Applied Numerical Mathematics最新文献

Given the Caputo-type fractional differential equation $D^{\alpha }y\left(t\right)=f(t,y(t\left)\right)$ with $\alpha \in (1,2)$, we consider two distinct solutions $y_1,y_2\in C[0,T]$ to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference $\left|y_1\right(t)-y_2(t\left)\right|$ for $t\in [0,T]$. The main emphasis is on describing how such bounds are related to the differences of the associated initial values.

In this paper we focus on the space of $C^2$ quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings.

We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a $O\left(h\right)$ term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method.

The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate $O\left(h^{\frac{3}{2}}\right)$. Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients.

Solution of sparse linear complementarity problem (LCP) has been widely discussed in many applications. In this paper, we consider the ${\ell }_p$ regularization problem with nonnegative constraint for sparse LCP, and propose algorithms based on the iterative reweighted method to approach a sparse solution of the LCP, and then show the convergence to the stationary point of ${\ell }_p$ regularization problem. Numerical results on simulated data exhibit an excellent performance of the proposed algorithms on approaching a sparse solution of the LCP. Finally, we apply this method to the frictional and frictionless contact problems. The numerical experiments demonstrate that the contact problems can be efficiently solved by the proposed algorithm.

In this paper, we study a class of nonconvex and nonsmooth fractional optimization problem, where the numerator of which is the sum of a nonsmooth and nonconvex function and a relative smooth nonconvex function, while the denominator is relative weakly convex nonsmooth function. We propose a Bregman proximal subgradient algorithm for solving this type of fractional optimization problems. Under moderate conditions, we prove that the subsequence generated by the proposed algorithm converges to a critical point, and the generated sequence globally converges to a critical point when the objective function satisfies the Kurdyka-Łojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, two numerical experiments illustrate the effectiveness and superiority of the algorithm. Our results give a positive answer to an open problem proposed by Bot et al. [14].

In this contribution it is shown how a recent quantum information-based theory of color perception permits to account in a natural way for several well-known properties and also to predict new ones. The quantum model is based on a completely different paradigm with respect to the one followed in classical colorimetry and it relies on the hypothesis that color sensations are the result of (perceptual) quantum measurements performed by human observers.

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate $O\left({\tau }^2\right)$ in $H^1$- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.