We introduce a new methodology to design uniformly accurate methods for oscil-latory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogs: the completely positive rank and the completely positive semidefinite rank. We study convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $\Omega \subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $\Omega$. In $\Omega\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(\Omega)$. We also prove that tensorization of a $P_1$-FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_\Omega$, the number of degrees of freedom in the domain $\Omega$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $\Omega$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_\Omega$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega}$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

Let K be an effective field of characteristic zero. An effective tribe is a subset of $$K [[z_1, z_2, \ldots ]] = K \cup K {[}[z_1]] \cup K [[z_1, z_2]] \cup \cdots $$ that is effectively stable under the K-algebra operations, restricted division, composition, the implicit function theorem, as well as restricted monomial transformations with arbitrary rational exponents. Given an effective tribe with an effective zero test, we will prove that an effective version of the Weierstrass division theorem holds inside the tribe and that this can be used for the computation of standard bases.

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green–Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte Carlo sampling and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.

This paper examines the problem of extrapolation of an analytic function for $$x > 1$$ given $$N+1$$ perturbed samples from an equally spaced grid on $$[-1,1]$$ . For a function f on $$[-1,1]$$ that is analytic in a Bernstein ellipse with parameter $$\rho > 1$$ , and for a uniform perturbation level $$\varepsilon $$ on the function samples, we construct an asymptotically best extrapolant e(x) as a least squares polynomial approximant of degree $$M^*$$ determined explicitly. We show that the extrapolant e(x) converges to f(x) pointwise in the interval $$I_\rho \in [1,(\rho +\rho ^{-1})/2)$$ as $$\varepsilon \rightarrow 0$$ , at a rate given by a x-dependent fractional power of $$\varepsilon $$ . More precisely, for each $$x \in I_{\rho }$$ we have $$\begin{aligned} |f(x) - e(x)| = \mathcal {O}\left( \varepsilon ^{-\log r(x) / \log \rho } \right) , \quad r(x) = \frac{x+\sqrt{x^2-1}}{\rho }, \end{aligned}$$ up to log factors, provided that an oversampling conditioning is satisfied, viz. $$\begin{aligned} M^* \le \frac{1}{2} \sqrt{N}, \end{aligned}$$ which is known to be needed from approximation theory. In short, extrapolation enjoys a weak form of stability, up to a fraction of the characteristic smoothness length. The number of function samples does not bear on the size of the extrapolation error provided that it obeys the oversampling condition. We also show that one cannot construct an asymptotically more accurate extrapolant from equally spaced samples than e(x), using any other linear or nonlinear procedure. The proofs involve original statements on the stability of polynomial approximation in the Chebyshev basis from equally spaced samples and these are expected to be of independent interest.

We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing over-determined systems. Along the way, we derive new Lipshitz estimates for polynomial maps from R^n to R^m, extending earlier work of Kellog on the case m=1, which may be of independent interest. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which monomial terms can appear) can help further improve earlier condition number estimates.

The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common generalization of the dimension polynomial and the (algebraic) counting polynomial. Under mild additional assumptions, the differential counting polynomial decides whether a given set of solutions of a system of differential equations is the complete set of solutions.

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order epsilon -approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order q ge 1 can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most O(epsilon ^{-(q+1)}) evaluations of f and its derivatives to compute an epsilon -approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp.