This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen–Cahn equations with space-time white noise which have been obtained in a recent previous

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential \(\exp (A)v\) has to be approximated by \(\left( p_m(s^{-1}A)\right) ^sv=\exp (A+\varDelta A)v\). We propose here a rigorous bound for the relative backward error \(\left\Vert \varDelta A\right\Vert _{2}/\left\Vert

We consider a subdiffusion problem described by a time fractional Riemann–Liouville derivative of order \(0<\alpha <1\). The main purpose of this work is to show how we can apply fractional splines of order \(0<\beta \le 1\) to approximate a fractional integral and hence how to solve the subdiffusion problem using this approach. To discuss the convergence of the numerical method we present the error

The methods used for extracting an approximate eigenpair are crucial in sparse iterative eigensolvers. Using least squares and line search techniques this paper devises a method for an approximate eigenpair extraction. Numerical comparison of the Jacobi–Davidson method using the suggested method of eigenpair extraction, Rayleigh–Ritz, and refined Ritz projections shows that the suggested method is

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of 2S dynamical symplectic-orthogonal deterministic basis functions with time-dependent stochastic coefficients. The reduced (low rank) dynamics is obtained by a

The semi-implicit Euler–Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change)

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented

In this paper, we study the numerical approximation of a coupled system of elliptic–parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications

In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a

The eigenvector-dependent nonlinear eigenvalue problem arises in many important applications, such as the discretized Kohn–Sham equation in electronic structure calculations and the trace ratio problem in linear discriminant analysis. In this paper, we perform a perturbation analysis for the eigenvector-dependent nonlinear eigenvalue problem, which gives upper bounds for the distance between the solution

A formulation of the Taylor expansion with symmetric polynomial algebra allows to compute the coefficients of compact finite difference schemes, which solve the Poisson equation at an arbitrary order of accuracy on a uniform Cartesian grid in arbitrary dimensions. This construction produces original high order schemes which respect the Discrete Maximum Principle: a tenth order scheme in dimension three

The maximum angles \(\vartheta _q\) for which the three-, four-, five- and six-step backward difference formula methods are A(\(\vartheta _q\))-stable, slight improvements of the well-known angles, are determined.

Image resampling is a widely used tool in image processing. The upsampling increases the number of pixels and introduces new information to the image which can have undesired effects, like ringing artifacts and oscillations, aliasing “jagged” lines effect, or introduces too much numerical diffusion. Histopolation upsampling methods produce much sharper images but are more prone to aliasing and ringing

In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and

We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to a Taylor series expansion approach previously used in the context of Nitsche’s method and, in the

In this study, a general formulation for the fractional-order general Lagrange scaling functions (FGLSFs) is introduced. These functions are employed for solving a class of fractional differential equations and a particular class of fractional delay differential equations. For this approach, we derive FGLSFs fractional integration and delay operational matrices. These operational matrices and collocation

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro–macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field method to preserve multiple invariants simultaneously. Based on the a prior estimate for high-order moments

The present article revisits the well-known stochastic theta methods (STMs) for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. Under a coupled monotonicity condition in a domain \(D \subset {{\mathbb {R}}}^d, d \in {{\mathbb {N}}}\), we propose a novel approach to achieve upper mean-square error bounds for STMs with the method parameters \(\theta

In this paper a special class of one-dimensional L-splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L-splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions

The article Jumping with variably scaled discontinuous kernels (VSDKs) written by S. De Marchi, F. Marchetti and E. Perracchione was originally published electronically on the publisher’s Internet portal on [date of OnlineFirst publication] without open access. With the author(s)’ decision to opt for Open Choice, the copyright of the article changed on January 2020 to © The Author(s) 2020 and the article

BDF formulas are among the most efficient methods for numerical integration, in particular of stiff equations (see e.g. Gear in Numerical initial value problems in ordinary differential equations, Prentice Hall, Upper Saddle River, 1971). Their excellent stability properties are known for precisely half a century, from the first calculation of their angles of \(A(\alpha )\)-stability by Nørsett (BIT

Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane. In an annulus, the Hadamard three-circles theorem implies that the ill-conditioning is not too severe, and we show how this explains the effectiveness of Chebfun and related numerical methods

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker

Local Fourier analysis is a commonly used tool to assess the quality and aid in the construction of geometric multigrid methods for translationally invariant operators. In this paper we automate the process of local Fourier analysis and present a framework that can be applied to arbitrary, including non-orthogonal, repetitive structures. To this end we introduce the notion of crystal structures and

The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed

We derive bounds for the constants in Poincaré–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions.

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian

In the originally published version, the author found some subsequent corrections. It should read correct.

Cellular reactions have a multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary across orders of magnitude. This diversity naturally leads to hybrid models that combine continuous and discrete modeling regimes. In order to capture this multi-scale nature, we proposed jump-diffusion approximations in a previous study. The key idea was to

A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form \(T(p) + U(q)\) is presented. The new integration methods are defined in terms of an explicitly defined generating function (of the third kind). They are implicit in q (but explicit in p and the internal states), and require the evaluation of the gradients of T(p) and U(q)

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing

Using perfectly matched layers for the computation of resonances in open systems typically produces artificial or spurious resonances. We analyze the dependency of these artificial resonances with respect to the discretization parameters and the complex scaling function. In particular, we study the differences between a standard frequency independent complex scaling and a frequency dependent one. While

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular

An interval union is a finite set of closed and disjoint intervals. In this paper we introduce the interval union Gauss-Seidel procedure to rigorously enclose the solution set of linear systems with uncertainties given by intervals or interval unions. We also present the interval union midpoint and Gauss-Jordan preconditioners. The Gauss-Jordan preconditioner is used in a mixed strategy to improve

In the originally published version of this article, the acknowledgement was unfortunately not yet in the final version. It should read as follows:

The Riemann–Liouville–Caputo (RLC) derivative is a new class of derivative that is motivated by modelling considerations; it lies between the more familiar Riemann–Liouville and Caputo derivatives. The present paper studies a two-point boundary value problem on the interval [0, L] whose highest-order derivative is an RLC derivative of order \(\alpha \in (1,2)\). It is shown that the choice of boundary

Two general operations are proposed on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of “nonstandard” convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity

This paper introduces a univariate interpolation scheme using a binary parameter called signature such that the graph of the interpolant—which we refer to as affine zipper fractal interpolation function—is obtained as an attractor of a suitable affine zipper. The scaling vector function is identified so that the graph of the corresponding affine zipper fractal interpolation function can be inscribed

Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size \(d\cdot n\), where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n-dimensional space. Jacobi–Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver