Anisotropic a posteriori error estimation for the mixed discontinuous Galerkin approximation of the Stokes problem

This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e., elements with very large aspect ratio. Our analysis covers two‐ and three‐dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh, which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.     

Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most \(n+d\). Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most \(n^{O(d)}\) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

     

Deterministic and probabilistic discrepancies

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the d-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension d under consideration.     

On the complexity of the multivariate Sturm–Liouville eigenvalue problem

We study the complexity of approximating the smallest eigenvalue of -Δ+q with Dirichlet boundary conditions on the d-dimensional unit cube. Here Δ is the Laplacian, and the function q is non-negative and has continuous first order partial derivatives. We consider deterministic and randomized classical algorithms, as well as quantum algorithms using quantum queries of two types: bit queries and power queries. We seek algorithms that solve the problem with accuracy . We exhibit lower and upper bounds for the problem complexity. The upper bounds follow from the cost of particular algorithms. The classical deterministic algorithm is optimal. Optimality is understood modulo constant factors that depend on d. The randomized algorithm uses an optimal number of function evaluations of q when d≤2. The classical algorithms have cost exponential in d since they need to solve an eigenvalue problem involving a matrix with size exponential in d. We show that the cost of quantum algorithms is not exponential in d, regardless of the type of queries they use. Power queries enjoy a clear advantage over bit queries and lead to an optimal complexity algorithm.     

A polynomial-time linear decision tree for the traveling salesman problem and other NP-complete problems

We show that the position of an input point in the Euclideand-dimensional space with respect to a given set of hyperplanes can be determined efficiently by linear decision trees. As an application, we prove that many concrete problems whose recognition versions are NP-complete, like the traveling salesman problem, many other shortest path problems, and integer programming, have polynomial-time upper bounds in the linear decision tree model of computation.     

Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ?^d ? Sp (d, ?), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d² + 2d, whereas if H ? Sp (d, R), then dim H ≤ d² + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions $f:\{ 0,1\} ^n \to \mathcal{R}$ can be tested with O(n/∈) queries, for any totally ordered range $\mathcal{R}$ . More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(∈µ(n))). We construct a family of instances of Ω(2 ⁿ ) pairs in n-dimensional hypercubes such that no more than roughly a $\frac{1} {{\sqrt n }}$ fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [16], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n ^3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not $n^{\tfrac{1} {2} - \delta }$ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/∈) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.     

On Some Special Classes of Sequential Dynamical Systems

Sequential Dynamical Systems (SDSs) are mathematical models for analyzing simulation systems. We investigate phase space properties of some special classes of SDSs obtained by restricting the local transition functions used at the nodes. We show that any SDS over the Boolean domain with symmetric Boolean local transition functions can be efficiently simulated by another SDS which uses only simple threshold and simple inverted threshold functions, where the same threshold value is used at each node and the underlying graph is d-regular for some integer d. We establish tight or nearly tight upper and lower bounds on the number of steps needed for SDSs over the Boolean domain with 1-, 2- or 3-threshold functions at each of the nodes to reach a fixed point. When the domain is a unitary semiring and each node computes a linear combination of its inputs, we present a polynomial time algorithm to determine whether such an SDS reaches a fixed point. We also show (through an explicit construction) that there are Boolean SDSs with the NOR function at each node such that their phase spaces contain directed cycles whose length is exponential in the number of nodes of the underlying graph of the SDS.AMS Subject Classification: 68Q10, 68Q17, 68Q80.     

Some properties of the exit measure for super Brownian motion

 We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ ^d . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4]. Received: 6 December 1999 / Revised version: 9 June 2000 / Published online: 11 December 2001     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On the Intrinsic Complexity of Elimination Theory

We consider the intrinsic complexity of selected algorithmic problems of classical elimination theory in algebraic geometry. The inputs and outputs of these problems are given by finite sets of polynomials which we represent alternatively in dense form or by straight line programs. We begin with an overview on the known upper bounds for the sequential and parallel time complexity of these problems and show then that in the most important cases these bounds are tight. Our lower bound results include both the relative and the absolute viewpoint of complexity theory. On one side we give reductions of fundamental questions of elimination theory to NP- and P^#-complete problems and on the other side we show that some of these questions may have exponential size outputs. In this way we confirm the intrinsically exponential character of algorithmic problems in elimination theory whatever the type of data structure may be.     

