Types Of Basis Functions Research Articles

An Empirical Analysis of Valuation Algorithms for Pricing Callable Snowball Floaters

In this paper we value a callable snowball floater, a complex interest rate instrument with variable coupon payments, which depend on the prevailing interest rates in arrears and recursively on previous coupon payments. The embedded option requires solving an optimal stopping problem using the dynamic programming principle. A well-known and widely used algorithm to estimate conditional expectations is a specific form of least squares Monte Carlo simulation introduced by Longstaff and Schwartz (2001), which we refer to as the LSM approach. Contrary to the standard approach, where discounted option values of the subsequent period are regressed on the current state variables, Longstaff and Schwartz (2001) use the ex post realized payoffs of in-the-money option scenarios from continuation instead. They argue that, in doing so, they get values less than or equal to the value implied by the optimal stopping rule, which provides an objective convergence criterion. We compare the LSM approach with the standard approach and use the price estimate from an elaborate nested Monte Carlo simulation as a benchmark. We empirically find that the LSM estimate of the embedded option might be considerably downward biased, whereas the standard estimate is much closer to the benchmark price. Moreover, we find that there is no optimal type of basis function that can generally be recommended for pricing interest rate instruments. Instead, we suggest using the LSM approach to determine the optimal type of basis function that results in the largest option value and rely on the standard approach to price the instrument. These are important issues to consider when pricing complex interest rate instruments, in general, and callable snowball floaters, in particular.

Recovering the spatial distribution of a groundwater contaminant at a prior time: The forward collocation method

We study the backward parabolic problem related to the convection-diffusion operator A u : ≡ u t − ( D ( x ) u x ) x + ( c ( x ) u ) x when the diffusion coefficient D ( x ) may be discontinuous. The forward collocation method (FC-method) is used for numerical solution of this backward transmission problem. According to the method, we approximate the unknown function ϕ ( x ) = u ( x , t 0 ) by the piecewise linear continuous, Lagrange type of basis functions. Moreover, we solve the obtained ill-conditioned system of algebraic equations by using truncated singular value decomposition (TSVD).

Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data

We introduce a stochastic Galerkin mixed formulation of the steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use with the minimum residual method. We derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters.

Functional principal component analysis via regularized Gaussian basis expansions and its application to unbalanced data

This paper introduces regularized functional principal component analysis for multidimensional functional data sets, utilizing Gaussian basis functions. An essential point in a functional approach via basis expansions is the evaluation of the matrix for the integral of the product of any two bases (cross-product matrix). Advantages of the use of the Gaussian type of basis functions in the functional approach are that its cross-product matrix can be easily calculated, and it creates a much more flexible instrument for transforming each individual's observation into a functional form. The proposed method is applied to the analysis of three-dimensional (3D) protein structural data that can be referred to as unbalanced data. It is shown that our method extracts useful information from unbalanced data through the application. Numerical experiments are conducted to investigate the effectiveness of our method via Gaussian basis functions, compared to the method based on B-splines. On performing regularized functional principal component analysis with B-splines, we also derive the exact form of its cross-product matrix. The numerical results show that our methodology is superior to the method based on B-splines for unbalanced data.

Evaluation of molecular integral of Cartesian Gaussian type basis function with complex‐valued center coordinates and exponent via the McMurchie–Davidson recursion formula, and its application to electron dynamics

AbstractMcMurchie–Davidson recursion formula is extended to derive the ab initio molecular integrals with higher angular quantum number complex Gaussian type basis function which has complex‐valued center coordinates and a complex‐valued exponent. Using the analytical recursion formulae, some calculations of electronic dynamics after beta decay of tritium hydride molecular ion HT+ are performed by a quantum wave packet method with thawed Gaussian basis functions of s‐ and p‐type. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Higher Order Frequency-Domain Computational Electromagnetics

