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A logarithmic complexity floating frame of reference formulation with interpolating splines for articulated multi-flexible-body dynamics

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2013年 57卷 146-153页

作者： Khan, I. M. Ahn, W. Anderson, K. S. De, S. Rensselaer Polytech Inst Dept Mech Aerosp & Nucl Engn Computat Dynam Lab Troy NY 12180 USA Rensselaer Polytech Inst Dept Mech Aerosp & Nucl Engn Ctr Modeling Simulat & Imaging Med Troy NY 12180 USA

An interpolating spline-based approach is presented for modeling multi-flexible-body systems in the divide-and-conquer (DCA) scheme. This algorithm uses the floating frame of reference formulation and piecewise spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body system undergoing large rotations and translations. The new approach is compared with the flexible DCA (FDCA) that uses the assumed modes method [1]. The FDCA, in many cases, must resort to sub-structuring to accurately model the deformation of the system. We demonstrate, through numerical examples, that the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The present approach is appropriate for modeling flexible mechanisms with thin 1D bodies undergoing large rotations and translations, including those with irregular shapes. As such, the present approach extends the current capability of the DCA to model deformable systems. The algorithm retains the theoretical logarithmic complexity inherent in the DCA when implemented in parallel. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Multi-flexible-body systems logarithmic complexity Divide-and-conquer algorithm Interpolating splines

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Efficient Computation of Collatz Sequence Stopping Times: A Novel Algorithmic Approach

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IEEE ACCESS 2025年 13卷 41210-41220页

作者： Getachew, Eyob Solomon Assefa, Beakal Gizachew Addis Ababa Inst Technol Sch Informat Technol & Engn Addis Ababa 1000 Ethiopia

The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute the stopping time of numbers in the Collatz sequence, achieving significant computational improvements. By leveraging structural patterns in the Collatz tree, the proposed algorithm minimizes redundant operations and optimizes computational steps. Unlike prior methods, it efficiently handles extremely large numbers without requiring advanced techniques such as memoization or parallelization. Experimental evaluations confirm computational efficiency improvements of approximately 28% over state-of-the-art methods. These findings underscore the algorithm's scalability and robustness, providing a foundation for future large-scale verification of the conjecture and potential applications in computational mathematics.

关键词： Computational efficiency Encryption Approximation algorithms Visualization Probabilistic logic Optimization complexity theory Watermarking Time complexity Testing Algorithm optimization bitwise operations Collatz conjecture Collatz tree computational mathematics high-performance algorithms logarithmic complexity memoization parallel computing stopping time computational complexity sequence analysis and large-scale verification

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Performance investigation and constraint stabilization approach for the orthogonal complement-based divide-and-conquer algorithm

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MECHANISM AND MACHINE THEORY 2013年 67卷 111-121页

作者： Khan, I. M. Anderson, K. S. Rensselaer Polytech Inst Computat Dynam Lab Dept Mech Aerosp & Nucl Engn Troy NY 12180 USA

The introductory paper on the orthogonal complement-based divide-and-conquer algorithm (ODCA) lacked in properly characterizing the growth rate of the constraint violation error and the singularity handling capabilities of the algorithm [1]. In this paper, we investigate the performance of the ODCA with regards to constraint error growth and singularity handling capabilities. Moreover we introduce a new inverse dynamics-based constraint stabilization approach. The proposed method is applicable to general multibody systems with arbitrary number of closed kinematic loops. We compare the ODCA with augmented [2] and reduction [3] methods. Our results indicate that the error growth rate for the ODCA falls between these two traditional techniques. Moreover, using benchmark numerical problems, we illustrate the effectiveness of the stabilization scheme. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Constrained multibody systems Orthogonal complement Divide-and-conquer algorithm logarithmic complexity Constraint stabilization

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Efficient methodology for multibody simulations with discontinuous changes in system definition

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MULTIBODY SYSTEM DYNAMICS 2007年第2期18卷 145-168页

作者： Mukherjee, Rudranarayan M. Anderson, Kurt S. Rensselaer Polytech Inst Dept Mech Aeronaut & Nucl Engn Troy NY 12180 USA

