BIT Numerical Mathematics最新文献

A pressure-robust space discretization of the incompressible Navier–Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, (H^1)-conforming mixed finite element methods like Scott–Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples support the theoretical results.

In this paper sharp lower error bounds for numerical methods for jump-diffusion stochastic differential equations (SDEs) with discontinuous drift are proven. The approximation of jump-diffusion SDEs with non-adaptive as well as jump-adapted approximation schemes is studied and lower error bounds of order 3/4 for both classes of approximation schemes are provided. This yields optimality of the transformation-based jump-adapted quasi-Milstein scheme.

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the (L^2)-norm, the Galerkin projection onto this generalized finite element space even yields (varepsilon )-independent error bounds, (varepsilon ) being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate (varepsilon )-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method.

The objective of this work is to develop a conforming virtual element method for viscoelastic wave equations with variable coefficients on polygonal meshes. For problems where the coefficients are variable, the standard virtual element discrete forms do not work efficiently and require modification. For the optimal convergence estimate of the semi-discrete approximation in the (L^{2}) norm, a special projection operator is used. In the fully discrete scheme, the implicit second-order Newmark method is employed to approximate the temporal derivatives. Numerical experiments are presented to support the theoretical results. The proposed numerical algorithm can be applied to various problems arising in the engineering and medical fields.

This paper suggests an hp-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the hp-discontinuous Galerkin method. We establish prior error bounds in the (L^{2})-norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.

In this paper, we consider large-scale ranking problems where one is given a set of (possibly non-redundant) pairwise comparisons and the underlying ranking explained by those comparisons is desired. We show that stochastic gradient descent approaches can be leveraged to offer convergence to a solution that reveals the underlying ranking while requiring low-memory operations. We introduce several variations of this approach that offer a tradeoff in speed and convergence when the pairwise comparisons are noisy (i.e., some comparisons do not respect the underlying ranking). We prove theoretical results for convergence almost surely and study several regimes including those with full observations, partial observations, and noisy observations. Our empirical results give insights into the number of observations required as well as how much noise in those measurements can be tolerated.