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A review of the higher order computational electromagnetics (CEM) for antenna, wireless, and microwave engineering applications is presented. Higher order CEM techniques use current/field basis functions of higher orders defined on large (e.g., on the order of a wavelength in each dimension) curvilinear geometrical elements, which greatly reduces the number of unknowns for a given problem. The paper reviews all major surface/volume integral- and differential-equation electromagnetic formulations within a higher order computational framework, focusing on frequency-domain solutions. With a systematic and unified review of generalized curved parametric quadrilateral, triangular, hexahedral, and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions, in both divergence- and curl-conforming arrangements, a large number of actual higher order techniques, representing various combinations of formulations, elements, bases, and solution procedures, are identified and discussed. The examples demonstrate the accuracy, efficiency, and versatility of higher order techniques, and their advantages over low-order discretizations, the most important one being a much faster (higher order) convergence of the solution. It is demonstrated that both components of the higher order modeling, namely, higher order geometrical modeling and higher order current/field modeling, are essential for accurate and efficient CEM analysis of general antenna, scattering, and microwave structures. </para>

Resonant frequency of a rectangular patch antenna using asymptotic basis functions

PurposeThe purpose of this paper is to illustrate three types of basis functions and their asymptotic forms defined over the patch to model the unknown currents.Design/methodology/approachA rigorous calculation of a microstrip structure that contains isotropic and anisotropic substrates via Galerkin's method is presented. The dyadic Green's functions of the problem are efficiently determined by the (TM, TE) representation.FindingsThe resonant frequency of a rectangular microstrip patch antenna using these different asymptotic basis functions is investigated. The effect of uniaxial anisotropy in the resonant frequency is presented for these asymptotic currents.Originality/valueComparisons are made, and show that the utilization of the asymptotic basis function provides a significant improvement in the computation time over the exact form in the evaluation of the resonant frequency of a microstrip patch antenna. Convergent solutions are in good agreement with the exact sinusoid basis function without edge condition and with those obtained from literature.

Integrated structure selection and parameter optimisation for eng-genes neural models

The eng-genes concept involves the use of fundamental known system functions as activation functions in a neural model to create a ‘grey-box’ neural network. One of the main issues in eng-genes modelling is to produce a parsimonious model given a model construction criterion. The challenges are that (1) the eng-genes model in most cases is a heterogenous network consisting of more than one type of nonlinear basis functions, and each basis function may have different set of parameters to be optimised; (2) the number of hidden nodes has to be chosen based on a model selection criterion. This is a mixed integer hard problem and this paper investigates the use of a forward selection algorithm to optimise both the network structure and the parameters of the system-derived activation functions. Results are included from case studies performed on a simulated continuously stirred tank reactor process, and using actual data from a pH neutralisation plant. The resulting eng-genes networks demonstrate superior simulation performance and transparency over a range of network sizes when compared to conventional neural models.

Adaptive multiresolution refinement with distance fields

AbstractThis paper describes a multiresolution approach to field modeling that can be used with any meshfree or mesh‐based method for adaptive solution refinement. The refined solution is represented as a superposition of a coarse (unrefined) solution and a sequence of refinements that provide additional degrees of freedom with higher spatial or functional resolution. Each refinement is treated as a solution to a boundary‐value problem within a specified refinement window. The proposed approach is based on the meshfree method with distance fields (Comput. Mech. 2000; 25:305–316, Eng. Comput. 2002; 18(4):295–311) and guarantees Cm continuity of the refined solutions with matching or non‐matching grids. The method does not restrict the shape of the refinement window and does not place any constraints on the type of basis functions, or relative position and resolution of the refinement grids. Combining the proposed approach with hierarchical space decompositions and a posteriori error estimators results in an effective tool for automatic solution refinement. Carefully chosen numerical examples illustrate the power and advantages of the proposed approach. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Accurate Analysis of Arbitrarily Shaped Wire Antenna-Dielectric Radome Structures

