Divide-and-conquer Approach Research Articles

Distributed Operational Space Formulation of Serial Manipulators

This paper presents a new parallel algorithm for the operational space dynamics of unconstrained serial manipulators, which outperforms contemporary sequential and parallel algorithms in the presence of two or more processors. The method employs a hybrid divide and conquer algorithm (DCA) multibody methodology which brings together the best features of the DCA and fast sequential techniques. The method achieves a logarithmic time complexity (O(log(n)) in the number of degrees of freedom (n) for computing the operational space inertia (Λe) of a serial manipulator in presence of O(n) processors. The paper also addresses the efficient sequential and parallel computation of the dynamically consistent generalized inverse (J¯e) of the task Jacobian, the associated null space projection matrix (Ne), and the joint actuator forces (τnull) which only affect the manipulator posture. The sequential algorithms for computing J¯e, Ne, and τnull are of O(n), O(n2), and O(n) computational complexity, respectively, while the corresponding parallel algorithms are of O(log(n)), O(n), and O(log(n)) time complexity in the presence of O(n) processors.

Model transitions and optimization problem in multi-flexible-body systems: Application to modeling molecular systems

This paper presents an efficient algorithm for the simulation of multi-flexible-body systems undergoing discontinuous changes in model definition. The equations governing the dynamics of the transitions from a higher to a lower fidelity model and vice versa are formulated through imposing/removing certain constraints on/from the system model. The issue of the non-uniqueness of the results associated with the transition from a lower to a higher fidelity model may be handled by solving an optimization problem. This optimization problem is subjected to the satisfaction of the constraint imposed by the generalized impulse–momentum equations. The divide-and-conquer algorithm (DCA) is applied to formulate the jumps in the system states resulting from the model transition. The DCA formulation in its basic form is both time and processor optimal and results in linear and logarithmic complexity when implemented in serial and parallel with O(n) processors, respectively. As such, its application can reduce the effective computational cost of formulating and solving the optimization problem in the transitions to the finer models. The principal aspects of the mathematics for the algorithm implementation is developed and numerical examples are provided to validate the method.

The bounds of the eigenvalues for rank-one modification of Hermitian matrix

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a spectral perturbation theorem for rank-one updated matrices of special structure in [7] and considered two applications. Cheng, Luo, and Li considered the bounds of the smallest and largest eigenvalues for rank-one modification of Hermitian matrices [8]. Eigenvalue bounds for perturbations of Hermitian matrices have been considered by Ipsen and Nadler in [9]. In this paper, we consider the bounds of the eigenvalues for rank-one modification of Hermitian matrices. The ideas of this paper were mainly motivated by one of [9]. We study the following form

Response to "On Mathematical Equivalence Between Vector OFDM and Quadrature OFDMA"

In, we introduced a novel layered inverse Fast Fourier Transform (IFFT) structure based on the Divide-andConquer approach for computing the IFFT. Based on the structure we proposed asymmetric OFDM systems which bridge OFDM and single carrier systems. Its multiuser access version, Quadrature OFDMA, is developed in. In the presented form, the basic transceiver of asymmetric OFDM is mathematically similar to the vector OFDM system. We regret for not citing the vector OFDM and related work in our papers. As a response to, we argue that there are actually significant differences between our work and the vector OFDM work.

Canonical ensemble simulation of biopolymers using a coarse-grained articulated generalized divide-and-conquer scheme

In this paper, a scheme for the canonical ensemble simulation of the coarse-grained articulated polymers is discussed. In this coarse-graining strategy, different subdomains of the system are considered as rigid and/or flexible bodies connected to each other via kinematic joints instead of stiff, but elastic bonds. Herein, the temperature of the simulation is controlled by a Nosé–Hoover thermostat. The dynamics of this feedback control system in the context of multibody dynamics may be represented and solved using traditional methods with computational complexity of O(n3) where n denotes the number of degrees of freedom of the system. In this paper, we extend the divide-and-conquer algorithm (DCA), and apply it to constant temperature molecular simulations. The DCA in its original form uses spatial forces to formulate the equations of motion. The Generalized-DCA applied here properly accommodates the thermostat generalized forces (from the thermostat), which control the temperature of the simulation, in the equations of motion. This algorithm can be implemented in serial and parallel with computational complexity of O(n) and O(logn), respectively.

An extended divide-and-conquer algorithm for a generalized class of multibody constraints

An extension to the divide-and-conquer algorithm (DCA) is presented in this paper to model constrained multibody systems. The constraints of interest are those applied to the system due to the inverse dynamics or control laws rather than the kinematically closed loops which have been studied in the literature. These imposed constraints are often expressed in terms of the generalized coordinates and speeds. A set of unknown generalized constraint forces must be considered in the equations of motion to enforce these algebraic constraints. In this paper dynamics of this class of multibody constrained systems is formulated using a Generalized-DCA. In this scheme, introducing dynamically equivalent forcing systems, each generalized constraint force is replaced by its dynamically equivalent spatial constraint force applied from the appropriate parent body to the associated child body at the connecting joint without violating the dynamics of the original system. The handle equations of motion are then formulated considering these dynamically equivalent spatial constraint forces. These equations in the GDCA scheme are used in the assembly and disassembly processes to solve for the states of the system, as well as the generalized constraint forces and/or Lagrange multipliers.

An efficient parallel dynamics algorithm for simulation of large articulated robotic systems

This paper presents a new parallel computer algorithm for the simulation of large articulated robotic systems. The method employs the Divide and Conquer Algorithm (DCA) multibody methodology and achieves significant increases in speed by using a variation of the Articulated Body Algorithm (ABA) to efficiently construct the DCA subsystems. The method outperforms contemporary parallel dynamics algorithms for a serial topology and degenerates to the ABA in the absence of parallel resources. The implementation and comparison with the ABA demonstrate useful speed increases with as few as two parallel processors and speed ups by a factor of 2.25 in the presence of four processors for serial manipulators. The paper also addresses the practical limitations on obtainable speedups due to the interprocessor communication costs and its consequences on choosing optimal DCA subsystems.

How does it become possible to treat delocalized and/or open-shell systems in fragmentation-based linear-scaling electronic structure calculations? The case of the divide-and-conquer method

The authors have developed a fragmentation-based linear-scaling electronic structure calculation strategy named the divide-and-conquer (DC) method, which has been implemented into the Gamess program package. Although there are many sorts of fragmentation-based linear-scaling schemes, most of them require the charge and spin multiplicity of each fragment a priori. Therefore, their applications to delocalized and/or open-shell systems have been limited. However, the DC method is a notable exception because the distribution of electrons in the entire system is automatically determined by the universal Fermi level. In this perspective, the authors have summarized the performance of the linear-scaling self-consistent field (SCF) and post-SCF calculations of delocalized and/or open-shell systems based on the DC method. Furthermore, some future prospects of the method have been discussed.

A divide-and-conquer Delaunay triangulation algorithm with a vertex array and flip operations in two-dimensional space

Divide-and-conquer algorithms provide computational efficiency for constructing Delaunay triangulations; however, their implementation is complicated. Most divide-and-conquer algorithms for Delaunay triangulation utilize edge-based structures, because triangles are frequently deleted and created during the merge process. However, as our proposed divide-and-conquer algorithm does not require existing edges to be deleted in the merge process, a simple array-based data structure can be used for the representation of the triangulation topology. Rather than deleting the edges of the conflicting triangles, which was used in previous methods, the Delaunay property is also preserved with a new flip propagation method in the merge phase. This array-based data structure is much simpler than the commonly used edge-based data structures and requires less memory allocation. The proposed algorithm arranges sites into a permutation vector that represents a kdtree with an array; thus, the space partitioning information is internally represented in the array without any additional data. Since the proposed divide-and-conquer algorithm is compact, the implementation complexity of conventional divideand-conquer triangulation algorithms can be avoided. Despite of the simplicity of this new algorithm, the experimental results indicate that the computational efficiency is comparable to the previous divide-and-conquer algorithms.

Divide-and-conquer self-consistent field calculation for open-shell systems: Implementation and application

In this Letter, the divide-and-conquer (DC) linear-scaling self-consistent field method is extended to the spin-unrestricted Hartree–Fock (UHF) method or Kohn–Sham density functional theory (UDFT) for treating large open-shell systems. Although the DC method is one of the fragmentation-based linear-scaling schemes, the present DC-UHF/UDFT framework can avoid specifying the number of up- and down-spin electrons in each fragment by introducing up- and down-spin Fermi levels. Test calculations for oligothiophenes demonstrate the high efficiency and accuracy of the DC-UHF/UDFT method even for spin-delocalized systems.

Dimerization free energy of vancomycin‐group antibiotics and the cooperative effect: A density functional approach

AbstractThe cooperative effects between the binding of model bacterial cell wall termini and the dimer forms of the glycopeptide antibiotic vancomycin and chloro‐eremomycin were investigated with conventional and divide‐and‐conquer (DC) ab initio quantum chemistry simulations. This feature is potentially important for drug design and controlling molecular recognition phenomena in these and other biochemical systems. In terms of the calculation, the vibrational contributions to the dimerization free energies of both molecules found to be the most costly and necessitated the use of the DC method. We specifically focused on calculating the relative strength of the cooperative effect for the two antibiotics. In good agreement with experiments, we find that the cooperative effect is stronger for chloro‐eremomycin by about 1.39 kcal/mol. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Divide and Conquer Hartree−Fock Calculations on Proteins

The ability to perform ab initio electronic structure calculations that scales linearly with the system size is one of the central aims in theoretical chemistry. In this study, the implementation of the divide-and-conquer (DC) algorithm, an algorithm with the potential to aid the achievement of true linear scaling within Hartree-Fock (HF) theory is revisited. Standard HF calculations solve the Roothaan-Hall equations for the whole system; in the DC-HF approach, the diagonalization of the Fock matrix is carried out on smaller subsystems. The DC algorithm for HF calculations was validated on polyglycines, polyalanines and eleven real three-dimensional proteins of up to 608 atoms in this work. We also found that a fragment-based initial guess using molecular fractionation with conjugated caps (MFCC) method significantly reduces the number of SCF cycles and even is capable of achieving convergence for some globular proteins where the simple superposition of atomic densities (SAD) initial guess fails.

Comparative Study on Serial and Parallel Forward Dynamics Algorithms for Kinematic Chains*

The main focus of this paper is to investigate the essential differences among four forward dynamics algorithms: the Articulated-Body Algorithm (ABA) and Divide-and-Conquer Algorithm (DCA) by Featherstone; the Constraint Force Algorithm (CFA) by Fijany et al.; and the Assembly—Disassembly Algorithm (ADA) by the present authors. All of the algorithms have O(N) asymptotic complexity for serial computation where N is the number of rigid bodies, and three of them can also be processed in parallel which results in O (log N ) complexity on O (N) processors. We start by summarizing two essential backgrounds of the forward dynamics algorithms, i.e. articulated-body inertias and joint constraint representation. We then present a new formulation of ADA as well as the outlines of the other three algorithms using the same notation. Finally, we perform qualitative as well as quantitative comparisons of the algorithms using our implementations of ABA, CFA, and ADA.

An Efficient Multibody Divide and Conquer Algorithm and Implementation

A new and efficient form of Featherstone’s multibody divide and conquer algorithm (DCA) is presented and evaluated. The DCA was the first algorithm to achieve theoretically the optimal logarithmic time complexity with a theoretical minimum of parallel computer resources for general problems of multibody dynamics; however, the DCA is extremely inefficient in the presence of small to modest parallel computers. This alternative efficient DCA (DCAe) approach demonstrates that large DCA subsystems can be constructed using fast sequential techniques to realize a substantial increase in speed. The usefulness of the DCAe is directly demonstrated in an application to a four processor workstation and compared with the results from the original DCA and a fast sequential recursive method. Previously the DCA was a tool intended for a future generation of parallel computers; this enhanced version delivers practical and competitive performance with the parallel computers of today.

Implementation of divide‐and‐conquer method including Hartree‐Fock exchange interaction

The divide-and-conquer (DC) method, which is one of the linear-scaling methods avoiding explicit diagonalization of the Fock matrix, has been applied mainly to pure density functional theory (DFT) or semiempirical molecular orbital calculations so far. The present study applies the DC method to such calculations including the Hartree-Fock (HF) exchange terms as the HF and hybrid HF/DFT. Reliability of the DC-HF and DC-hybrid HF/DFT is found to be strongly dependent on the cut-off radius, which defines the localization region in the DC formalism. This dependence on the cut-off radius is assessed from various points of view: that is, total energy, energy components, local energies, and density of states. Additionally, to accelerate the self-consistent field convergence in DC calculations, a new convergence technique is proposed.

Formulation and analysis of in-place MSD radix sort algorithms

We present a unified treatment of a number of related inplace MSD radix sort algorithms with varying radices, collectively referred to here as 'Matesort' algorithms. These algorithms use the idea of in-place partitioning which is a considerable improvement over the traditional linked list implementation of radix sort that uses O( n) space. The binary Matesort algorithm is a recast of the classical radixexchange sort, emphasizing the role of in-place partitioning and efficient implementation of bit processing operations. This algorithm is O(k) space and has O( kn) worst-case order of running time, where k is the number of bits needed to encode an element value and n is the number of elements to be sorted. The binary Matesort algorithm is evolved into a number of other algorithms including 'continuous Matesort' for handling floating point numbers, and a number of 'general radix Matesort' algorithms. We present formulation and analysis for three different approaches (sequential, divide-and-conquer and permutation-loop) for partitioning by the general radix Matesort algorithm. The divide-andconquer approach leads to an elegantly coded algorithm with better performance than the permutation-loop-based American Flag Sort algorithm.

Divide and conquer algorithms for computing the eigendecomposition of symmetric diagonal-plus-semiseparable matrices

Three fast and stable divide and conquer algorithms to compute the eigendecomposition of symmetric diagonal-plus-semiseparable matrices are considered.

Fast semiempirical calculations for nuclear magnetic resonance chemical shifts: a divide-and-conquer approach.

A new approach to calculate nuclear magnetic resonance chemical shifts has been implemented at the semiempirical modified neglect of diatomic overlap level using gauge-including atomic orbitals. The perturbed density matrix with respect to the magnetic field is obtained by the diagonalization of the complex Fock matrix using the divide and conquer (DC) method, instead of by solving the computationally expensive coupled perturbed Hartree-Fock equations. Adopting the Patchkovskii and Thiel parameters [S. Patchkovskii and W. Thiel J. Comput. Chem. 20, 1220 (1999)], we were able to reproduce their results for small organic molecules. The errors introduced by DC method are negligible, as shown by the calculations on a series of polyalaine structures. Test calculations on proteins have demonstrated that our approach makes it possible to calculate chemical shifts routinely on systems with hundreds of atoms with good accuracy.

Intraprotein electrostatics derived from first principles: divide-and-conquer approaches for QM/MM calculations.

Two divide-and-conquer (DAQ) approaches for building multipole-based molecular electrostatic potentials of proteins are presented and evaluated for use in QM/MM calculations. One approach is a further development of the neutralization method of Bellido and Rullmann (J Comput Chem 1989, 10, 479-487) while the other is based on removing part of the electron density before performing the multipole expansion. Both methods create systems with integer charges without using charge renormalization. To determine their performance in terms of location of cuts and distance to QM region, the new DAQ approaches are tested in calculations of the proton affinity of N(zeta) of Lys55 in the inhibitor turkey ovomucoid third domain. Finally, the two methods are used to build a variety of MM regions, applied to calculations of the pK(a) of Lys55, and compared to other computational methodologies in which force field charges are employed.

Construction of genetic network using evolutionary algorithm and combined fitness function.

This paper proposes a method to capture the dynamics in gene expression data using S-system formalism and construct genetic network models. Our purposed method exploits the probabilistic heuristic search and divide-and-conquer approach to estimate the network structure. In evaluating the network structure, we attempt a primitive integration of other knowledge to the statistical criterion. The Z-score is used to analyze the robust and significant parameters from stochastic search results. We evaluated the proposed method on artificially generated data and E.coli mRNA expression data.