Choice Of Basis Functions Research Articles

THE ELUCIDATION OF MECHANISMS IN HOMOGENEOUS CATALYSIS

After a reminder of the definition of common terms used in catalysis and reaction mechanisms, different chapters highlight the various approaches used for the elucidation of catalytic mechanisms, each with its own advantages and limitations. Kinetics investigations define the rate law and provide information on certain features of the reaction pathway between the resting state and the transition state of the rate-determining step. The use of isotope labels is useful in three ways: through the measurement of kinetic isotope effects, through the study of the incorporation into the products (regio-, stereochemical features), and through the incorporation into the incompletely converted reactants (reversibility). The investigation of isolated intermediates, if available, gives experimental structural information and thermodynamic/activation data on selected stoichiometric steps of the catalytic cycle. In situ spectroscopic investigations help determine the chemical nature of the resting state and other low-energy intermediates if carried out under catalytic conditions, or assist in the identification of other intermediates if performed under modified conditions on stoichiometric steps. Effects of the ligand modification on catalytic activity and selectivity also provide information on kinetically relevant steps. Finally, the application of computational methods is highlighted, with emphasis on the advantages and limitations of different computational strategies, the choice of basis functions and chemical model, and the handling of solvent models for the computation of thermodynamically and kinetically relevant parameters in solution.

Besov spaces with variable smoothness and integrability

In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder–Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.

The multiscale coarse-graining method. III. A test of pairwise additivity of the coarse-grained potential and of new basis functions for the variational calculation

The multiscale coarse-graining (MS-CG) method, proposed by Izvekov and Voth [J. Phys. Chem. B 109, 2469 (2005); Izvekov and VothJ. Chem. Phys. 123, 134105 (2005)], is a method for determining the effective potential energy function for a coarse-grained model of a fluid using data obtained from molecular dynamics (MD) simulation of the corresponding atomically detailed model. The method has been given a rigorous statistical mechanical basis [Noid et al. J. Chem. Phys. 128, 244114 (2008); Noid et al., J. Chem. Phys. 128, 244115 (2008)]. The coarse-grained (CG) potentials obtained using the MS-CG method are an approximate variational solution for the exact many-body potential of mean force for the coarse-grained sites. In this paper we apply this method to study the many-body potential of mean force among solutes in a simple model of a solution of Lennard-Jones particles. We use a new set of basis functions for the variational calculation that is useful when the coarse-grained potential is approximately equal to an arbitrarily complicated pairwise additive, central interaction among the sites of the coarse-grained model. For this model, pairwise additivity of the many-body potential of mean force is a very good approximation when the solute concentration is low, and it becomes less accurate for high concentrations, indicating the importance of many-body contributions to the coarse-grained potential. The best possible pairwise additive CG potential of the solute particles is found to be quite long ranged for all concentrations except those for which the mole fraction of solute is very close to unity. We discuss strategies for construction of short-ranged potentials for efficient but accurate CG MD simulation. We also discuss how the choice of basis functions for the variational calculation can be used to provide smoothing of the calculated CG potential function to overcome statistical sampling error in the atomistic simulation data used for the generation of the potential.

Mole. A system for quantum chemistry I. general description

A Quantum Chemistry System is proposed as a solution to the problem of implementing new techniques to study molecular structure. The criteria for such a system are described and the MOLE Quantum Chemistry System is presented as a demonstration of the utility of this approach. No particular bias towards any one technique (including choice of basis functions) has enabled MOLE to be quite general. It is designed for use on all current large-scale computing equipment. A general description of the applications sub-programs is also provided to show the range of application of this concept.

