Nonlinear Regression with Logistic Product Basis Networks

A technique for estimating a plasma drift velocity distribution in the ionosphere is presented. This technique is based on a framework for representing a global vector field on a sphere by using a set of localized basis functions which is newly derived as a variant of the spherical elementary current system (SECS). A vector field on a sphere can be divided into its divergence-free (DF) component and curl-free (CF) component. The DF and CF components can then be represented by weighted sums of the DF and CF vector-valued basis functions, respectively. While the SECS basis functions have a singular point, the new basis functions do not diverge over a sphere. This property of the new basis function allows us to achieve robust prediction of the drift velocity at any point in the ionosphere. Assuming that the ionospheric plasma drift velocity has no divergence, its distribution can be represented by a weighted sum of the DF basis functions. The proposed technique estimates the ionospheric plasma drift velocity distribution from the SuperDARN data by using the DF basis functions. Since there are some wide gaps in the spatial coverage of the SuperDARN, an empirical convection model is combined with the framework based on the new basis functions. It is demonstrated that the proposed technique is useful for the estimation and modeling of the ionospheric plasma velocity distribution.

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.

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. >

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 order to describe the change in electronic structure arising from the presence of impurities in a metal or semiconductor it is convenient to have a localized basis representation of the host. To optimize such impurity calculations it is also necessary that this set of localized basis functions be as small as possible. The best representation of the electronic structure of the diamond and zinc‐blende semiconductors in terms of a small set of localized basis functions has been obtained by Chadi. In this method Bloch sums for describing the pseudo‐wave‐functions and energy bands are constructed from Slater orbitals of different symmetry centred on each atomic site. Using a variational approach to determine the decay constants parametrizing these various orbitals Chadi obtains an accurate description of the bandstructures with a simple four state per atom s–p basis set and reliable pseudo‐charge densities with only ten states per atom. These original calculations of Chadi are not, however, carried through to full convergence. Therefore this analysis is repeated for the case of Si and it is found that, while Chadi's results are indeed fortuitous, it is possible to accurately describe both the bandstructure and valence electron charge density of these semiconductors in terms of a minimum set of well localized basis functions.

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.

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.

The possibility of reconstructing two‐dimensional electron‐density profiles in the ionosphere with ionospheric tomography is significant. However, due to the nature of the imaging system, there are several resolution degradation parameters. In order to compensate for these degradation parameters, a priori information must be used. This article introduces the orthogonal decomposition algorithm for image reconstruction, which uses the a priori information to generate a set of orthogonal basis functions for the source domain. This algorithm consists of two simple steps: orthogonal decomposition and recombination. In the development of the algorithm, it is shown that the degradation parameters of the imaging system result in correlations among projections of orthogonal functions. Gram–Schmidt orthogonalization is used to compensate for these correlations, producing a matrix that measures the degradation of the system. Any set of basis functions can be used, and depending upon this choice, the nature of the algorithm varies greatly. Choosing the basis functions of the source domain to be the Fourier kernels produces an algorithm capable of isolating individual frequency components of individual projections. This particular choice of basis functions also results in an algorithm that strongly resembles the direct Fourier method, but without requiring the use of inverse Fourier transforms.

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.

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

A deep learning based reduced order modeling for stochastic underground flow problems

Adaptive Basis Function Construction: An Approach for Adaptive Building of Sparse Polynomial Regression Models

Unsteady mixed convective heat transfer flow past an inclined square cylinder is numerically investigated using a local multi-quadric radial basis function (MQRBF) interpolation. The blockage parameter (ratio of square cylinder length d to height of the computational domain H) varies from 0.025 to 0.2. Air is considered as the working fluid and Prandtl number is fixed at 0.71. Richardson number generally affects the heat transfer efficiency ranging from 0 to 20. Inclined angle of square cylinder ranges from 0° to 45°. The inlet flow is assumed to be laminar and uniform. At the outlet of the computational domain, a convective boundary condition is compared with a traditional Neumann condition. A study of the shape parameter of MQRBF which is sensitive to the distribution of inhomogeneous supporting nodes is provided. The representative streamlines, vortex structures and isotherm patterns are presented and discussed. In addition, the overall lift and drag coefficients, average Nusselt number and Strouhal number for unsteady flow are analyzed for various Reynolds number and Richardson number.

In this paper, adaptive meshfree methods using local nodes and radial basis functions (RBFs) which based on strong-form formulation is presented. In this present formulation, radial basis functions are used in the function approximation for the discretization of the governing system equations. Regularization techniques are suggested and examined to stabilize the solutions in order to obtain stable and accurate results. Different schemes for constructing regularization matrix are compared and discussed. As stability is restored, meshfree strong-form method using local nodes and RBFs can facilitate an easier implementation for adaptive analysis to achieve desired accuracy. Residual based error indicator is devised in the adaptive scheme. A simple and practical node refinement procedure is presented for node insertion at each adaptive step.

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.