Point process simulation of generalised inverse Gaussian processes and estimation of the Jaeger integral

In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) L\'{e}vy process. Such processes are intractable for simulation except in certain special edge cases, since the L\'{e}vy density associated with the GIG process is expressed as an integral involving certain Bessel Functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct calculation of the integrals involved. The resulting augmented bivariate process is still intractable and so we propose a novel thinning method based on upper bounds on the intractable integrand. Moreover our approach leads to lower and upper bounds on the Jaeger integral itself, which may be compared with other approximation methods. The shot noise method involves a truncated infinite series of decreasing random variables, and as such is approximate, although the series are found to be rapidly convergent in most cases. We note that the GIG process is the required Brownian motion subordinator for the generalised hyperbolic (GH) L\'{e}vy process and so our simulation approach will straightforwardly extend also to the simulation of these intractable proceses. Our new methods will find application in forward simulation of processes of GIG and GH type, in financial and engineering data, for example, as well as inference for states and parameters of stochastic processes driven by GIG and GH L\'{e}vy processes

Learning Theory and Approximation

Learning theory studies data structures from samples and aims at understanding unknown function relations behind them. This leads to interesting theoretical problems which can be often attacked with methods from Approximation Theory. This workshop - the second one of this type at the MFO - has concentrated on the following recent topics: Learning of manifolds and the geometry of data; sparsity and dimension reduction; error analysis and algorithmic aspects, including kernel based methods for regression and classification; application of multiscale aspects and of refinement algorithms to learning

On Poroelastic and Poro-elasto-plastic Hertzian Contact Problems

A class of contact problems between a rigid sphere and a linear poroelastic or poro-elasto-plastic half-space is examined through an integrated theoretical and numerical analysis to understand the feasibility of spherical indentation as a testing technique for poroelasticity characterization of geomaterials. Fully coupled theoretical solutions are first derived for spherical indentation into a poroelastic half-space with three distinct cases of surface drainage conditions when the indenter is subjected to step displacement loading. The solutions are obtained within the framework of Biot's theory using the McNamee-Gibson displacement function method. Specifically, we overcome the mathematical difficulties associated with evaluating integrals with highly oscillatory kernels by using techniques such as special functions and the method of contour integration. Moreover, the method of successive substitution, instead of the method of quadrature previously used in the literature, is employed to solve the Fredholm integral equation of the second kind to improve solution accuracy. The theoretical analyses show that the normalized transient indentation force response has a relatively weak dependence on material properties through a single derived material constant only. Master curves of indentation force relaxation can be constructed by fitting the full solution with an elementary function for convenient use of poroelasticity characterization in the laboratory. A hydromechanically coupled finite element method (FEM) algorithm following a mixed continuous Galerkin formulation for displacement and pore pressure and incorporating a frictionless contact scheme is then constructed for modeling of spherical indentation in a poro-elasto-plastic medium. Numerical simulations with the step displacement loading condition realized with or without ramping show that the normalized force relaxation responses from the numerical simulations agree very well with the theoretical solutions if the indentation strain and ramping duration are relatively small. For indentation in a poro-elasto-plastic medium, it is shown that even though plasticity could occur immediately at the undrained limit, if the indentation strain and material strength are such that there is no additional plastic strain accumulation during the transient period, the normalized force relaxation behavior could be approximated as poroelastic. Finally, a combined theoretical and numerical analysis is performed to examine the step force loading condition. Results show that the normalized transient displacement response has a relatively stronger dependence on material properties, suggesting that force-controlled tests may be less reliable than displacement-controlled tests for poroelasticity characterization.Ph.D

