Foundations of Computational Mathematics (FOUND COMPUT MATH)

Publications in this journal

• Article: Jean Bernard Lasserre: Moments, Positive Polynomials and Their Applications

  Mihai Putinar
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  ABSTRACT: Reviews of the classical moment problem and the loose ends of its multivariate analog are linked to recent developments in global polynomial optimization. KeywordsMoment problem–Positive polynomial–Semi-algebraic set–Non-linear optimization–Semi-definite programming
  Foundations of Computational Mathematics 05/2012; 11(4):489-497.
• Article: H. Attouch, G. Buttazzo, and G. Michaille. Variational Analysis in Sobolev and BV Spaces, in MPS–SIAM Series on Optimization

  D. Azé
  Foundations of Computational Mathematics 05/2012; 9(4):515-516.
• Article: Gene H. Golub and Gérard Meurant: Matrices, Moments and Quadrature with Applications

  Zdeněk Strakoš
  Foundations of Computational Mathematics 04/2012; 11(2):241-255.
• Source

  Available from: columbia.edu

  Article: Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

  Michael Gnewuch, Henryk Woźniakowski
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  ABSTRACT: We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J.Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S d given as the d-fold tensor product of a compact linear operator S 1 for d=1,2,…, with ‖S 1‖=1 and S 1 having at least two positive singular values. Let n(ε,S d ) be the minimal number of information evaluations needed to approximate S d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S d ) can be bounded by a multiple of a power of T(ε −1,d) for all (ε −1,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε −1,d) whose multiple bounds n(ε,S d ). We also study weak tractability, i.e., when lime-1+d®¥,(e-1,d) Î \varOmegalnn(e,Sd)/(e-1+d)=0\lim_{\varepsilon^{-1}+d\to\infty,(\varepsilon^{-1},d)\in \varOmega}\ln n(\varepsilon,S_{d})/(\varepsilon^{-1}+d)=0 . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S 1, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S 1 decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the functionT such that generalized tractability holds. These conditions are obtained in terms of the singular values of S 1 and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x 1+ln y . We also study tractability functions T of product form, T(x,y)=f 1(x)f 2(x). Assume that a i =lim inf  x→∞(ln ln f i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff $a_{i}>1\quad\mbox{and}\quad(a_{1}-1)(a_{2}-1)\ge1,$a_{i}>1\quad\mbox{and}\quad(a_{1}-1)(a_{2}-1)\ge1, and if (a 1−1)(a 2−1)=1 then we need to assume one more condition given in the paper. If (a 1−1)(a 2−1)>1 then the exponent of tractability is zero, and if (a 1−1)(a 2−1)=1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S 1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω unr with the results from our previous paper obtained for the restricted domain Ω res=[1,∞)×{1,2,…,d *}∪[1,ε 0−1)×ℕ with d *≥1 and ε 0∈(0,1). In general, the tractability results are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T(x,y)=xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability.
  Foundations of Computational Mathematics 04/2012; 9(4):431-460.
• Article: Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation

  Klaus Meer, Martin Ziegler
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  ABSTRACT: The word problem for discrete groups is well known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. As a main difference to discrete groups these groups may be generated by uncountably many generators with index running over certain sets of real numbers. We study the word problem for such groups within the Blum–Shub–Smale (BSS) model of real number computation. The main result establishes the word problem to be computationally equivalent to the Halting Problem for such machines. It thus gives the first non-trivial example of a problem complete, that is, computationally universal for this model.
  Foundations of Computational Mathematics 04/2012; 9(5):599-609.
• Article: Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting

  Nira Dyn, Ron Goldman
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  ABSTRACT: We investigate the Lane–Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive some of its relevant properties. We then provide sufficient conditions on the nonlinear binary averaging rules used in the Lane–Riesenfeld algorithm that ensure the convergence of the algorithm to a continuous function. We also show that, when the averaging rules are C 2 with uniformly bounded second derivatives, then the limit is a C 1 function. Acanonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane–Riesenfeld algorithm with these averages is investigated. KeywordsNonlinear subdivision schemes–Lane–Riesenfeld algorithm–Data refinement–Nonlinear symmetric averages– p-averages–Convergence and smoothness analysis
  Foundations of Computational Mathematics 04/2012; 11(1):79-94.
• Source

  Available from: eprints.maths.ox.ac.uk

  Article: Numerical Solution of Riemann–Hilbert Problems: Painlevé II

  Sheehan Olver
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  ABSTRACT: We describe a new, spectrally accurate method for solving matrix-valued Riemann–Hilbert problems numerically. The effectiveness of this approach is demonstrated by computing solutions to the homogeneous Painlevé II equation. This can be used to relate initial conditions with asymptotic behavior. KeywordsRiemann–Hilbert problems–Spectral methods–Collocation methods–Painlevé transcendents
  Foundations of Computational Mathematics 04/2012; 11(2):153-179.
• Source

