Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

In the present paper we introduce a new concept of Hardy type space naturally defined on the Klein-Dirac quadric. We study different properties of the functions belonging to these spaces, in particular boundary value problems. We apply these new spaces to polyharmonic interpolation and to interpolatory cubature formulas.Comment: 32 page

Fast methods for modelling fluid flow and characterising petroleum reservoirs

This thesis tackles three kinds of computationally efficient methods widely applicable in the fields of engineering, simulation and numerical modelling. First, the Non-Intrusive Reduced Order Modelling (NIROM) is discussed, reframed, generalised and tested. While NIROM is a general methodology, the main focus of this work is to evaluate its potential in the field of reservoir modelling. For this purpose a new method for constructing parameterised NIROMs is proposed and the POD-RBF approach is investigated on a number of representative test cases. A detailed analysis concludes with NIROM not being a viable practical solution at this stage; the underlying issues, their causes and future development the method are discussed in detail. Second, a method for classifying well log data is given. The method is an alternative to typical machine learning (ML) approaches, which up to date have been the only tools utilised for the purpose. Our approach is motivated by (and mitigates a number of) issues with applying ML in practical applications, in particular the lack of explainability. Instead of being a complex surrogate with a large number of degrees of freedom (cf ML), our model consists of the automatically re-scaled training set and a single additional number extracted during the training procedure. The technique proposed is characterised by a case-independent design, very high computational efficiency and relies on an intuitively meaningful operating principle; it also provides additional functionality in comparison with alternatives. It is demonstrated that (out of the box) the method outperforms the vast majority of alternatives on a realistic data set in terms of efficiency and accuracy, even when implemented in serial in an interpreted programming language. Finally, the last part of the thesis addresses the issue of efficient semi-analytical modelling of solid boundaries in Smoothed Particle Hydrodynamics (SPH) simulations. More precisely, this work focuses on the purely technical aspect of efficient evaluation of correction factors governing the boundary effects; the framework utilising their values is already well established. Mathematically, the problem is described as efficiently integrating a spherically symmetric function over its compact spherical support truncated by a surface (or a collection of surfaces) representing a solid boundary (wall). Three types of boundary geometries are considered, namely piecewise-planar, spherical and super-ellipsoid/super-toroid surfaces, with the latter two categories addressed for the first time in the literature. All methods provided are characterised by an arbitrary degree of accuracy and simplicity of implementation, especially in comparison with all to up to date alternatives. A number of representative test cases is studied.Open Acces

Forward modelling 3-D geophysical electromagnetic field data with meshfree methods

Simulating geophysical electromagnetic (EM) data over real-life conductivity models requires numerical algorithms that can incorporate realistically complex geometry and topography. The most successful way to incorporate them is to use unstructured meshes in the discretization of an Earth model. Current mesh-based numerical methods that are capable of using such meshes have inherent drawbacks caused by generating 3-D unstructured meshes conforming to irregular geometries. Such a mesh generation process may become computationally expensive and unstable, and particularly so for EM inversion computations in which the forward modelling may be required many times. In this thesis I investigate the feasibility and applicability of radial basis function-based finite difference (RBF-FD), a meshfree method, in forward modelling 3-D EM data. In the meshfree method, the physical model is represented using only a set of unconnected points, effectively overcoming the issues related to the mesh generation. To improve numerical efficiency, unstructured point sets are used in the computation for the first time for EM problems. The computation is further accelerated by introducing a new type of radial basis function in the RBFFD method. The convergence and accuracy of the proposed RBF-FD method are demonstrated first via forward modelling gravity and gravity gradient data. The computational efficiency of the meshfree method is compared with that of using a more traditional finite element method. The meshfree method is then applied to forward model magnetotelluric data of which the effectiveness is demonstrated using three benchmark conductivity models from the literature. Faithful reproduction of the physics in the EM fields, e.g. discontinuous electric fields across the conductivity contrasts, is achieved by proposing a hybrid meshfree scheme which is a modification to standard meshfree algorithms. The hybrid method is also applied to simulate controlled-source EM data in the frame of both total-field and primary-secondary field approaches, in which the problems in dealing with singular source functions that cause singularities in the EM fields are addressed. For these two approaches, the accuracies of the proposed hybrid meshfree method in forward modelling the controlled-source EM data are demonstrated by using idealized 1-D layered models and a 3-D marine canonical disk model. The successful applications of the proposed meshfree method in modelling the above EM data suggest that the meshfree technique has the potential of becoming an important numerical method for simulating EM responses over complicated conductivity models

Gaussian extended cubature formulae for polyharmonic functions

Abstract. The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses m values of linear functionals (integrals over hyperspheres) and is exact for all 2m-harmonic functions, and consequently, for all algebraic polynomials of n variables of degree 4m − 1

Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses ℳ values of linear functional (integrals over hyperspheres) and is exact for all 2ℳ-harmonic functions, and consequently, for all algebraic polynomials of n variables of degree 4ℳ - 1

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal