Development Of Approximate Methods Research Articles

Distributed Implementation of Full Configuration Interaction for One Trillion Determinants.

Full configuration interaction (FCI) can provide an exact molecular ground-state energy within a given basis set and serve as a benchmark for approximate methods in quantum chemical calculations, including the emerging variational quantum eigensolver. However, its exponential computational and memory requirements easily exceed the capability of a single server and limit its applicability to large molecules. In this paper, we present a distributed FCI implementation employing a hybrid parallelization scheme with multithreading and multiprocessing to expand FCI's applicability. We optimize this scheme to minimize the bottlenecks arising from interprocess communications and interthread data management. Our implementation achieves higher scalability than the naive combination of prior works and successfully calculates the exact energy of C3H8/STO-3G with 1.31 trillion determinants, which is the largest FCI calculation to the best of our knowledge. Furthermore, we provide a comprehensive list of FCI results with 136 combinations of molecules and basis sets for future evaluation and development of approximate methods.

On Construction of Projection Operators.

The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.

Development of approximation method to evaluate isotopic composition of burnt fuel

Abstract This paper deals with the development of an approximation method to evaluate the isotopic composition of burnt VVER fuel rods (with U-Gd) by engineering codes (TVS-M, PERMAK-A) and taking into account fuel rods depletion conditions. Burn-up conditions for fuel rods differ from each other during operation due to the presence of water cavities, absorber and other within FA. Therefore, the isotopes evolution during burn-up in different fuel rods goes on in different manner even with identical values of fuel burn-up (MW · day/kg U). The isotopic content evolution can be described in detail by spectral codes. However, a full-size core calculation takes much running time. The approximation method, which takes into account fuel rods spectral differences during operation, was considered. The method is based on spectral functionality (a non asymptotic factor) usage, which allows to take into account the spectral conditions and history of exposure.

Substructure synthesis method for buckling analysis in frames

An everlasting objective of Structural Stability and Structural Dynamics is the development of approximate methods that yield highly accurate results (critical loads, and natural frequencies and modes) and that involve low numbers of degrees of freedom. The Rayleigh–Ritz–Meirovitch substructure synthesis method (RRMSSM) has proved to be more efficient than the finite element method (FEM) in dynamic analysis of structures that can be modeled as assemblages of simple substructures, and the question is if it would be also more efficient in stability analysis of these structures. The RRMSSM is the classical Rayleigh–Ritz method with Meirovitch’s introduction of quasicomparison functions [1] and extension to multibody systems [2]. This Rayleigh-Ritz methodology with both developments was presented by Meirovitch and Kwak [3]. Lately, it was extended to multiply supported or framed structures [4, 5]. In these investigations, the

Asymptotic analysis and effect of reaction order on a reversible gas-solid system with fast reaction

The development of approximate methods for the equations describing noncatalytic reversible gas-solid systems with fast reactions are investigated by means of perturbation techniques and series solutions. The effect of the reaction orders, when the back reaction is not neglected, on the existence of a moving boundary is also discussed. It is shown that a moving boundary exists only when the reversible reaction rate is independent of the reactant solid concentration. Estimates for conversion times are also obtained for the cases where all the reaction orders are non-zero and where the kinetics is independent of the reactant and product solids. These approximate solutions may find application, for example, in computational models of ironmaking blast furnaces.

The principle of interval constraints: A generalization of the symmetric dirichlet distribution