In this letter, the performance of wire antennas enclosed by a small/moderate-size dielectric radome with arbitrary shape is investigated by using the coupled surface integral equation (CSIE). PMCHWT formula is applied on the surfaces of the radome, while the EFIE is employed to model the equivalent strip structure of wire antennas. The advantages of this approach over other methods are (1) all of the mutual interactions between antennas and radome are taken into account so that the performance of the composite antenna-radome system can be predicted rigorously and accurately; (2) one can use only one type of basis function (e.g. RWG functions) to represent the equivalent currents on both antenna and radome surfaces; (3) A simple delta-gap feed model can be conveniently incorporated in the method of moments (MoM) with good accuracy. Numerical results of wire antenna covered by radomes with both spherical shell and Von Karman shapes are presented to validate the proposed method.

AIM Analysis of Scattering and Radiation by Arbitrary Surface-Wire Configurations

The adaptive integral method is utilized to solve electromagnetic scattering and radiation problems of conducting surface-wire configurations. The method of moments (MoM) is applied to establish the integral equations where triangular type basis functions are used to represent the currents on surfaces and wires. Attachment mode has been used to model the surface-wire junction to ensure the current continuity. The resultant matrix system is then solved by an iterative solver where the adaptive integral method (AIM) is employed to reduce memory requirements and to accelerate the matrix-vector multiplications. Numerical results are presented to demonstrate the accuracy and efficiency of the present technique for the arbitrary surface-wire configurations

Application of modal wavelets to PIV measurements by selecting basis function

A modal wavelet transform, which overcomes the intrinsic data number limitation of power of two to conventional wavelet transform, has been applied to analysis of axial and eddy pseudo velocity fields, standard PIV velocity field and experimental PIV measurement. The modal wavelet transform is compared with the discrete wavelet transform in order to select the optimum basis function among Neumann, Dirichlet and Green function types basis functions. Consequently, it is verified that Neumann type function is the best basis because the correlation of Neumann type basis function is higher and the root mean square is lower than the other basis functions. Also, the decomposition vector patterns by Neumann type are similar to that by conventional Daubechies basis function of 4th order.

Attosecond electron dynamics with linear combination of floating gaussian type basis function

A new approach to electron dynamics in molecular system is developed, which combines ab initio molecular orbital (MO) scheme and Gaussian wave packet dynamics. The time-dependent electronic wavefunction is described by the linear combination of atomic orbitals (LCAO) using frozen-width floating Gaussian wave packet basis functions. The equations of motion for the time-dependent variables, such as the LCAO coefficients, the Gaussian centers, and their phase factors of the respective basis functions, are determined based on time-dependent variational principle. This technique is demonstrated for the attosecond propagation of electronic wavefunction of hydrogen and helium atoms in response to electrostatic field.

Selection of a quantum-chemical method and basis set for optimization of the complex ion Cu(H2O)+

Calculations of the open-shell van der Waals complex Cu(H2O)+ were carried out at 15 theoretical levels basing on the DFT theory in its variants B3LYP and BLYP. Five types of basis function were used (3-21G*, 3-21G**, 6-31G(d,p), 6-311G(d,p) and CEP-4G) each combined with one (+) or two (++) diffuse functions. The aim of the research was to find out the most accurate combination of method and basis set which predicts structural, energetic and vibration parameters closest to the experimentally found ones. Several experimental parameters were used as reference values.

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

We investigate the viability of dynamical pruning of localized basis sets in time-dependent quantum wave packet methods. Basis functions that have a very small population at any given time are removed from the active set. The basis functions themselves are time independent, but the set of active functions changes in time. Two different types of localized basis functions are tested: discrete variable representation (DVR) functions, which are localized in position space, and phase-space localized (PSL) functions, which are localized in both position and momentum. The number of functions active at each point in time can be as much as an order of magnitude less for dynamical pruning than for static pruning, in reactive scattering calculations of H2 on the Pt(211) stepped surface. Scaling of the dynamically pruned PSL (DP-PSL) bases with dimension is considerably more favorable than for either the primitive (direct product) or DVR bases, and the DP-PSL basis set is predicted to be three orders of magnitude smaller than the primitive basis set in the current state-of-the-art six-dimensional reactive scattering calculations.

