Advances in Computational Mathematics最新文献

In this work, we investigate a neural network-based solver for optimal control problems (without/with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme and provide (L^2(Omega )) error bounds on the state, control, and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.

A nonconforming (P_3) finite element is constructed by enriching the conforming (P_3) finite element space with nine (P_4) nonconforming bubbles, on each tetrahedron. Here, the divergence of the (P_4) bubble is not a (P_3) polynomial, but a (P_2) polynomial. This nonconforming (P_3) finite element, combined with the discontinuous (P_2) finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently, such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special (P_4) bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.

We consider Maxwell’s equations in a decoupled formulation by introducing Lagrange multipliers and obtain the magnetic field given the known electric field. The proposed formulation combines the decoupled weak form with the four equations of Maxwell’s model. The decoupled system reduces the computational complexity by restricting the degrees of freedom of the electric or magnetic fields. We present the construction of mixed weak Galerkin finite element methods for electric field and magnetic field, utilizing backward Euler time discretization in fully discrete schemes. We analyze the error estimate of the electric and magnetic field in the energy norm. Finally, we present numerical results for the proposed schemes in three-dimensional space to validate our theory.

We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove that functions in ( L^2(K) ) with ( K subseteq {{mathbb {R}}^d}) compact, are uniquely determined by phaseless lattice-samples of its short-time Fourier transform with window function g, provided that specific density properties of translates of g are met. By proving completeness statements for systems of discrete translates in Banach function spaces on compact sets, we obtain new uniqueness statements for phaseless sampling on lattices beyond the known Gaussian window regime. Our results apply to a large class of window functions which are relevant in time-frequency analysis and applications.

For a window ( gin L^2(mathbb {R}) ), the subset of all lattice parameters ( (a, b)in mathbb {R}^2_+ ) such that ( mathcal {G}(g,a,b)={e^{2pi ib mcdot }g(cdot -a k): k, min mathbb {Z}} ) forms a frame for ( L^2(mathbb {R}) ) is known as the frame set of g. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in (J. Fourier Anal. Appl. 22, 1440–1451, 2016) conjectured that if

then the Gabor system ( mathcal {G}(Q_2, a, b) ) of the second-order B-spline ( Q_2 ) is not a frame along the hyperbolas

for every ( a_0 ), ( b_0 ). Nielsen in (2015) also conjectured that ( mathcal {G}(Q_2, a,b) ) is not a frame for

In this paper, we prove that both Conjectures are true.

The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization at each time step. The scheme is fully explicit and relies on the computation of matrix exponential vector products. To accelerate such computation, we further construct a noniterative, nonoverlapping domain decomposition algorithm, namely localized ETDRK2, which loosely decouples the system at each time step via suitable interface conditions. Temporal error analysis of the proposed global and localized ETDRK2 schemes is rigorously proved; moreover, the schemes are shown to be conservative under periodic boundary conditions. Numerical results for the Burgers’ equation in one and two dimensions (with moving shocks) are presented to verify the theoretical results and illustrate the performance of the global and localized ETDRK2 methods where large time step sizes can be used without affecting numerical stability.

The nonlocal cubic Gross-Pitaevskii equation, in comparison to the cubic Gross-Pitaevskii equation, incorporates a nonlocal diffusion operator and can capture a wider range of practical phenomena. However, this nonlocal formulation significantly increases the computational expenses in numerical simulations, necessitating the development of efficient and accurate time integration schemes. This paper uses the relaxation method to present two linearly implicit conservative exponential schemes for the nonlocal cubic Gross-Pitaevskii equation. One proposed scheme can inherit the discrete energy while the other preserves the mass in the discrete scene. We first apply the Fourier pseudo-spectral method to the equation and derive a conservative semi-discrete system. Then, based on the ideas of the traditional relaxation method, adopting the exponential time difference method to approximate the system in time can lead to an energy-preserving exponential scheme. The mass-preserving scheme is derived by using the integral factor method to discretize the system in the temporal direction. The stability results of the constructed schemes are given. In addition, all schemes are linearly implicit and can be implemented efficiently with a large time step. Finally, numerical results show that both proposed methods are remarkably efficient and have better stability than the original relaxation scheme.

The demagnetization field in micromagnetism is given as the gradient of a potential that solves a partial differential equation (PDE) posed in (mathbb {R}^d). In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain, and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem rely on the representation of the potential via the Green’s function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green’s function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs are obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings: periodic magnetization and high-frequency magnetization. Numerical examples are given to verify the convergence rates.

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Building upon the ideas from the article (Arnold et al. SIAM J. Numer. Anal. 49, 1436–1460, 2011), we first analytically transform the given equation into a smoother (i.e., less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

In this paper, we investigate the optimal control problem governed by parabolic PDEs in random cylindrical domains, where the random domains are independent of time. We introduce a random mapping to transform the original problem in the random domain into the stochastic problem in the reference domain. The randomness of the transformed problem is reflected in the random coefficient matrix of the elliptic operator, the random time-derivative term, and the random forcing term. We make the finite-dimensional noise assumption on the random mapping in order to represent the random source of the transformed problem. Then, we use the perturbation method to expand the random functions in the transformed problem and establish the decoupled first-order and second-order optimality systems. Further, we combine the finite element method and the backward Euler scheme to obtain the fully discrete schemes for these two systems. Finally, the error analyses are respectively performed for the first-order and second-order schemes, and some examples are provided to verify the theoretical results.