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65 Numerical Analysis e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 65 NUMERICAL ANALYSIS MR2918625 65-06 FRecent advances in scientific computing and matrix analysis. Proceedings of the International Workshop held at the University of Macau, Macau, December 28{30, 2009. Edited by Xiao-Qing Jin, Hai-Wei Sun and Seak-Weng Vong. International Press, Somerville, MA; Higher Education Press, Beijing, 2011. xii+126 pp. ISBN 978-1-57146-202-2 Contents: Zheng-jian Bai and Xiao-qing Jin [Xiao Qing Jin1], A note on the Ulm-like method for inverse eigenvalue problems (1{7) MR2908437; Che-man Cheng [Che-Man Cheng], Kin-sio Fong [Kin-Sio Fong] and Io-kei Lok [Io-Kei Lok], Another proof for commutators with maximal Frobenius norm (9{14) MR2908438; Wai-ki Ching [Wai-Ki Ching] and Dong-mei Zhu [Dong Mei Zhu1], On high-dimensional Markov chain models for categorical data sequences with applications (15{34) MR2908439; Yan-nei Law [Yan Nei Law], Hwee-kuan Lee [Hwee Kuan Lee], Chao-qiang Liu [Chaoqiang Liu] and Andy M. Yip, An additive variational model for image segmentation (35{48) MR2908440; Hai-yong Liao [Haiyong Liao] and Michael K. Ng, Total variation image restoration with automatic selection of regularization parameters (49{59) MR2908441; Franklin T. Luk and San-zheng Qiao [San Zheng Qiao], Matrices and the LLL algorithm (61{69) MR2908442; Mila Nikolova, Michael K. Ng and Chi-pan Tam [Chi-Pan Tam], A fast nonconvex nonsmooth minimization method for image restoration and reconstruction (71{83) MR2908443; Gang Wu [Gang Wu1], Eigenvalues of certain augmented complex stochastic matrices with applications to PageRank (85{92) MR2908444; Yan Xuan and Fu-rong Lin, Clenshaw-Curtis-rational quadrature rule for Wiener-Hopf equations of the second kind (93{110) MR2908445; Man-chung Yeung [Man-Chung Yeung], On the solution of singular systems by Krylov subspace methods (111{116) MR2908446; Qi- fang Yu, San-zheng Qiao [San Zheng Qiao] and Yi-min Wei, A comparative study of the LLL algorithm (117{126) MR2908447. fThe papers are being reviewed individually.g 65C Probabilistic methods, simulation and stochastic differential equations MR2871782 65C05 11B39 65D30 Bilyk, Dmitriy (1-SC; Columbia, SC); Temlyakov, V. N. [Temlyakov, Vladimir N.] (1-SC; Columbia, SC); Yu, Rui [Yu, Rui1] (1-SC; Columbia, SC) Fibonacci sets and symmetrization in discrepancy theory. (English summary) J. Complexity 28 (2012), no. 1, 18{36. Summary: \We study the Fibonacci sets from the point of view of their quality 1 with respect to discrepancy and numerical integration. Let fbngn=0 be the sequence 2 of Fibonacci numbers. The bn-point Fibonacci set Fn ⊂ [0; 1] is defined as Fn := bn f(µ/bn; fµbn−1=bng)gµ=1, where fxg is the fractional part of a number x 2 R. It is known that cubature formulas based on the Fibonacci set Fn give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. \We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set 0 Fn = Fn [ f(p1; 1 − p2): (pq; p2) 2 Fng has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets 1 e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 have the smallest currently known L2 discrepancy among two-dimensional point sets. \We also introduce quartered Lp discrepancy, which is a modification of the Lp discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set Fn has minimal in the sense of order quartered Lp discrepancy for all p 2 (1; 1). This in turn implies that certain two-fold symmetrizations of the Fibonacci set Fn are optimal with respect to the standard Lp discrepancy." MR2871781 65C05 46E22 65D30 Gnewuch, Michael (D-KIEL-II; Kiel) Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces. (English summary) J. Complexity 28 (2012), no. 1, 2{17. Summary: \We extend the notion of L2-B-discrepancy introduced in [E. Novak and H. Wo´zniakowski, in Analytic number theory, 359{388, Cambridge Univ. Press, Cam- bridge, 2009; MR2508657 (2010d:11088)] to what we shall call weighted geometric L2-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures dif- ferent from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces. \We relate the weighted geometric L2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wo´zniakowski. \Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms." MR2822407 65C05 60G70 62G20 62G32 90B25 Guyader, Arnaud (F-RENN2-NDM; Rennes); Hengartner, Nicholas [Hengartner, Nicolas W.] (1-LANL-IF; Los Alamos, NM); Matzner-Løber, Eric [Matzner-Løber, Eric]´ (F-RENN2-NDM; Rennes) Simulation and estimation of extreme quantiles and extreme probabilities. (English summary) Appl. Math. Optim. 