Learned infinite elements for helioseismology

This thesis presents efficient techniques for integrating the information contained in the Dirichlet-to-Neumann (DtN) map of time-harmonic waves propagating in a stratified medium into finite element discretizations. This task arises in the context of domain decomposition methods, e.g. when reducing a problem posed on an unbounded domain to a bounded computational domain on which the problem can then be discretized. Our focus is on stratified media like the Sun, that allow for strong reflection of waves and for which suitable methods are lacking. We present learned infinite elements as a possible approach to deal with such media utilizing the assumption of a separable geometry. In this case, the DtN map is separable, however, it remains a non-local operator with a dense matrix representation, which renders its direct use computationally inefficient. Therefore, we approximate the DtN only indirectly by adding additional degrees of freedom to the linear system in such a way that the Schur complement w.r.t. the latter provides an optimal approximation of DtN and sparsity of the linear system is preserved. This optimality is ensured via the solution of a small minimization problem, which incorporates solutions of one-dimensional time-harmonic wave equations and allows for great flexibility w.r.t. properties of the medium. In the first half of the thesis we provide an error analysis of the proposed method in a generic framework which demonstrates that exponentially fast convergence rates can be expected. Numerical experiments for the Helmholtz equation and an in-depth study on modelling the solar atmosphere with learned infinite elements demonstrate the high accuracy and flexibility of the proposed method in practical applications. In the second half of the thesis, the potential of learned infinite elements in the context of sweeping preconditioners for the efficient iterative solution of large linear systems is investigated. Even though learned infinite elements are very suitable for separable media, they can only be used for tiny perturbations thereof since the corresponding DtN maps turn out to be extremely sensitive to perturbations in the presence of strong reflections.2021-12-2

Self-consistent charge densities at isolated planar defects in metals

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Advances in Quantum Theory

The quantum theory is the first theoretical approach that helps one to successfully understand the atomic and sub-atomic worlds which are too far from the cognition based on the common intuition or the experience of the daily-life. This is a very coherent theory in which a good system of hypotheses and appropriate mathematical methods allow one to describe exactly the dynamics of the quantum systems whose measurements are systematically affected by objective uncertainties. Thanks to the quantum theory we are able now to use and control new quantum devices and technologies in quantum optics and lasers, quantum electronics and quantum computing or in the modern field of nano-technologies

Material invariant properties of shales : nanoindentation and microporoelastic analysis

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, February 2005.Includes bibliographical references (p. 228-236).Shales compose the major part of sedimentary rocks and cover most of hydrocarbon bearing reservoirs. Shale materials are probably one of the most complex natural composites, and their mechanical properties are still an enigma that has deceived many decoding attempts from experimental and theoretical sides. Advanced experimental techniques, such as nanoindentation, and theoretical microporomechanics make it possible today to break such a heterogeneous material down to a scale where physical chemistry meets mechanics, to extract intrinsic material properties that do not change from one material to another, and to upscale the intrinsic material behavior from the submicroscale to the macroscale of engineering application. This thesis identifies material invariant properties of shales by investigating the elastic properties of shales at multiple scales. We combine new experimental data of shale microstructure and mechanical properties, with nanoindentation analysis and microporomechanics. This leads to the development of a novel multiscale upscaling model for shale poroelasticity. The proposed model relies on a few quantities that can be easily obtained from mineralogy and porosity data. This model is calibrated and validated, and its domain of application and limitations are discussed. The strong predictive capabilities of the model are particularly important for the Oil and Gas Industry, which can apply our predictive model of shale elasticity for geophysics and exploitation engineering applications.by A. Delafargue.S.M

Heat Transfer

Over the past few decades there has been a prolific increase in research and development in area of heat transfer, heat exchangers and their associated technologies. This book is a collection of current research in the above mentioned areas and describes modelling, numerical methods, simulation and information technology with modern ideas and methods to analyse and enhance heat transfer for single and multiphase systems. The topics considered include various basic concepts of heat transfer, the fundamental modes of heat transfer (namely conduction, convection and radiation), thermophysical properties, computational methodologies, control, stabilization and optimization problems, condensation, boiling and freezing, with many real-world problems and important modern applications. The book is divided in four sections : "Inverse, Stabilization and Optimization Problems", "Numerical Methods and Calculations", "Heat Transfer in Mini/Micro Systems", "Energy Transfer and Solid Materials", and each section discusses various issues, methods and applications in accordance with the subjects. The combination of fundamental approach with many important practical applications of current interest will make this book of interest to researchers, scientists, engineers and graduate students in many disciplines, who make use of mathematical modelling, inverse problems, implementation of recently developed numerical methods in this multidisciplinary field as well as to experimental and theoretical researchers in the field of heat and mass transfer