A structure for representing problems in decision analysis and in expert systems, which reason under uncertainty, is the influence diagram or causal network. A causal network consists of an underlying joint probability distribution and a directed acyclic graph in which a propositional variable that represents a marginal distribution is stored at each vertex in the graph. This paper is concerned with two of the problems in applications that use causal networks. The first problem is the determination of the conditional probabilities of the values of remaining propositional variables in the network given that certain variables are instantiated for particular values. This is called probability propagation. The second problem is the determination of the most probable, second most probable, third most probable, and so on sets of values of a particular set of variables (called the explanation set) given that certain variables are instantiated for particular values. This problem is called abductive inference. There exists a class of causal networks in which each variable has only two parents, for which the time required, by any known method, for probability propagation is exponential relative to the number of vertices in the network. The determination of a new method that would be efficient for all causal networks appears unlikely, because probability propagation has been shown to be #P-complete. In many medical applications, networks are often large and not sparsely connected. Therefore a method for the exact determination of probability values appears unlikely for such applications, and the development of approximation methods seems to be the best solution. The current approximation methods obtain interval bounds for the probability values. When such intervals are obtained, it is not possible in general to rank the alternatives. In this paper, a method is developed for obtaining expected values for the point probabilities from interval constraints on the probabilities. The method is based on an application of the principle of indifference to the probability values themselves. The distributions obtained with the principle of indifference are a generalization of the symmetric Dirichlet distribution in which prior ignorance is assumed.

Development of approximate methods of calculating residual stresses formed in the heat treatment of glass components of complicated form

Development of approximate methods of calculating residual stresses formed in the heat treatment of glass components of complicated form

An approximate method for calculating the boundary layer of two-temperature plasma at electrodes with large values of the hall parameter

The processes occurring in the plasma boundary layer at electrodes are described by a complicated system of differential equations with boundary conditions specified on two boundaries. Even when a computer is used, solution of a boundary-value problem of this type presents serious difficulties. Besides, a method is desirable whereby fairly fast estimates can be obtained and the influence of different gross factors analyzed. From this point of view, the development of approximate methods is of great value for theory of the plasma boundary layer.

Frequency measurements with the cathode ray oscillograph

The cathode ray oscillograph frequency measurement circuit described differs from previous circuits in the use of by-pass condensers and plate leaks which permit the connection of the oscillograph to a-c. circuits having large d-c. components and which permit the use of biasing controls for shifting the position of patterns on the screen. Reference oscillators are used in conjunction with the frequency standards. They are of a type chosen for their high stability. The well-known properties of Lissajous' figures are reviewed briefly and then developed more fully for the cases in which only one term of their ratios may be determined from the oscillograph pattern. Following a general discussion of the accuracy of syntonization, there is discussed a detailed method of calibrating oscillators. The patterns used may be interpreted from one term of their ratio. Interpolation formulas are derived for use in making interpolations on the reference oscillators. The methods of interpolaion, discussed may be used for intervals up to 0.5 per cent frequetncy with high degrees of accuracy. The interpolation formulas are extended further in the development of approximation methods by which any unknown frequency may be quickly determined within a few per cent. Several special circuits are described for use in frequency measurement work with the cathode ray oscillograph. The methods and apparatus described are suitable not only for the technical measurements of a development and research nature but are equally adaptable for routine commercial work. The advantages which particularly commend themselves are the rapidity with which such work may be done and the ease with which the average man can learn the work.

Frequency Measurements with the Cathode Ray Oscillograph

The cathode ray oscillograph frequency measurement circuit described differs from previous circuits in the use of by-pass condensers and plate leaks which permit connection of the oscillograph to a-c. circuits having large d-c. components and which permit the use of biasing controls for shifting the position of patterns on the screen. Reference oscillators are used in conjunction with the frequency standards. They are of a type chosen. for their high stability. The well-known properties of Lissajous' figures are reviewed briefly and then developed more fully for the cases in which only one term of their ratios may be determined from the oscillograph pattern. Following a general discussion of the accuracy of syntonization, there is discussed a detailed method of calibrating oscillators. The patterns used may be interpreted from one term of their ratio. Interpolation formulas are derived for use in making interpolations on the reference oscillators. The methods of interpolation discussed may be used for intervals up to 0.5 per cent frequency with high degrees of accuracy. The interpolation formulas are extended further in the development of approximation methods by which any unknown frequency may be quickly determined within a few per cent. Several special circuits are described for use infrequency measurements work with the cathode ray oscillograph. The methods and apparatus described are suitable not only for the technical measurements of a development and research nature but are equally adaptable for routine commercial work.