter $f$. For the posterior distribution with a class of Gaussian process priors, we study the statistical performance of variational Bayesian inference to the posterior with variational sets consisting of Gaussian measures or a mean-field family. We propose certain conditions on the forward map $\mathcal{G}$, the variational set $\mathcal{Q}$ and the prior such that, as the number $N$ of measurements increases, the resulting variational posterior distributions contract to the ground truth $f_0$ generating the data, and derive a convergence rate with polynomial order or logarithmic order. As specific examples, we consider a collection of non-linear inverse problems, including the Darcy flow problem, the inverse potential problem for a subdiffusion equation, and the inverse medium scattering problem. Besides, we show that our convergence rates are minimax optimal for these inverse problems.

In this paper we develop a $C^0$-conforming virtual element method (VEM) for a class of second-order quasilinear elliptic PDEs in two dimensions. We present a posteriori error analysis for this problem and derive a residual based error estimator. The estimator is fully computable and we prove upper and lower bounds of the error estimator which are explicit in the local mesh size. We use the estimator to drive an adaptive mesh refinement algorithm. A handful of numerical test problems are carried out to study the performance of the proposed error indicator.

To solve the Cahn-Hilliard equation numerically, a new time integration algorithm is proposed, which is based on a combination of the Eyre splitting and the local iteration modified (LIM) scheme. The latter is employed to tackle the implicit system arising each time integration step. The proposed method is gradient-stable and allows to use large time steps, whereas, regarding its computational structure, it is an explicit time integration scheme. Numerical tests are presented to demonstrate abilities of the new method and to compare it with other time integration methods for Cahn-Hilliard equation.

Approximating solutions of ordinary and partial differential equations constitutes a significant challenge. Based on functional expressions that inherently depend on neural networks, neural forms are specifically designed to precisely satisfy the prescribed initial or boundary conditions of the problem, while providing the approximate solutions in closed form. Departing from the important class of ordinary differential equations, the present work aims to refine and validate the neural forms methodology, paving the ground for further developments in more challenging fields. The main contributions are as follows. First, it introduces a formalism for systematically crafting proper neural forms with adaptable boundary matches that are amenable to optimization. Second, it describes a novel technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. Third, it outlines a method for determining an upper bound on the absolute deviation from the exact solution. The proposed augmented neural forms approach was tested on a set of diverse problems, encompassing first- and second-order ordinary differential equations, as well as first-order systems. Stiff differential equations have been considered as well. The resulting solutions were subjected to assessment against existing exact solutions, solutions derived through the common penalized neural method, and solutions obtained via contemporary numerical analysis methods. The reported results demonstrate that the augmented neural forms not only satisfy the boundary and initial conditions exactly, but also provide closed-form solutions that facilitate high-quality interpolation and controllable overall precision. These attributes are essential for expanding the application field of neural forms to more challenging problems that are described by partial differential equations.

Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $ h $ and $ \Delta t $ respectively, we prove an error estimate $ O(\Delta t^3 + \frac{h^4}{\Delta t}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(\Delta t) $. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $ L^2 $ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.

We study the decidability and complexity of non-cooperative rational synthesis problem (abbreviated as NCRSP) for some classes of probabilistic strategies. We show that NCRSP for stationary strategies and Muller objectives is in 3-EXPTIME, and if we restrict the strategies of environment players to be positional, NCRSP becomes NEXPSPACE solvable. On the other hand, NCRSP_>, which is a variant of NCRSP, is shown to be undecidable even for pure finite-state strategies and terminal reachability objectives. Finally, we show that NCRSP becomes EXPTIME solvable if we restrict the memory of a strategy to be the most recently visited t vertices where t is linear in the size of the game.

We consider the configuration space of points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that they satisfy semi-discrete analogues of mKdV and sine-Gordon equations. The configuration space we investigate corresponds to the state space of a linkage mechanism known as the Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system generates an orbit in the space of real solutions to polynomial equations defined by geometric constraints.

The inverse problems about fractional Calder\'on problem and fractional Schr\"odinger equations are of interest in the study of mathematics. In this paper, we propose the inverse problem to simultaneously reconstruct potentials and sources for fractional Schr\"odinger equations with internal source terms. We show the uniqueness for reconstructing the two terms under measurements from two different nonhomogeneous boundary conditions. By introducing the variational Tikhonov regularization functional, numerical method based on conjugate gradient method(CGM) is provided to realize this inverse problem. Numerical experiments are given to gauge the performance of the numerical method.