We propose a differential geometric approach for building families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of “anchor” matrices for interpolation, for instance over a grid of relevant conditions describing the underlying

This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme is A-stable, satisfies a B-convergence property, and that the correction on a scheme DC2j of order 2j of accuracy leads to a scheme DC2j + 2 of order 2j + 2. The

Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive overlapping Schwarz domain decomposition methods. The numerical error in a user-specified functional of the solution (quantity of interest) is decomposed into contributions

This paper is concerned with the parameterized least squares inverse eigenvalue problems for the case that the number of parameters to be constructed is greater than the number of prescribed realizable eigenvalues. Intrinsically, this is a specific problem of finding a zero of an under-determined nonlinear map defined between a Riemannian product manifold and a matrix space. To solve this problem,

In this paper a numerical scheme for partitioned systems of index 2 DAEs, such as those arising from nonholonomic mechanical problems, is proposed and its order for a certain class of Runge–Kutta methods we call of Lobatto-type is proven.

We consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman–Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (1) we approximate small jumps by a diffusion; (2) we use

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis for the method is presented for a not explicitly known

A time-fractional partial differential equation with a hidden-memory space-time dependent variable order is studied. The well-posedness and high-order regularity estimates of the problem are proved via the resolvent estimates. Accordingly, an error estimate of the L-1 discretization without any artificial regularity assumption of the true solution is analyzed. Numerical experiments are performed to

Various applications in numerical linear algebra and computer science are related to selecting the \(r\times r\) submatrix of maximum volume contained in a given matrix \(A\in \mathbb R^{n\times n}\). We propose a new greedy algorithm of cost \(\mathcal O(n)\), for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio

We extend some Nielson type interpolation operators to the cases of standard and arbitrary triangles with one curved side. The correspondence between the operators defined on standard triangles and arbitrary triangles is made using barycentric coordinates. We study the interpolation properties of the obtained operators and the interpolation errors. For illustration, we give some numerical examples

In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time

The main aim of this article is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods. In Gaddam and Gudi (Comput Methods Appl Math 18:223–236, 2018. https://doi.org/10.1515/cmam-2017-0018), a bubble enriched conforming quadratic finite element method is introduced and analyzed for the obstacle problem in dimension 3. In this article,

In this paper we analyze a greedy procedure to approximate a linear functional defined in a reproducing kernel Hilbert space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals. For a large class of functionals, that includes integration functionals and other interesting cases, but does not include differentiation, we prove convergence results for

A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential

We study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states

Randomized methods can be competitive for the solution of problems with a large matrix of low rank. They also have been applied successfully to the solution of large-scale linear discrete ill-posed problems by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails the computation of an approximation of a partial singular value decomposition of a large matrix A that is

In this paper, we consider diagonal-times-nonsymmetric-Toeplitz (DNT) preconditioner for Toeplitz-like linear systems arising from one-sided space fractional diffusion equations (OSFDE) with variable coefficients. We first correct the result of Lemma 3.3 in Lin et al. (BIT Numer Math 58, 729–748, 2018) and prove the corrected result. We then extend the DNT preconditioner to Toeplitz-like linear systems

In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter \(0<\varepsilon \le 1\). The solution of this system propagates wave with \(O(\varepsilon ^2)\) wavelength and \(O(\varepsilon ^4)/O(\varepsilon ^2)\) amplitude for well-prepared/ill-prepared initial data. Based on the NPI and the exponential

A complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in Claeys and Hiptmair (Integr Equ Oper Theory 77:167–197, 2013. https://doi.org/10.1007/s00020-013-2085-x) for the Helmholtz equation, and in Claeys and Hiptmair

We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic

We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied

The modified Filon–Clenshaw–Curtis rules, proposed earlier by the author, are combined with the (double) exponential transformations in such a way that (1) the necessity of computing the inverse of the oscillator function is released, (2) possible inaccuracy due to rounding error, when the amplitude function has endpoint singularities, is treated, (3) difficulties raised by the stationary points of

We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block

We present randomized algorithms based on block Krylov subspace methods for estimating the trace and log-determinant of Hermitian positive semi-definite matrices. Using the properties of Chebyshev polynomials and Gaussian random matrix, we provide the error analysis of the proposed estimators and obtain the expectation and concentration error bounds. These bounds improve the corresponding ones given

