The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.

A generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

The present work constitutes a fraction of a more extensive study that is devoted to numerical methods in acoustics. More precisely, we address here the interpolation process, which is more and more frequently used in Computational Acoustics–whether it is for enabling multi-stage hybrid calculations, or for easing the proper handling of complex configurations via advanced techniques such as Chimera grids or Immersed Boundary Conditions. In that regard, we focus on high-order interpolation schemes, so as to analyze their intrinsic features and to assess their effective accuracy. Taking advantage of specific insights that had been previously achieved by the present authors regarding standard high-order interpolation schemes (of centered nature), we here focus on their so-called spectral-like optimized counterparts (of both centered and noncentered nature). The latter spectral-like optimized schemes are analyzed thoroughly thanks to dedicated theoretical developments, which allow highlighting better what their strengths and weaknesses are. Among others, the various ways such interpolation schemes can degrade acoustic signals they are applied to are carefully investigated from a theoretical point-of-view. Besides that, specific criteria that could help in optimizing interpolation schemes better are provided, along with generic rules about how to minimize the signal degradation induced by existing interpolation schemes, in practice.

This paper extends an algorithm of P1-conservative interpolation on triangular meshes to tetrahedral meshes and thus constructs an approach of solution reconstruction for three-dimensional problems. The conservation property is achieved by local mesh intersection and the mass of a tetrahedron of the current mesh is calculated by the integral on its intersection with the background mesh. For each current tetrahedron, the overlapped background tetrahedrons are detected efficiently. A mesh intersection algorithm is proposed to construct the intersection of a current tetrahedron with the overlapped background tetrahedron and mesh the intersection region by tetrahedrons. A localization algorithm is employed to search the host units in background mesh for each vertex of the current mesh. In order to enforce the maximum principle and avoid the loss of monotonicity, correction of nodal interpolated solution on tetrahedral meshes is given. The performance of the present solution reconstruction method is verified by numerical experiments on several analytic functions and the solution of the flow around a sphere.

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using l1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via l1-minimisation.

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’, Comm. Inst. Sci. Kharkov13 (1936), 9–95] bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of $n$ variables’, Math. USSR Izv.37 (1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’, Israel. J. Math.186 (2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’, J. Approx. Theory162 (2010), 72–93]). Still, given a subset $Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here ${\mathcal{P}}_{d}(\mathbb{R}^{n})$ is the space of real polynomials of degree at most $d$ on $\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality: $\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$ for all $P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$ with $C$ independent of $P$. (Although ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree $d$ in $\mathbb{R}^{n}$, there are many apparently unrelated reasons for $Z\subset [-1,1]^{n}$ to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces $V$ of continuous functions on $[-1,1]^{n}$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for $Z$ to be $V$-norming, partly known, partly new, restricting ourselves to the simplest nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of the set $Z$, showing that it is Lipschitz in the Hausdorff metric.

Given a closed set $C$ in a Banach space $(X,\Vert \cdot \Vert )$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_{C}(x)=\Vert x-z\Vert$, where $d_{C}$ is the distance of $x$ from $C$. We survey the problem of studying the size of the set of points in $X$ which have nearest points in $C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

In many-server systems it is crucial to staff the right number of servers so that targeted service levels are met. These staffing problems typically lead to constraint satisfaction problems that are difficult to solve. During the last decade, a powerful many-server asymptotic theory has been developed to solve such problems and optimal staffing rules are known to obey the square-root staffing principle. In this paper we develop many-server asymptotics in the so-called quality and efficiency driven regime, and present refinements to many-server asymptotics and square-root staffing for a Markovian queueing model with admission control and retrials.

The relative projection constant${\it\lambda}(Y,X)$ of normed spaces $Y\subset X$ is ${\it\lambda}(Y,X)=\inf \{\Vert P\Vert :P\in {\mathcal{P}}(X,Y)\}$, where ${\mathcal{P}}(X,Y)$ denotes the set of all continuous projections from $X$ onto $Y$. By the well-known result of Bohnenblust, for every $n$-dimensional normed space $X$ and a subspace $Y\subset X$ of codimension one, ${\it\lambda}(Y,X)\leq 2-2/n$. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality ${\it\lambda}(Y,X)=2-2/n$ and present several applications. We prove that every three-dimensional space has a subspace with the projection constant less than $\frac{4}{3}-0.0007$. This gives a nontrivial upper bound in the problem posed by Bosznay and Garay. In the general case, we give an upper bound for the number of ($n-1$)-dimensional subspaces with the maximal relative projection constant in terms of the facets of the unit ball of $X$. As a consequence, every $n$-dimensional normed space $X$ has an ($n-1$)-dimensional subspace $Y$ with ${\it\lambda}(Y,X)<2-2/n$. This contrasts with the separable case in which it is possible that every hyperplane has a maximal possible projection constant.

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

We find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min–max–min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^{s}$ with $s>0.$

We give an approximation for the value of the Bessel function $J_\nu (x)$ in the transition region with an explicit sharp error term.

This paper proposes a new family of symmetric $4$ -point ternary non-stationary subdivision schemes  that can generate the limit curves of $C^3$ continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter $0<\alpha <\pi /3 $ , to show the usefulness of the proposed scheme. Moreover, the Hölder regularity and the reproduction property are also being calculated.

We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$ , $2\le p<\infty $ .

We study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$ . Such functions arise in the application of implicit $s$ -stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$ , disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

We consider the classical problem of finding the best uniform approximation by polynomials of $1/(x-a)^2,$ where $a>1$ is given, on the interval $[-\! 1,1]$ . First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.