An efficient implementation of a least squares Monte Carlo method for valuing American-style options

Several methods for valuing high-dimensional American-style options were proposed in the last years. Longstaff and Schwartz (LS) have suggested a regression-based Monte Carlo approach, namely the least squares Monte Carlo method. This article is devoted to an efficient implementation of this algorithm. First, we suggest a code for faster runs. Regression-based Monte Carlo methods are sensitive to the choice of basis functions for pricing high-dimensional American-style options and, like all Monte Carlo methods, to the underlying random number generator. For this reason, we secondly propose an optimal selection of basis functions and a random number generator to guarantee stable results. Our basis depends on the payoff of the high-dimensional option and consists of only three functions. We give a guideline for an efficient option price calculation of high-dimensional American-style options with the LS algorithm, and we test it in examples with up to 10 dimensions.

Function spaces of variable smoothness and integrability

In this article we introduce Triebel–Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.

FLAPW: applications and implementations

Modern material design involves a close collaboration between experimental andcomputational materials scientists. To be useful, the theory must be able to accuratelypredict the stability and properties of new materials, describe the physics of theexperiments, and be applicable to new and complex structures—the all-electronfull-potential linearized augmented plane wave (FLAPW) is one such method that providesthe requisite level of numerical accuracy, albeit at the cost of complexity. Technical aspectsand modifications related to the choice of basis functions (energy parameters,core–valence orthogonality, extended local orbitals) that affect the applicabilityand accuracy of the method are described, as well as an approach for obtainingk-independent matrix elements. The inclusion of external electric fields is illustrated byresults for the induced densities at the surfaces of both magnetic and non-magnetic metals,and the relationship to image planes and to nonlinear effects such as second harmonicgeneration. The magnetic coupling of core hole excitations in Fe, the calculation of intrinsicdefect formation energies, the concentration-dependent chemical potentials, entropiccontributions, and the relative phase stability of Zr-rich Zr–Al alloys are also discussed.

Spectral approximation of solutions to the chemical master equation

The master equation of chemical reactions is an accurate stochastic description of general systems in chemistry. For D reacting species this is a differential-difference equation in D dimensions, exactly soluble for very simple systems only. We propose and analyze a novel solution strategy in the form of a Galerkin spectral method with a favorable choice of basis functions. A spectral approximation theory in the corresponding spaces is developed and the issue of stability is discussed. The convergence properties of the method are demonstrated by the numerical solution of two model problems with known solutions and a third problem for which no solution is known. It is shown that the method is effective and accurate, providing a viable alternative to other solution methods when the dimensionality is not too high.

Direct and iterative solution of the generalized Dirichlet–Neumann map for elliptic PDEs on square domains

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet–Neumann map for Laplace’s equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.

Formulation of improved basis sets for the study of polymer dynamics through diffusion theory methods

In this work a new method is proposed for the choice of basis functions in diffusion theory (DT) calculations. This method, named hybrid basis approach (HBA), combines the two previously adopted long time sorting procedure (LTSP) and maximum correlation approximation (MCA) techniques; the first emphasizing contributions from the long time dynamics, the latter being based on the local correlations along the chain. In order to fulfill this task, the HBA procedure employs a first order basis set corresponding to a high order MCA one and generates upper order approximations according to LTSP. A test of the method is made first on a melt of cis-1,4-polyisoprene decamers where HBA and LTSP are compared in terms of efficiency. Both convergence properties and numerical stability are improved by the use of the HBA basis set whose performance is evaluated on local dynamics, by computing the correlation times of selected bond vectors along the chain, and on global ones, through the eigenvalues of the diffusion operator L. Further use of the DT with a HBA basis set has been made on a 71-mer of syndiotactic trans-1,2-polypentadiene in toluene solution, whose dynamical properties have been computed with a high order calculation and compared to the "numerical experiment" provided by the molecular dynamics (MD) simulation in explicit solvent. The necessary equilibrium averages have been obtained by a vacuum trajectory of the chain where solvent effects on conformational properties have been reproduced with a proper screening of the nonbonded interactions, corresponding to a definite value of the mean radius of gyration of the polymer in vacuum. Results show a very good agreement between DT calculations and the MD numerical experiment. This suggests a further use of DT methods with the necessary input quantities obtained by the only knowledge of some experimental values, i.e., the mean radius of gyration of the chain and the viscosity of the solution, and by a suitable vacuum trajectory, with great savings in computational time required. This offers a theoretical bridge between the experimental static and dynamical properties of polymers.

Examination and Analysis of Implementation Choices within the Material Point Method (MPM)

The Material Point Method (MPM) has shown itself to be a powerful tool in the sim- ulation of large deformation problems, especially those involving complex geometries and contact where typical finite element type methods fre- quently fail. While these large complex problems lead to some impressive simulations and solu- tions, there has been a lack of basic analysis char- acterizing the errors present in the method, even on the simplest of problems. The large number of choices one has when implementing the method, suchas thechoiceofbasisfunctionsandboundary treatments,furthercomplicates thiserroranalysis. In thispaperweexploresomeofthemany choices one can make when implementing an MPM al- gorithm and the numerical ramifications of these choices. Specifically, we analyze and demonstrate how the smoothing length within the General- ized InterpolationMaterial Point Method (GIMP) can affect the error and stability properties of the method. We also demonstrate how various choices of basis functions and boundary treat- ments affect the spatial convergence properties of MPM.

Identification of age‐structured models: Cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

Appropriate SCF basis sets for orbital studies of galaxies and a ‘quantum-mechanical’ method to compute them

We address the question of an appropriate choice of basis functions for the self-consistent field (SCF) method of simulation of the N-body problem. Our criterion is based on a comparison of the orbits found in N-body realizations of analytical potential-density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen gamma-models for values of $\gamma$ in the range 0<=gamma<=1. When an N-body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces, or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton-Brock and the Hernquist-Ostriker (HO) basis sets, as well as a family of numerical basis sets which are `close' to the HO basis set. The family of numerical basis sets is parametrized in terms of a quantity $\epsilon$ which appears in the kernel functions of the Sturm-Liouville (SL) equation defining each basis set. The HO basis set is the $\epsilon=0$ member of the family. We demonstrate that grid solutions of the SL equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum-mechanical method of solution of the SL equation which overcomes these errors. We finally give criteria for a choice of optimal value of $\epsilon$ and calculate the latter as a function of the value of gamma.

Calculation of the Properties of Reentrant Cylindrical Cavity Resonators

The lowest resonant frequencies of reentrant cylindrical cavity resonators are calculated using the method of moments to obtain upper and lower bounds. The accuracy and convergence of the results are investigated and the factors and shunt impedances calculated. A simple empirical assumption about the choice of basis functions leads to results that are of good accuracy and readily computed. The results obtained are compared with those of experiment and from calculations using MAFIA and Microwave Studio.

On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper analyses the robustness of Least - Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least - squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices.

The significance of the choice of basis functions in the problems of acoustic tomography of the ocean

Different approaches to the parametric description of the ocean inhomogeneities of both refraction and kinetic types are discussed in the context of the inhomogeneity reconstruction with the use of tomographic techniques. In addition to bases commonly known and widely used in oceanological problems (such as specification of the inhomogeneity parameters at the grid nodes or in nonoverlapping shapes compactly covering the region of interest), a new nonorthogonal and redundant basis consisting of a set of overlapping bands (and, presumably, more convenient for solving tomographic problems) is considered. The abilities of different bases to reconstruct the ocean inhomogeneities are compared with the use of special theoretical approach. The quality of reconstruction on the band and cell bases is investigated against the relationship between the number and composition of the basis elements. Examples of the reconstruction of ocean inhomogeneities with the use of the aforementioned basis functions are given together with the results of comparison.

Reduced Atmospheric Models Using Dynamically Motivated Basis Functions

Abstract Nonlinear deterministic reduced models of large-scale atmospheric dynamics are constructed. The dynamical framework is a quasigeostrophic three-level spectral model with realistic mean state and variability as well as Pacific–North America (PNA) and North Atlantic Oscillation (NAO) patterns. The study addresses the problem of finding appropriate basis functions for efficiently capturing the dynamics and a comparison between different choices of basis functions; it focuses on highly truncated models, keeping only 10–15 modes. The reduced model is obtained by a projection of the equations of motion onto a truncated basis spanned by empirically determined modes. The total energy metric is used in the projection; the nonlinear terms of the low-order model then conserve total energy. Apart from retuning the coefficient of horizontal diffusion, no empirical terms are fitted in the dynamical equations of the low-order model in order to properly preserve the physics of the system. Using the methodology of principal interaction patterns (PIPs), a basis is derived that carefully compromises minimizing tendency error with maximizing explained variance in the resolved modes. A new PIP algorithm is introduced that is more compact and robust than earlier PIP algorithms; a top-down approach is adopted, removing modes from the system one by one. The mean state and standard deviation of the streamfunction as well as transient momentum fluxes are well reproduced by a PIP model with only 10 modes. Probability density functions are accurately modeled and autocorrelation functions are captured fairly well using 15 modes. Reduced models based on PIPs are substantially superior to reduced models based on empirical orthogonal functions (EOFs). The leading PIPs have a higher projection onto the PNA and NAO teleconnection patterns than the corresponding EOFs. Both with EOFs and PIPs, the interactions between the resolved modes are predominantly linear and the improvement of PIP models on EOF models stems entirely from better modeling these linear interactions although the full nonlinear tendencies are optimized. There is considerable influence of smaller-scale modes on the large-scale modes due to nonlinear coupling that is not well captured by either EOFs or PIPs. This nonlinear backscattering possibly plays a role in generating the low-frequency variability of the model. The results call for a nonlinear and/or stochastic closure scheme in which PIPs may be suitable basis functions.

Finite difference operators from moving least squares interpolation

In a foregoing paper [Sonar, ESAIM: M2AN 39 (2005) 883–908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Salkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLS method we address the question of getting admissible results at the boundary by introducing “ghost points”. Most interesting in IMLS methods are the finite difference operators which can be computed from different choices of basis functions and weight functions. We present a way of deriving these discrete operators in the spatially one-dimensional case. Multidimensional operators can be constructed by simply extending our approach to higher dimensions. Numerical results ranging from 1-d interpolation to the solution of PDEs are given.

Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes

The angular factors of atomic orbitals are real spherical harmonics. This is independent of the choice of basis function. In the course of molecular electronic structure calculations, numerous rotations of real spherical harmonics are required in a suitably defined space-fixed coordinate system. The origin and axes are space-fixed and rotation matrices defined on the basis of spherical harmonics. Typically, rotations are used to align the local axes for atoms with a unique interatomic direction in a fixed frame for integral evaluations. In this work, a highly compact expression and efficient evaluation of the rotation matrices are given for a real spherical harmonic basis. Relations to Gaunt coefficients are shown explicitly as are recurrence formulae for rotation matrices. This leads to extremely rapid and precise rotation algorithms. The Wigner rotation matrices which are still often used in orbital rotations are shown to be completely surpassed by this approach. The present work is related to a method described by Kautz in the field of image processing but significant improvements have been made, especially in the study of structure and storage of the rotation matrices. After complete testing using computer algebra, a numerical program was written in C. Numerical tests are cited in the closing sections of this work.

Solving Maxwell’s equations using the ultra weak variational formulation

We investigate the ultra weak variational formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an electromagnetic UWVF solver. In particular, we propose a method to avoid the conditioning problems that are known to hamper the use of the UWVF for problems in general geometries and inhomogeneous media. In addition, we show how to implement the PML in the UWVF to accurately approximate physically unbounded problems and discuss the parallelization of the UWVF. Three-dimensional numerical simulations are used to examine the feasibility of the UWVF for simulating wave propagation in inhomogeneous media and scattering from complex structures.