Asymptotically derived boundary elements for the Helmholtz equation in high frequencies

We present an asymptotically derived boundary element method for the Helmholtz equation in exterior domains. Each basis function is the product of a smooth amplitude and an oscillatory phase factor, like the asymptotic solution. The phase factor is determined a priori by using arguments from geometrical optics and the geometrical theory of diffraction, while the smooth amplitude is represented by high-order splines. This yields a high-order method in which the number of unknowns is virtually independent of the wavenumber k. Two types of diffracted basis functions are presented: the first accounts for the dominant oscillatory behavior in the shadow region while the second also accounts for the decay of the amplitude there. We show that the matrix A, associated with the discrete problem, has only O ( N ) significant entries as k → ∞ , where N is the number of basis functions. Hence it can be approximated with a matrix A ^ having O ( N ) terms, and the relative error between A and A ^ rapidly converges to zero as k → ∞ . Although the method is applicable to a variety of scatterers, we focus our attention here on scattering from smooth closed convex bodies in two dimensions. Computations on a circular cylinder illustrate our results.

Numerical solution of the CFIE using vector bases and dual interlocking meshes

A numerical solution of the combined-field integral equation for wave scattering from homogeneous dielectric bodies is proposed. The approach uses a simultaneous representation of the equivalent surface current densities in both curl-conforming and divergence-conforming bases, defined on a dual interlocking mesh representing the scatterer surface. The dual-mesh based approach is stable at internal resonances and allows the use of the optimum type of basis function for each integral operator within the combined field equation. The procedure can also be used to obtain a radiation boundary condition for differential equation formulations.

An improved MoM model for line-fed patch antennas and printed circuits

In this paper, an improved model is proposed to analyze the edge-connected line-fed patch antennas and printed circuits based on the method of moments (MoM), where the number of unknowns can be significantly reduced using simplified meshes. In the presented model, three types of basis functions are used to describe the currents on the antenna patch and circuit, the feedline and the feedline-patch junction. A new feedline-patch junction basis function is proposed based on the conventional wire-surface junction basis function. Numerical results are given to illustrate the accuracy and efficiency of the improved MoM model.

Artificial neural network to translate offshore satellite wave data to coastal locations

Owing to the spatial averaging involved in satellite sensing, use of observations so collected is often restricted to offshore regions. This paper discusses a technique to obtain significant wave heights at a specified coastal site from their values gathered by a satellite at deeper offshore locations. The technique is based on the approach of Artificial Neural Network (ANN) of Radial Basis Function (RBF) and Feed-forward Back-propagation (FFBP) type. The satellite-sensed data of significant wave height; average wave period and the wind speed were given as input to the network in order to obtain significant wave heights at a coastal site situated along the west coast of India. Qualitative as well as quantitative comparison of the network output with target observations showed usefulness of the selected networks in such an application vis-à-vis simpler techniques like statistical regression. The basic FFBP network predicted the higher waves more correctly although such a network was less attractive from the point of overall accuracy. Unlike satellite observations collection of buoy data is costly and hence, it is generally resorted to fewer locations and for a smaller period of time. As shown in this study the network can be trained with samples of buoy data and can be further used for routine wave forecasting at coastal locations based on more permanent flow of satellite observations.

Study on the EM scattering of rivets on the plate

The electromagnetic (EM) scattering by rivets on the conducting plate is studied for the first time by using electric field integral equation (EFIE) in conjunction with the moment method. The surfaces of the rivets and the plate are partitioned into triangular cells, the current distribution on the patches is represented by sub-domain type basis function, the EFIE is translated into matrix equation by the Galerkin method, then the current coefficient is obtained. The results show the properties of radar cross section (RCS) varying with the incident angle when there are rivets on the plate.