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties

The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: (i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; (ii)

We study a priori error estimates of discontinuous Galerkin (DG) methods for solving a quasi-variational inequality, which models a frictional contact problem with normal compliance. In Xiao et al. (Numer Funct Anal Optim 39:1248–1264, 2018), several DG methods are applied to solve quasi-variational inequality, but no error analysis is given. In this paper, the unified numerical analysis of these DG

A class of block boundary value methods (BBVMs) is constructed for linear weakly singular Volterra integro-differential equations (VIDEs). The convergence and stability of these methods is analysed. It is shown that optimal convergence rates can be obtained by using special graded meshes. Numerical examples are given to illustrate the sharpness of our theoretical results and the computational effectiveness

The space fractional Cahn–Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn–Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn–Hilliard model. The scheme is based on the second-order backward differentiation formula in time and a finite difference

This paper is devoted to the numerical approximation of the spatially extended FitzHugh–Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude

We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective

The simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the

In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction

In this work, Abel’s integral operator is represented based on Alpert’s multiwavelets as a sparse matrix and then a non-linear Abel’s integral equation of the second kind is solved by multiwavelets Galerkin method. Nonlinearity and singularity make the numerical procedure more challenging. But the proposed scheme overcomes these problems. Convergence analysis is investigated and some numerical examples

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of treating eigenvalue \(\lambda \) and eigenvector u separately, we regard the eigenpair \((\lambda , u)\) as one element in the composite space \({\mathbb {R}} \times H_0^1(\varOmega )\) and then Newton iteration step is adopted for the nonlinear

Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential integrator for solving evolution equations, where matrix functions must be computed repeatedly, have been investigated and refined. In this paper, we propose a new method for computing these matrix functions which is called an inexact

In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the rth derivative is positive within precision \(\varepsilon >0\) are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge–Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger

We show that integrating a polynomial f of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous related Bombieri polynomials of degree \(j=1,2,\ldots ,t\), each at a unique point \(\varvec{\xi }_j\) of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such

In this paper, a piecewise quadratic finite element method on rectangular grids for \(H^1\) problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be \(O(h^2)\) in the energy norm on uniform grids over a convex domain. A lower bound of the \(L^2\)-norm error is also proved, which makes

For an \(m \times n\) matrix A, the mathematical property that the rank of A is equal to r for \(0< r < \min (m,n)\) is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer

The randomized Kaczmarz (RK) method is an iterative method for approximating the least-squares solution of large linear systems of equations. The standard RK method uses sequential updates, making parallel computation difficult. Here, we study a parallel version of RK where a weighted average of independent updates is used. We analyze the convergence of RK with averaging and demonstrate its performance

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain

Classical a posteriori error analysis for differential equations quantifies the error in a Quantity of Interest which is represented as a bounded linear functional of the solution. In this work we consider a posteriori error estimates of a quantity of interest that cannot be represented in this fashion, namely the time at which a threshold is crossed for the first time. We derive two representations

Evaluating the action of a matrix function on a vector, that is \(x=f({\mathcal {M}})v\), is an ubiquitous task in applications. When \({\mathcal {M}}\) is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent

In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in space-time with transmission conditions of Dirichlet and

This paper introduces an adaptive preconditioner for iterative solution of sparse linear systems arising from partial differential equations with self-adjoint operators. This preconditioner allows to control the growth rate of a dominant part of the algebraic error within a fixed point iteration scheme. Several numerical results that illustrate the efficiency of this adaptive preconditioner with a

This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. When the interface geometries are complex, the dominant term in the computational cost of creating the direct solver scales O(NI) where N is the number of discretization points on each interface and I is the number of interfaces. The bulk of the

This paper introduces the Runge–Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves

The numerical method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation

We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime \(\vartheta = 1\) as well as in the classical regime \(\vartheta \rightarrow 0\) without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit \(\vartheta

The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to \(L^2\) anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup

In this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal \(L^{\infty }(H^1)\)-error estimates is still open, both in the

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary

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