Calcolo最新文献

In this paper, we introduce two conservative Crank–Nicolson type finite difference schemes and a Chebyshev collocation scheme for the nonlinear Schrödinger equation with a Dirac delta potential in 1D. The key to the proposed methods is to transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal (H^1) error estimates and the conservative properties of the finite difference schemes are investigated. Both Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time, and the first-order/second-order convergence rates in space, depending on the approximation of the interface condition. Furthermore, the Chebyshev collocation method has been established by the domain-decomposition techniques, and it is numerically verified to be second-order convergent in time and spectrally accurate in space. Numerical examples are provided to support our analysis and study the orbital stability and the motion of the solitary solutions.

Elliptic polytopes are convex hulls of several concentric plane ellipses in ({{mathbb {R}}}^d). They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for (d=2, 3), it admits an explicit solution. For larger d, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.

Embedding randomization procedures in the Alternating Direction Method of Multipliers (ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in practice, the introduction of such techniques could mitigate the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the convergence in expectation, which may not be a good indicator of its robustness and efficiency. In this work, analysing the strongly convex quadratic programming case from a linear algebra perspective, we interpret the block Gauss–Seidel sweep performed by the multi-block ADMM in the context of the inexact Augmented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative technique to those present in the literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only ({mathcal {O}}(N)) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.