Computational efficiency algorithms

Unless very few experimental data are available, the dimensionality of the problem is extremely large and hence difficult to treat with standard nonlinear minimization methods. Schwetlick and Tiller (1985), Salazar-Sotelo et al. (1986) and Valko and Vajda (1987) have exploited the structure of the problem and proposed computationally efficient algorithms. 

Although the definition of the inertia matrix is simple in the ]4iysical sense, its calculation is quite complex. A number of different ai roaches have be investigated in the search for computationally efficient algorithms. Lee and Lee [24] use the generalized d Alemtert equations of motion to describe the dynamic behavior of robot manipulators with revolute joints. Specific expressions... 

The purpose of this book is to present computationally efficient algorithms for the dynamic simulation of closed-chain robotic systems. In particular, the simulation of single closed chains and simple closed-chain mechanisms (such as multilegged vehicles or dexterous hands) is investi ted in detail. In conjunction with the simulation algorithms, efficient algorithms are also derived for the computation of the joint space and operational space inntia matrices of a manipulator. These two inertial quantities are important factors in a variety of robotics applications, including both simulation and control. 

Hussain, J. and Rea, C. (2010) Computationally efficient algorithm to identify matched molecular parrs (MMPs) in large data sets. Journal of Chemical Information and Modeling 50 (3), 339-348. 

Hussain J, Rea C. Computationally efficient algorithm to identify matched molecular... 

There are two major approaches to elaboration of computationally efficient algorithms. These are based on Lagrangian relaxation and Bender s decomposition. A short overview of these methods is provided here. Readers are referred to Avriel and Golany (1996) for a detailed coverage of mathematical programming. 

Studies of proton transport in PEMs or at interfaces, as well as studies of processes at the electrified interface, usually demand quantum mechanical simulations to incorporate electronic structure effects and hydrogen bond dynamics. Studies of structure formation and transport properties in heterogeneous media demand computationally efficient algorithms that enable simulations of sufficient length (>20 nm) and time... 

In this subsection, a computationally efficient algorithm is presented for online updating of the noise parameter vector O +ui+i and the associated covariance matrix 2e, +i k+ Starting with an arbitrary initial condition oio and arbitrarily positive definite matrix 2e,oio> th noise parameter vector and its associated covariance matrix can be updated based on Eqs. 28 and 29. As aforementioned, the noise parameter vector can be updated by solving the optimization problem in Eq. 28. However, due to the large prior uncertainty and numerical considerations, a training process is necessary for a preliminary solution... 

Some work on spin effects and in particular, spin-controlled reactivity has already been presented in the literature [14-21] which have highlighted the importance of coherent and incoherent effects in the modelling of spur kinetics. As a result, one of the major aims of this work is to develop computationally efficient algorithms which are capable of modelling both the kinetics and spin-dynamics explicitly for any radiation chemical system. Using these simulation programs, this work then aims to ... 

The construction of an efficient algorithm rests on the ability to separate the Hamiltonian into parts which are themselves integrable and also efficiently computable. Suppose that the MD Hamiltonian H defined by (6) is split into two parts as... 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

The first summation requires electron repulsion integrals with four virtuaJ indices. Efficient algorithms that avoid the storage of these integrals have been discussed in detail [20]. For every orbital index, p, this OV contraction must be repeated for each energy considered in the pole search it is usually the computational bottleneck. 

The paper-and-pencir method of eigenvector decomposition can only be performed on small matrices, such as illustrated above. For matrices with larger dimensions one needs a computer for which efficient algorithms have been designed (Section 31.4). 

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