  Available from: mathematik.uni-kl.de

  Article: Normal Multi-scale Transforms for Curves

  S. Harizanov, P. Oswald, T. Shingel
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  ABSTRACT: Extending upon Daubechies et al. (Constr. Approx. 20:399–463, 2004) and Runborg (Multiscale Methods in Science and Engineering, pp. 205–224, 2005), we provide the theoretical analysis of normal multi-scale transforms for curves with general linear predictor S, and a more flexible choice of normal directions. The main parameters influencing the asymptotic properties (convergence, decay estimates for detail coefficients, smoothness of normal re-parametrization) of this transform are the smoothness of the curve, the smoothness of S, and its order of exact polynomial reproduction. Our results give another indication why approximating S may not be the first choice in compression applications of normal multi-scale transforms. KeywordsNonlinear geometric multi-scale transforms–Approximating subdivision schemes–Lipschitz smoothness–Curve representation–Detail decay estimate
  Foundations of Computational Mathematics 04/2012; 11(6):617-656.
• Source

  Available from: math.univ-toulouse.fr

  Article: Jean-Marc Azaïs and Mario Wschebor: Level Sets and Extrema of Random Processes and Fields

  Didier Dacunha-Castelle
  Foundations of Computational Mathematics 04/2012; 10(4):481-484.
• Source

  Available from: pawelpilarczyk.com

  Article: Parallelization Method for a Continuous Property

  Paweł Pilarczyk
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  ABSTRACT: An automated general purpose method is introduced for computing a rigorous estimate of a bounded region in ℝ n whose points satisfy a given property. The method is based on calculations conducted in interval arithmetic and the constructed approximation is built of rectangular boxes of variable sizes. An efficient strategy is proposed, which makes use of parallel computations on multiple machines and refines the estimate gradually. It is proved that under certain assumptions the result of computations converges to the exact result as the precision of calculations increases. The time complexity of the algorithm is analyzed, and the effectiveness of this approach is illustrated by constructing a lower bound on the set of parameters for which an overcompensatory nonlinear Leslie population model exhibits more than one attractor, which is of interest from the biological point of view. This paper is accompanied by efficient and flexible software written in C++ whose source code is freely available at http://www.pawelpilarczyk.com/parallel/.
  Foundations of Computational Mathematics 04/2012; 10(1):93-114.
• Source

  Available from: homepages.cwi.nl

  Article: Conditioning of Random Conic Systems Under a General Family of Input Distributions

  Raphael Hauser, Tobias Müller
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  ABSTRACT: We consider the conic feasibility problem associated with the linear homogeneous system Ax≤0, x≠0. The complexity of iterative algorithms for solving this problem depends on a condition number C(A). When studying the typical behavior of algorithms under stochastic input, one is therefore naturally led to investigate the fatness of the tails of the distribution of C(A). Introducing the very general class of uniformly absolutely continuous probability models for the random matrixA, we show that the distribution tails of C(A) decrease at algebraic rates, both for the Goffin–Cheung–Cucker number C G and the Renegar number C R . The exponent that drives the decay arises naturally in the theory of uniform absolute continuity, which we also develop in this paper. In the case of C G , we also discuss lower bounds on the tail probabilities and show that there exist absolutely continuous input models for which the tail decay is subalgebraic.
  Foundations of Computational Mathematics 04/2012; 9(3):335-358.
• Source

  Available from: ArXiv

  Article: Finite Resolution Dynamics

  Stefano Luzzatto, Paweł Pilarczyk
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  ABSTRACT: We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Hénon attractor is mixing at all resolutions coarser than10−5. KeywordsDynamical system–Finite resolution–Open cover–Combinatorial dynamics–Rigorous numerics–Directed graph–Transitivity–Mixing–Algorithm
  Foundations of Computational Mathematics 04/2012; 11(2):211-239.
• Article: Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

  Saugata Basu, Thierry Zell
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  ABSTRACT: Toda (in SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines in Bull. Am. Math. Soc. (NS) 21(1): 1–46, 1989) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry. KeywordsPolynomial hierarchy-Betti numbers-Semi-algebraic sets-Toda’s theorem Mathematics Subject Classification (2000)14P10-14P25-68W30
  Foundations of Computational Mathematics 04/2012; 10(4):429-454.
• Source

  Available from: ArXiv

  Article: Scattering in Flatland: Efficient Representations via Wave Atoms

  Laurent Demanet, Lexing Ying
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  ABSTRACT: This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero. This phenomenon was perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context of local Fourier bases (Bradie etal. in Appl. Comput. Harmon. Anal. 1:94–99, 1993). Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the non-standard wave atom form, provides a compression of the acoustic single- and double-layer potentials with wave number k as O(k)-by-O(k) matrices with C ε δ k 1+δ non-negligible entries, with δ>0 arbitrarily small, and ε the desired accuracy. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that allow this result to be written as a theorem are a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling (wavelength of the wave packet) ∼ (essential diameter)2. Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections. KeywordsFast algorithm-Wave propagation-Boundary integral equation-Computational harmonic analysis Mathematics Subject Classification (2000)65R20-65T99
  Foundations of Computational Mathematics 04/2012; 10(5):569-613.
• Source

  Available from: arxiv.org

  Article: The Chow Rings of Generalized Grassmannians

  Haibao Duan, Xuezhi Zhao
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  ABSTRACT: Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol.56 (part1), pp.1–26, AMS, Providence, 1994) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, 2005) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, 2006), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in (Duan and Zhao, arXiv:math.AT/0801.2444 and arXiv:math.AT/0711.2541) to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes. KeywordsFlag manifolds-Schubert varieties-Cohomology Mathematics Subject Classification (2000)14M15-57T15
  Foundations of Computational Mathematics 04/2012; 10(3):245-274.