64 (2011), no. 2, 171{196. The paper presents an efficient algorithm for estimating (i) a tail probability given a quantile, or (ii) a quantile given a tail probability. The algorithm improves the multilevel splitting methods known in the literature. By means of Poisson process tools the exact distribution of the estimated probabilities and quantiles are established. Bernd Heidergott MR2788863 65C05 Lee, Jeong Eun (5-QUT-SM; Brisbane); McVinish, Ross [McVinish, R¨oss] (5-QLD; Brisbane); Mengersen, Kerrie [Mengersen, Kerrie L.] (5-QUT-SM; Brisbane) Population Monte Carlo algorithm in high dimensions. (English summary) Methodol. Comput. Appl. Probab. 13 (2011), no. 2, 369{389. Summary: \The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental 2 e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix for the importance function can be estimated in special cases." MR2839113 65C05 93D09 Nurges, Ulo¨ (ES-TALL; Tallinn) Discussion on: \On the generation of random stable polynomials" [MR2839112]. Eur. J. Control 17 (2011), no. 2, 160{161. This paper presents a discussion of [P. S. Shcherbakov and F. Dabbene, Eur. J. Control 17 (2011), no. 2, 145{159; MR2839112] on the generation of stable random polynomials. Two issues regarding the paper are raised. The first concerns the terminology assigned to a set of parameters appearing in a particular method (based on the Schur-Cohn stability test, and called Levinson-Durbin (LD) parameters in [op. cit.]; the authors of the original paper and of this discussion both acknowledge that these parameters go by many names in many different fields). The second relates to the fact that the uniform distribution for LD parameters for high-order polynomials leads to the generation of polynomials whose roots cluster close to the stability boundary. A modified algorithm avoiding this behavior is presented, together with simulation results. fFor further information pertaining to this item see [P. Shcherbakov and F. Dabbene, Eur. J. Control 17 (2011), no. 2, 161; MR2839120].g David C. Saunders MR2839120 65C05 93D05 Shcherbakov, P.; Dabbene, F. [Dabbene, Fabrizio] Final comments by the authors: \On the generation of random stable polynomials" [MR2839112; MR2839113]. Eur. J. Control 17 (2011), no. 2, 161. The authors respond to the two points raised in [U.¨ Nurges, Eur. J. Control 17 (2011), no. 2, 160{161; MR2839113]. They justify their choice of terminology on historical grounds. They raise some issues regarding the algorithm proposed in [op. cit.], and further note that clustering of roots near the stability boundary may not necessarily be a bad thing from the point of view of practical applications, pointing to the motivating examples in [P. S. Shcherbakov and F. Dabbene, Eur. J. Control 17 (2011), no. 2, 145{ 159; MR2839112] for justification. David C. Saunders MR2839112 65C05 93D09 Shcherbakov, Pavel [Shcherbakov, Pavel S.] (RS-AOS-CN; Moscow); Dabbene, Fabrizio (I-TRNP-IEI; Turin) On the generation of random stable polynomials. (English summary) Eur. J. Control 17 (2011), no. 2, 145{159. This paper presents a survey of various techniques for the random generation of stable polynomials. Both the case where the roots must lie in the open unit disk (Schur stable), and the case where they must lie in the left half-plane C− = fs 2 C : Re s < 0g (Hurwitz stable), are considered. The paper begins with an introduction presenting motivating applications. Next, how general random sampling schemes may be adapted to the problem of generating stable random polynomials is discussed. Random generation schemes based on classical tests for stability of polynomials are then presented. Finally, the paper concludes with a list of interesting open problems in the area. Simulation results are presented for many of the schemes discussed in the paper. 3 e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 fFor further information pertaining to this item see [U.¨ Nurges, Eur.
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