Code constructions and existence bounds for relative generalized Hamming weight

The relative generalized Hamming weight (RGHW) of a linear code C and a subcode C ¹ is an extension of generalized Hamming weight. The concept was firstly used to protect messages from an adversary in the wiretap channel of type II with illegitimate parties. It was also applied to the wiretap network II for secrecy control of network coding and to trellis-based decoding algorithms for complexity estimation. For RGHW, bounds and code constructions are two related issues. Upper bounds on RGHW show the possible optimality for the applications, and code constructions meeting upper bounds are for designing optimal schemes. In this article, we show indirect and direct code constructions for known upper bounds on RGHW. When upper bounds are not tight or constructions are hard to find, we provide two asymptotically equivalent existence bounds about good code pairs for designing suboptimal schemes. Particularly, most code pairs (C, C ¹) are good when the length n of C is sufficiently large, the dimension k of C is proportional to n and other parameters are fixed. Moreover, the first existence bound yields an implicit lower bound on RGHW, and the asymptotic form of this existence bound generalizes the usual asymptotic Gilbert–Varshamov bound.     

Piecewise linear paths among convex obstacles

Let ℬ be a set ofn arbitrary (possibly intersecting) convex obstacles in ℝ ^d . It is shown that any two points which can be connected by a path avoiding the obstacles can also be connected by a path consisting ofO(n ^{(d−1)[d/2+1]}) segments. The bound cannot be improved below Ω(n ^d ); thus, in ℝ³, the answer is betweenn ³ andn ⁴. For open disjoint convex obstacles, a Θ(n) bound is proved. By a well-known reduction, the general case result also upper bounds the complexity for a translational motion of an arbitrary convex robot among convex obstacles. Asymptotically tight bounds and efficient algorithms are given in the planar case. This research was supported by The Netherlands' Organization for Scientific Research (NWO) and partially by the ESPRIT III Basic Research Action 6546 (PROMotion). J. M. acknowledges support by a Humboldt Research Fellowship. Part of this research was done while he visited Utrecht University.     

On crossing numbers of hypercubes and cube connected cycles

We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs.The research of both authors was supported by Alexander von Humboldt Foundation, Bonn, Germany.     

Spectral properties of threshold functions

We examine the spectra of boolean functions obtained as the sign of a real polynomial of degreed. A tight lower bound on various norms of the lowerd levels of the function's Fourier transform is established. The result is applied to derive best possible lower bounds on the influences of variables on linear threshold functions. Some conjectures are posed concerning upper and lower bounds on influences of variables in higher order threshold functions.Supported by an Eshkol fellowship, administered by the National Council for Research and Development—Israel Ministry of Science and Development.     

Probabilistic analysis of some euclidean clustering problems

We are given n points distributed randomly in a compact region D of R^m. We consider various optimisation problems associated with partitioning this set of points into k subsets. For each problem we demonstrate lower bounds which are satisfied with high probability. For the case where D is a hypercube we use a partitioning technique to give deterministic upper bounds and to construct algorithms which with high probability can be made arbitrarily accurate in polynomial time for a given required accuracy.     

The carvingwidth of hypercubes

The notion of the carvingwidth of a graph was introduced by Seymour and Thomas [Call routing and the ratcatcher, Combinatorica 14 (1994) 217-241]. In this note, we show that the carvingwidth of a d-dimensional hypercube equals 2^d-1.     

Sets of unit vectors with small pairwise sums

We study the sizes of δ-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most δ. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. Füredi, J.C. Lagarias, F. Morgan, 1991). We show that the maximum size of a δ-additive set over all normed spaces of dimension d grows exponentially in d for fixed δ > 2/3, stays bounded for δ < 2/3, and grows linearly at the threshold δ = 2/3. Furthermore, the maximum size of a 2/3-additive set in d-dimensional normed space has the sharp upper bound of d, with the single exception of spaces isometric to three-dimensional l ¹ space, where there exists a 2/3-additive set of four unit vectors.