A new method is presented for accurately and efficiently simulating multi-scale multibody systems with discontinuous changes in system definitions as encountered in adaptive switching between models with different resolutions as well as models with different system topologies. An example of model resolution change is a transition of a system from a discrete particle model to a reduced order articulated multi-rigid body model. The discontinuous changes in system definition may be viewed as an instantaneous change (release or impulsive application of) the system constraints. The method uses a spatial impulse-momentum formulation in a divide and conquer scheme. The approach utilizes a hierarchic assembly-disassembly process by traversing the system topology in a binary tree map to solve for the jumps in the system generalized speeds and the constraint impulsive loads in linear and logarithmic cost in serial and parallel implementations, respectively. The method is applicable for systems in serial chain as well as kinematical loop topologies. The coupling between the unilateral and bilateral constraints is handled efficiently through the use of kinematic joint definitions. The equations of motion for the system are produced in a hierarchic sub-structured form. This has the advantage that changes in sub-structure definitions/models results in a change to the system equations only within the associated sub-structure. This allows for significant changes in model types and definitions without having to reformulate the equations for the whole system.

关键词： impulse momentum formulation divide and conquer scheme logarithmic complexity course-graining of models discontinuous system definitions

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On the complexity of aperiodic Fourier modal methods for finite periodic structures

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JOURNAL OF COMPUTATIONAL PHYSICS 2014年第0期261卷 130-144页

作者： Pisarenco, M. Setija, I. D. ASML Netherlands BV Dept Res NL-5504 DR Veldhoven Netherlands

The Fourier modal method (FMM) is based on Fourier expansions of the electromagnetic field and is inherently built for infinitely periodic structures. When the infinite periodicity assumption is not realistic, the finiteness of the structure has to be incorporated into the model. In this paper we discuss the recent extensions of the FMM for finite periodic structures and analyze their complexity both with respect to the main discretization parameter (N) over cap as well as with respect to the number of periods R. We show that among the three FMM-based approaches able to represent finiteness, the aperiodic Fourier modal method with alternative discretization has the lowest computational cost given by either O((N) over cap (3) log(2) R) or O((N) over cap R-2) depending on the values of (N) over cap and R. This result demonstrates that the method is highly suited for rigorous modeling of scattering from large periodic structures. For instance, for (N) over cap = 100 and R < 1000 the complexity of the aperiodic Fourier modal method with alternative discretization is comparable to the complexity of the standard FMM. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Fourier-modal method FMM Rigorous coupled-wave analysis RCWA Perfectly matched layers PML Computational costs logarithmic complexity Aperiodic Fourier-modal method AFMM-CFF Alternative discretization

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Vision pyramids that do not grow too high

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PATTERN RECOGNITION LETTERS 2005年第3期26卷 319-337页

作者： Kropatsch, WG Haxhimusa, Y Pizlo, Z Langs, G Vienna Univ Technol Inst Comp Aided Automat Pattern Recognit & Image Proc Grp 183 2 A-1040 Vienna Austria Purdue Univ Dept Psychol Sci Sch Elect & Comp Engn W Lafayette IN 47907 USA Graz Univ Technol Inst Comp Graph & Vis A-8010 Graz Austria

In irregular pyramids, their vertical structure is not determined beforehand as in regular pyramids. We present three methods, all based on maximal independent sets from graph theory, with the aim to simulate the major sampling properties of the regular counterparts: good coverage of the higher resolution level, not too large sampling gaps and, most importantly, the resulting height, e.g. the number of levels to reach the apex. We show both theoretically and experimentally that the number of vertices can be reduced by a factor of 2.0 at each level. The plausibility of log (diameter) pyramids is supported by psychological and psychophysical considerations. Their technical relevance is demonstrated by enhancing appearance-based object recognition. An irregular pyramid hypothesis generation for robust PCA through top-down attention mechanisms achieves higher speed and quality than regular pyramids and non-pyramidal approaches. (C) 2004 Elsevier B.V. All rights reserved.

