Advances in Computational Mathematics最新文献

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.

We introduce a two-parameter Tikhonov regularization method to approximate an ill-posed problem with an unbounded matrix operator. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence analysis of the regularized solution is presented. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, and numerical experiments are given to demonstrate the effectiveness of the proposed method.

This paper develops fast and efficient algorithms for computing Tucker decomposition with a given multilinear rank. By combining random projection and the power scheme, we propose two efficient randomized versions for the truncated high-order singular value decomposition (T-HOSVD) and the sequentially T-HOSVD (ST-HOSVD), which are two common algorithms for approximating Tucker decomposition. To reduce the complexities of these two algorithms, fast and efficient algorithms are designed by combining two algorithms and approximate matrix multiplication. The theoretical results are also achieved based on the bounds of singular values of standard Gaussian matrices and the theoretical results for approximate matrix multiplication. Finally, the efficiency of these algorithms is illustrated via some test tensors from synthetic and real datasets.

This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say (q(tau ,A)), on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of (q(tau ,w)) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.

In this paper, we propose a discontinuous plane wave neural network (DPWNN) method with (hp-)refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with (h-)refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.

Exponential integrators are an efficient alternative to implicit schemes for the time integration of stiff system of differential equations. In this paper, low-rank exponential integrators of orders one and two for stiff differential Riccati equations are proposed and investigated. The error estimates of the proposed schemes are established. The proposed approach allows to overcome the main difficulties that lay in the interplay of time integration and low-rank approximation in the numerical schemes, which is uncommon in standard discretization of differential equations. Results of numerical experiments demonstrate the validity of the convergence analysis and show the performance of the proposed low-rank approximations with different settings.