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Algorithms
Special Issues
Algorithms for Applied Mathematics
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Algorithms for Applied Mathematics

A special issue of Algorithms (ISSN 1999-4893).

Deadline for manuscript submissions: closed (30 April 2010) | Viewed by 55078

Special Issue Editor

Prof. Dr. Doron Levy

E-Mail Website
Guest Editor
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Keywords

  • discrete models
  • stochastic equations
  • ordinary differential equations
  • time-delayed equations
  • partial differential equations
  • finite-volume, finite difference, finite elements methods
  • particle methods
  • spectral methods
  • linear and nonlinear optimization
  • linear algebra

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Published Papers (6 papers)

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Research

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18 pages, 327 KiB  
Article
Univariate Cubic L1 Interpolating Splines: Spline Functional, Window Size and Analysis-based Algorithm
by Lu Yu, Qingwei Jin, John E. Lavery and Shu-Cherng Fang
Algorithms 2010, 3(3), 311-328; https://doi.org/10.3390/a3030311 - 20 Aug 2010
Cited by 12 | Viewed by 8719
Abstract
We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that [...] Read more.
We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that second-derivative-based 5-point-window L1 splines preserve shape as well as or better than the other types of L1 splines. To calculate second-derivative-based 5-point-window L1 splines, we introduce an analysis-based, parallelizable algorithm. This algorithm is orders of magnitude faster than the previously widely used primal affine algorithm. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
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18 pages, 150 KiB  
Article
Univariate Cubic L1 Interpolating Splines: Analytical Results for Linearity, Convexity and Oscillation on 5-PointWindows
by Qingwei Jin, John E. Lavery and Shu-Cherng Fang
Algorithms 2010, 3(3), 276-293; https://doi.org/10.3390/a3030276 - 30 Jul 2010
Cited by 8 | Viewed by 7407
Abstract
We analytically investigate univariate C1 continuous cubic L1 interpolating splines calculated by minimizing an L1 spline functional based on the second derivative on 5-point windows. Specifically, we link geometric properties of the data points in the windows with linearity, convexity [...] Read more.
We analytically investigate univariate C1 continuous cubic L1 interpolating splines calculated by minimizing an L1 spline functional based on the second derivative on 5-point windows. Specifically, we link geometric properties of the data points in the windows with linearity, convexity and oscillation properties of the resulting L1 spline. These analytical results provide the basis for a computationally efficient algorithm for calculation of L1 splines on 5-point windows. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
11 pages, 247 KiB  
Article
Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries
by Shay Kels, Nira Dyn and Evgeny Lipovetsky
Algorithms 2010, 3(3), 265-275; https://doi.org/10.3390/a3030265 - 26 Jul 2010
Cited by 1 | Viewed by 7145
Abstract
The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise [...] Read more.
The metric average is a binary operation between sets in Rn which is used in the approximation of set-valued functions. We introduce an algorithm that applies tools of computational geometry to the computation of the metric average of 2D sets with piecewise linear boundaries. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
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20 pages, 396 KiB  
Article
Segment LLL Reduction of Lattice Bases Using Modular Arithmetic
by Sanjay Mehrotra and Zhifeng Li
Algorithms 2010, 3(3), 224-243; https://doi.org/10.3390/a3030224 - 12 Jul 2010
Cited by 2 | Viewed by 10297
Abstract
The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction [...] Read more.
The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction algorithm that generates lattice basis satisfying a weaker condition than the LLL reduced basis with O(n) improvement than the LLL algorithm. In this paper we combine Storjohann’s modular arithmetic approach with the segment-LLL approach to further improve the worst case complexity of the segment-LLL algorithms by a factor of n0.5. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
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8 pages, 207 KiB  
Article
Algorithmic Solution of Stochastic Differential Equations
by Henri Schurz
Algorithms 2010, 3(3), 216-223; https://doi.org/10.3390/a3030216 - 1 Jul 2010
Viewed by 7997
Abstract
This brief note presents an algorithm to solve ordinary stochastic differential equations (SDEs). The algorithm is based on the joint solution of a system of two partial differential equations and provides strong solutions for finite-dimensional systems of SDEs driven by standard Wiener processes [...] Read more.
This brief note presents an algorithm to solve ordinary stochastic differential equations (SDEs). The algorithm is based on the joint solution of a system of two partial differential equations and provides strong solutions for finite-dimensional systems of SDEs driven by standard Wiener processes and with adapted initial data. Several examples illustrate its use. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)

Review

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19 pages, 164 KiB  
Review
An Introduction to Clique Minimal Separator Decomposition
by Anne Berry, Romain Pogorelcnik and Geneviève Simonet
Algorithms 2010, 3(2), 197-215; https://doi.org/10.3390/a3020197 - 14 May 2010
Cited by 39 | Viewed by 11801
Abstract
This paper is a review which presents and explains the decomposition of graphs by clique minimal separators. The pace is leisurely, we give many examples and figures. Easy algorithms are provided to implement this decomposition. The historical and theoretical background is given, as [...] Read more.
This paper is a review which presents and explains the decomposition of graphs by clique minimal separators. The pace is leisurely, we give many examples and figures. Easy algorithms are provided to implement this decomposition. The historical and theoretical background is given, as well as sketches of proofs of the structural results involved. Full article
(This article belongs to the Special Issue Algorithms for Applied Mathematics)
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