A framework for estimating spherical vector fields using localized basis functions and its application to SuperDARN data processing

Abstract. The geoelectric field drives geomagnetically induced currents (GICs) in technological conductor networks, which can affect the performance of critical ground infrastructure such as electric power transmission grids. The three-dimensional (3-D) electric field at the Earth's surface consists of an external divergence-free (DF) part due to temporally and spatially varying ionospheric and magnetospheric currents, an internal DF part due to temporally and spatially varying telluric currents, and a curl-free (CF) part due to charge accumulation at ground conductivity gradients. We have developed a new method for estimating these contributions. The external and internal parts of the DF electric field are calculated from the time derivative of the external and internal parts of the observed ground magnetic field, respectively, using DF two-dimensional (2-D) spherical elementary current systems (SECSs). The horizontal surface CF electric field is calculated from the known surface DF electric field using coefficients that linearly relate the DF electric field to the CF electric field. The coefficients were obtained from the 3-D induction model PGIEM2G (Polynomial Galerkin Integral Equation Modelling in ElectroMagnetic Geophysics). The calculations are carried out in the time domain, and only two consecutive time steps of the observed magnetic field are needed to compute the surface electric field. The external part of the DF electric field is valid at and below the ionosphere, the internal part at and above the Earth's surface, and the CF part at the Earth's surface. A dense magnetometer network is a requirement for reliable results. The external and internal parts of the DF electric field are generally oppositely directed and have comparable amplitudes, both on the ground and in the ionosphere, indicating that both contributions are significant for the total DF electric field. The largest peaks of the total DF electric field tend to occur when either the external or internal contribution is temporarily suppressed. At a given location, a DF electric field with a given amplitude can result in a total surface electric field amplitude with a difference of orders of magnitude depending on the direction of the DF electric field with respect to the locally dominant conductivity gradient structure. The electric field calculation is computationally light, facilitating operational implementation of a near-real-time 3-D surface electric field monitoring and derivation of long electric field time series.

A discrete representation of vibrational molecular wavefunctions in coordinate space offers many advantages for the solution of problems in quantum dynamics, in particular if realistic, multidimensional potential energy surfaces are involved. This leads to the problem of how to construct localized representations of operators. While any set of grid points might be defined for a discrete variable representation (DVR) of local operators this is not true for the Hamiltonian operator as a whole. As a consequence of the non-local nature of the kinetic energy operator a correspondence between the localized picture and a delocalized variational basis representation (VBR) or momentum space, respectively, must be established. The arrangement of grid points is therefore determined by the properties of the VBR basis functions. It is shown that linear matrix transformations between localized and delocalized representations and the so-called Fourier method for the evaluation of non-local operations can essentially be reduced to the same idea: localization transformations of orthogonal functions. Gaussian quadrature is a classical way to obtain localized basis functions derived from polynomials. We investigate the properties of the localized functions related to important polynomials and obtain a new relation for Legendre polynomials. VBR basis functions related to equidistant quadrature points, like particle-in-a-box or Fourier functions, are frequently applied and the associated localized basis functions are shown to possess interesting symmetry features. We give a mathematically consistent theory of matrix transformations between localized and delocalized pictures: a new representation, called nearly discrete variable representation (NDVR), is introduced, that is equivalent to the VBR, in analogy to the correspondence of DVR and finite basis representation (FBR). DVR basis functions are defined in a new way via delta sequences. Finally, a brief analysis of the error made in calculations with a DVR potential matrix is given for two simple one-dimensional models.

In general, there is a great difference between usual three-layer feedforward neural networks with local basis functions in the hidden processing elements and those with standard sigmoidal transfer functions (in the following often called global basis functions). The reason for this difference in nature can be seen in the ridge-type arguments which are commonly used. It is the aim of this paper to show that the situation completely changes when instead of ridge-type arguments were so-called hyperbolic-type arguments. In detail, we show that usual sigmoidal transfer functions evaluated at hyperbolic-type arguments—usually called sigma—pi units—can be used to construct local basis functions which vanish at infinity and, moreover, are integrable and give rise to a partition of unity, both in Cauchy's principal value sense. At this point, standard strategies for approximation with local basis functions can be used without giving up the concept of non-local sigmoidal transfer functions.

The problem of identification of a nonstationary stochastic system is considered and solved using local basis function approximation of system parameter trajectories. Unlike the classical basis function approach, which yields parameter estimates in the entire analysis interval, the proposed new identification procedure is operated in a sliding window mode and provides a sequence of point (rather than interval) estimates. It is shown that for the polynomial basis all computations can be carried out recursively and that two important design parameters – the number of basis functions and the size of the local analysis window – can be chosen in an adaptive way.

Summary Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use both a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features such as faults, fractures, facies, or other geobodies; or a particular computational challenge such as propagating displacement and chemical fronts, wells being turned on or off, and others. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a nonoverlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse gridblock. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators—computed on different coarse grids and possibly also by different basis-function formulations—to be combined into one iterative procedure. Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators that have different spatial resolution and that each individual operator can be designed to target, among others, challenging grids, including faults, pinchouts, and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.

Natural basis functions (NBF), also known as natural pixels in the literature, have been applied in tomographic reconstructions of soft x-ray (SXR) measurements in the JET (Joint European Torus) tokamak. The results are compared with those obtained with local basis functions (LBF), and those obtained with a conventional constrained-optimization tomography method. Truncated singular value decomposition is used as the inversion method. Reconstructions without a priori information, such as the NBF reconstructions, are, as can be expected, less good than reconstructions in which a priori information is used, as in the conventional method, but the reconstructions are shown to be reliable by means of phantom simulations. Reconstructions with the same number of LBFs as NBFs without a priori information are comparable to NBF reconstructions, although the latter seem to be somewhat better. No significant changes in results, apart from smaller reconstruction errors, are obtained if the measuring system has more regular or complete coverage than the JET SXR system. The various tomography methods are used to assess whether a newly observed in-out asymmetry in the SXR emission during the injection of nickel into an RF-heated plasma, with the peak on the inboard side, is real. A possible explanation for the asymmetry in emissivity is an increased nickel density on the inboard side as a result of an RF-induced increase of the hydrogen-minority density on the outboard side.

There is significant interest in the interplay between fuzzy systems and neural networks. Jang and Sun (1993) established the functional equivalence of Gaussian radial basis function (RBF) networks and a restricted class of Takagi-Sugeno-type (1985) fuzzy systems. This result was extended to the full TS-model by Hunt et al. (1994) who employed networks with local models and ellipsoidal basis functions. The restriction to Gaussian type basis functions, and therefore to Gaussian-shaped fuzzy membership functions, was later removed by Hunt et al. through employment of spline-based networks. This covers fuzzy systems with a broad range of membership function shapes (triangular and trapezoidal shapes are common special cases). In this paper we present a generalised form of the functional equivalence theorem and discuss its relevance for the direct implementation of fuzzy control systems in the form of neural networks. >

We introduce a novel general regression model that is based on a linear combination of a new set of non-local basis functions that forms an effective feature space. We propose a training algorithm that learns all the model parameters simultaneously and offer an initialization scheme for parameters of the basis functions. We show through several experiments that the proposed method offers better coverage for high-dimensional space compared to local Gaussian basis functions and provides competitive performance in comparison to other state-of-the-art regression methods.

Singular value decomposition has served as a diagnostic tool in optical computed tomography by using its capability to provide insight into the condition of ill-posed inverse problems. Various tomographic geometries are compared to one another through the singular value spectrum of their weight matrices. The number of significant singular values in the singular value spectrum of a weight matrix is a quantitative measure of the condition of the system of linear equations defined by a tomographic geometery. The analysis involves variation of the following five parameters, characterizing a tomographic geometry: 1) the spatial resolution of the reconstruction domain, 2) the number of views, 3) the number of projection rays per view, 4) the total observation angle spanned by the views, and 5) the selected basis function. Five local basis functions are considered: the square pulse, the triangle, the cubic B-spline, the Hanning window, and the Gaussian distribution. Also items like the presence of noise in the views, the coding accuracy of the weight matrix, as well as the accuracy of the accuracy of the singular value decomposition procedure itself are assessed.

We study families of holomorphic vector fields, holomorphically depending on parameters, in a neighborhood of an isolated singular point. When the singular point is in the Poincaré domain for every vector field of the family we prove, through a modification of classical Sternberg's linearization argument, cf. Nelson (1969) [7] too, analytic dependence on parameters of the linearizing maps and geometric bounds on the linearization domain: each vector field of the family is linearizable inside the smallest Euclidean sphere which is not transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko (2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6], a version of Brjuno's Theorem in the case of linearization of families of vector fields near a singular point of Siegel type, and apply it to study some 1-parameter families of vector fields in two dimensions.

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

A quadratic trigonometric B-Spline as an alternate to cubic B-spline

Free vibration analysis of non-axisymmetric and axisymmetric structures by the dual reciprocity BEM

This article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

Linear embedding via Green’s operators (LEGO) is a domain decomposition method particularly well suited for the solution of scattering and radiation problems comprised of many objects. The latter are enclosed in simple-shaped subdomains (electromagnetic bricks) which are in turn described by means of scattering operators. In this paper we outline the extension of the LEGO approach to the case of penetrable objects with dyadic permittivity or permeability. Since a volume integral equation is only required to solve the scattering problem inside a brick and the scattering operators are inherently surface operators, the LEGO procedureper secan afford a reduction of the number of unknowns in the numerical solution with the Method of Moments and subsectional basis functions. Further substantial reduction is achieved with the eigencurrents expansion method (EEM) which employs the eigenvectors of the scattering operator as local entire-domain basis functions over a brick’s surface. Through a few selected numerical examples we discuss the validation and the efficiency of the LEGO-EEM technique applied to clusters of anisotropic bodies.