关键词： image pyramid logarithmic complexity graph pyramid maximal independent set robust PCA

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Fast Distance Transforms in Graphs and in Gmaps

Fast Distance Transforms in Graphs and in Gmaps

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joint IAPR International Workshop on Structural and Syntactic Pattern Recognition (SSPR) / Statistical Techniques in Pattern Recognition (SPR)

作者： Banaeyan, Majid Carratu, Carmine Kropatsch, Walter G. Hladuvka, Jiri TU Wien Pattern Recognit & Image Proc Grp Vienna Austria

ISBN: (纸本)9783031230271;9783031230288

Distance Transform (DT) as a fundamental operation in pattern recognition computes how far inside a shape a point is located. In this paper, at first a novel method is proposed to compute the DT in a graph. By using the edge classification and a total order [1], the spanning forest of the foreground is created where distances are propagated through it. Second, in contrast to common linear DT methods, by exploiting the hierarchical structure of the irregular pyramid, the geodesic DT (GDT) is calculated with parallel logarithmic complexity. Third, we introduce the DT in the nD generalized map (n-Gmap) leading to a more precise and smoother DT. Forth, in the n-Gmap we define n different distances and the relation between these distances. Finally, we sketch how the newly introduced concepts can be used to simulate gas propagation in 2D sections of plant leaves.

关键词： nD distance transform Generalized maps Irregular pyramids Parallel processing logarithmic complexity Geodesic distance transform (GDT)

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Digital line recognition, convex hull, thickness, a unified and logarithmic technique

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11th International Workshop on Combinatorial Image Analysis

作者： Buzer, Lilian ESIEE Inst Gaspard Monge Lab A2SI F-93162 Noisy Le Grand France

ISBN: (纸本)3540351531

The recognition of discrete primitives as digital straight segments (DSS) is a deeply studied problem in digital geometry (see a review in [6]). One characterization of the DSS is purely geometrical: all the points must lie between two lines whose distance (relative to the infinite norni) is less than 1. A common approach used to solve this question is to compute the convex hull of the given points. Recent papers explain how to update the minimum distance when a point is inserted during an online (incremental) recognition in O(log n) time in the general case [2] or in O(1) time with assumption [2,4]. Nevertheless, for other cases like insertions mixed with deletions or the union of two DSS, we have no optimal method to compute the resulting width. Thus, we propose a unified, simple and optimal approach applicable for any configuration. Moreover, our function is called independently from the convex hull processing. This allows to reuse any existing library without any modification. Thereby, we offer an efficient tool that opens a new horizon for the applications.

关键词： digital line DSS online incremental dynamic union recognition convex hull logarithmic complexity

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fastJT: An R package for robust and efficient feature selection for machine learning and genome-wide association studies

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BMC BIOINFORMATICS 2019年第1期20卷 333-333页

作者： Lin, Jiaxing Sibley, Alexander Shterev, Ivo Nixon, Andrew Innocenti, Federico Chan, Cliburn Owzar, Kouros Duke Univ Dept Biostat & Bioinformat Durham NC 27708 USA Duke Univ Duke Canc Inst Med Ctr Durham NC 27708 USA Duke Univ Med Ctr Duke Human Vaccine Inst Durham NC USA Div Pharmacotherapy & Expt Therapeut Chapel Hill NC USA

BackgroundParametric feature selection methods for machine learning and association studies based on genetic data are not robust with respect to outliers or influential observations. While rank-based, distribution-free statistics offer a robust alternative to parametric methods, their practical utility can be limited, as they demand significant computational resources when analyzing high-dimensional data. For genetic studies that seek to identify variants, the hypothesis is constrained, since it is typically assumed that the effect of the genotype on the phenotype is monotone (e.g., an additive genetic effect). Similarly, predictors for machine learning applications may have natural ordering constraints. Cross-validation for feature selection in these high-dimensional contexts necessitates highly efficient computational algorithms for the robust evaluation of many *** have developed an R extension package, fastJT, for conducting genome-wide association studies and feature selection for machine learning using the Jonckheere-Terpstra statistic for constrained hypotheses. The kernel of the package features an efficient algorithm for calculating the statistics, replacing the pairwise comparison and counting processes with a data sorting and searching procedure, reducing computational complexity from O(n(2)) to O(n log(n)). The computational efficiency is demonstrated through extensive benchmarking, and example applications to real data are *** is an open-source R extension package, applying the Jonckheere-Terpstra statistic for robust feature selection for machine learning and association studies. The package implements an efficient algorithm which leverages internal information among the samples to avoid unnecessary computations, and incorporates shared-memory parallel programming to further boost performance on multi-core machines.

关键词： Jonckheere-Terpstra Linear rank statistic Constrained inference Robust statistic logarithmic complexity Genome-wide association studies Feature selection Machine learning Parallel processing

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