A generalization of general two-point formula with applications in numerical integration

We derive a general two-point integral quadrature formula using the concept of harmonic polynomials. An improved version of Guessab and Schmeisser’s result is given with new integral inequalities involving functions whose derivatives belong to various classes of functions (_Lp$L_p$ spaces, convex, concave, bounded functions). Furthermore, several special cases of polynomials are considered, and the generalization of well-known two-point quadrature formulae, such as trapezoid, perturbed trapezoid, two-point Newton–Cotes formula, two-point Maclaurin formula, midpoint, are obtained.     

The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H¹$H^1$ error estimate of the finite volume solution _uh$u_h$ and L²$L^2$ error estimate for _ph$p_h$ are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM.     

Galerkin/Runge–Kutta discretizations of nonlinear parabolic equations

Global error bounds are derived for full Galerkin/Runge–Kutta discretizations of nonlinear parabolic problems, including the evolution governed by the p  -Laplacian with p?2$p?2$. The analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants and an extended B  -convergence theory. The global error is bounded in _L2$L_2$ by Δx^r/2+Δt^q$\mathrm{\Delta }{x}^{r/2}+\mathrm{\Delta }{t}^q$, where r is the convergence order of the Galerkin method applied to the underlying stationary problem and q is the stiff order of the algebraically stable Runge–Kutta method.     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

Higher order weakly over-penalized symmetric interior penalty methods

In this paper we study higher order weakly over-penalized symmetric interior penalty methods for second-order elliptic boundary value problems in two dimensions. We derive h$h$–p$p$ error estimates in both the energy norm and the _L2$L_2$ norm and present numerical results that corroborate the theoretical results.     

A remark on least-squares Galerkin procedures for pseudohyperbolic equations

In this paper, we introduce two split least-squares Galerkin finite element procedures for pseudohyperbolic equations arising in the modelling of nerve conduction process. By selecting the least-squares functional properly, the procedures can be split into two sub-procedures, one of which is for the primitive unknown variable and the other is for the flux. The convergence analysis shows that both the two methods yield the approximate solutions with optimal accuracy in L²(Ω)$L^2\left(\Omega \right)$ norm for u$u$ and _ut$u_t$ and (L²(Ω))²${\left(L^2\left(\Omega \right)\right)}^2$ norm for the flux σ$\sigma$. Moreover, the two methods get approximate solutions with first-order and second-order accuracy in time increment, respectively. A numerical example is given to show the efficiency of the introduced schemes.     

Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations

One of the key problems in chance constrained programming for nonlinear optimization problems is the evaluation of derivatives of joint probability functions of the form P(x)=P(_gp(x,Λ)?_cp,p=1,…,_Nc)$P(x)=\mathbb{P}(g_p(x\text{,}\Lambda )?c_p\text{,}p=1\text{,}\dots \text{,}{N}_c)$. Here x∈R^_Nx$x\in {\mathbb{R}}^{N_x}$ is the vector of physical parameters, Λ∈R^_NΛ$\Lambda \in {\mathbb{R}}^{N_{\Lambda }}$ is a random vector describing the uncertainty of the model, g:R^_Nx×R^_NΛ→R^_Nc$g:{\mathbb{R}}^{N_x}\times {\mathbb{R}}^{N_{\Lambda }}\to {\mathbb{R}}^{N_c}$ is the constraints mapping, and c∈R^_Nc$c\in {\mathbb{R}}^{N_c}$ is the vector of constraint levels. In this paper specific Monte Carlo tools for the estimations of the gradient and Hessian of P(x)$P(x)$ are proposed when the input random vector Λ$\Lambda$ has a multivariate normal distribution and small variances. Using the small variance hypothesis, approximate expressions for the first- and second-order derivatives are obtained, whose Monte Carlo estimations have low computational costs. The number of calls of the constraints mapping g   for the proposed estimators of the gradient and Hessian of P(x)$P(x)$ is only 1+2_Nx+2_NΛ$1+2N_x+2N_{\Lambda }$.     

A connection between extreme value theory and long time approximation of SDEs

We consider a sequence _(ξn)n≥1${\left({\xi }_n\right)}_{n\ge 1}$ of i.i.d.   random values residing in the domain of attraction of an extreme value distribution. For such a sequence, there exist (_an)$\left(a_n\right)$ and (_bn)$\left(b_n\right)$, with _an>0$a_n>0$ and _bn∈R$b_n\in \mathbb{R}$ for every n≥1$n\ge 1$, such that the sequence (_Xn)$\left(X_n\right)$ defined by _Xn=(max(_ξ1,…,_ξn)−_bn)/_an$X_n=(max({\xi }_1,\dots ,{\xi }_n)-b_n)/a_n$ converges in distribution to a non-degenerated distribution.     

Simplified analytical expressions for numerical differentiation via cycle index

Based on the Lagrange interpolation to the function f[_x0,⋅]$f[x_0,\cdot ]$ for arbitrarily chosen _x0$x_0$ and logarithmic differentiation, we give a simple approach to analytical expressions for numerical differentiation using cycle index. A detailed analysis for the remainder is also included.     

The natural Banach space for version independent risk measures

Risk measures, or coherent measures of risk, are often considered on the space L^∞$L^{\infty }$, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure–the Average Value-at-Risk–are well defined on the larger space L¹$L^1$ and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some L^p$L^p$ space. But in many situations this is possibly unnatural, because any L^p$L^p$ with p>_p0$p>p_0$, say, is suitable to define the spectral risk measure as well. In addition to that, risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is?     

Frequency-dependent interpolation rules using first derivatives for oscillatory functions

We construct frequency-dependent rules to interpolate oscillatory functions y(x)$y(x)$ with frequency ω$\omega$ of the form,

y(x)=_f1(x)cos(ωx)+_f2(x)sin(ωx),$y(x)=f_1(x)\cos (\omega x)+f_2(x)\sin (\omega x)\text{,}$