Numerische Mathematik最新文献

In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel (mu ) or the flux f. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of (sqrt{Delta t}) in (L^1(mathbb {R})). To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.

Given a positive integer d and ({{varvec{a}}}_{1},dots ,{{varvec{a}}}_{r}) a family of vectors in ({{mathbb {R}}}^d), ({k_1{{varvec{a}}}_{1}+dots +k_r{{varvec{a}}}_{r}: k_1,dots ,k_r in {{mathbb {Z}}}}subset {{mathbb {R}}}^d) is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube ([0,1)^d) or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named FLE-LP).

This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the (L^infty ) norm from the theory of viscosity solutions which are independent of the regularization parameter (varepsilon ). They allow for the uniform convergence of the solution (u_varepsilon ) to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative (L^n) right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the (L^infty ) norm for continuously differentiable finite element approximations of u or (u_varepsilon ).

We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.

We investigate the use of two-layer networks with the rectified power unit, which is called the (text {ReLU}^k) activation function, for function and derivative approximation. By extending and calibrating the corresponding Barron space, we show that two-layer networks with the (text {ReLU}^k) activation function are well-designed to simultaneously approximate an unknown function and its derivatives. When the measurement is noisy, we propose a Tikhonov type regularization method, and provide error bounds when the regularization parameter is chosen appropriately. Several numerical examples support the efficiency of the proposed approach.