Calcolo最新文献

This paper investigates the analytical stability region and the asymptotic stability of linear fractional neutral delay differential equations. Employing boundary locus techniques, the stability region of this problem is analyzed. Furthermore, we derive the fundamental solution of linear fractional neutral delay differential equations, and prove the exponential boundedness, the asymptotic stability and the algebraic decay rate. Finally, numerical tests are conducted to verify the theoretical results.

This paper deals with the numerical solution of Fredholm integral equations of the second kind with endpoint algebraic singularities and with a kernel of Green’s function type. Novel Nyström schemes employing the Gauss quadrature rule are presented. These methods take into account the lack of smoothness along the diagonal of the kernel and may recover the full convergence rate of smooth kernels. A complete analysis of the stability and convergence is provided, and several numerical tests that illustrate the efficiency and accuracy of various approaches are considered.

We propose a lightning Virtual Element Method that eliminates the stabilisation term by actually computing the virtual component of the local VEM basis functions using a lightning approximation. In particular, the lightning VEM approximates the virtual part of the basis functions using rational functions with poles clustered exponentially close to the corners of each element of the polygonal tessellation. This results in two great advantages. First, the mathematical analysis of a priori error estimates is much easier and essentially identical to the one for any other non-conforming Galerkin discretisation. Second, the fact that the lightning VEM truly computes the basis functions allows the user to access the point-wise value of the numerical solution without needing any reconstruction techniques. The cost of the local construction of the VEM basis is the implementation price that one has to pay for the advantages of the lightning VEM method, but the embarrassingly parallelizable nature of this operation will ultimately result in a cost-efficient scheme almost comparable to standard VEM and FEM.

We provide numerical bounds on the Crouzeix ratio for KMS matrices A which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials p of the spectral norm of p(A) divided by the maximum absolute value of p on the numerical range of A. Our bounds satisfy the conjecture that this ratio is less than or equal to 2. We also give a precise description of these numerical ranges.

We consider the quaternion linear system (AX = B) for the unknown matrix X, where A, B are given (ntimes n), (ntimes s) matrices with quaternion entries, motivated by applications that arise from fields such as quantum mechanics and signal processing. Our primary concern is the large-scale setting when n is large so that direct solutions are not feasible. We describe a block Krylov subspace method for the iterative solution of these quaternion linear systems. One difference compared to usual block Krylov subspace methods over complex Euclidean spaces is that the multiplication of quaternion scalars is not commutative. We describe a block quaternion Arnoldi process, taking noncommutativity features of quaternions into account, to generate an orthonormal basis for the quaternion Krylov space (text {blockspan} { R_0, A R_0, dots , A^k R_0 }), where (R_0 = B - A X_0) and (X_0) is an initial guess for the solution. Then the best solution of (AX = B) in the least-squares sense is sought in the generated Krylov space. We explain how these least-squares problems over quaternion Krylov spaces can be solved efficiently by means of Householder reflectors. Most notably, we analyze rigorously the convergence of the proposed block quaternion GMRES approach when A is diagonalizable, and in the more general setting when A is not necessarily diagonalizable by making use of the Jordan form of A. Finally, we report numerical results that confirm the validity of the deduced theoretical convergence results, in particular illustrate that the proposed block quaternion Krylov subspace method converges quickly when A has clustered eigenvalues.

To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard–Lindelöf iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, Liouville–Bratu–Gelfand, and nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.

Accelerating slowly convergent sequences is one of the main purposes of extrapolation methods. In this paper, we present a new tensor polynomial extrapolation method, which is based on a modified minimisation problem and some ideas leading to the recent Tensor Global Minimal Extrapolation Method (TG-MPE). We discuss the application of our method to fixed-point iterative process. An efficient algorithm via the higher order Singular Value Decomposition (HOSVD) is proposed for its implementation. The numerical tests show clearly the effectiveness and performance of the proposed method.