Uncertainty Quantification for low-frequency Maxwell equations with stochastic conductivity models

Uncertainty Quantification (UQ) has been an active area of research in recent years with a wide range of applications in data and imaging sciences. In many problems, the source of uncertainty stems from an unknown parameter in the model. In physical and engineering systems for example, the parameters of the partial differential equation (PDE) that model the observed data may be unknown or incompletely specified. In such cases, one may use a probabilistic description based on prior information and formulate a forward UQ problem of characterising the uncertainty in the PDE solution and observations in response to that in the parameters. Conversely, inverse UQ encompasses the statistical estimation of the unknown parameters from the available observations, which can be cast as a Bayesian inverse problem. The contributions of the thesis focus on examining the aforementioned forward and inverse UQ problems for the low-frequency, time-harmonic Maxwell equations, where the model uncertainty emanates from the lack of knowledge of the material conductivity parameter. The motivation comes from the Controlled-Source Electromagnetic Method (CSEM) that aims to detect and image hydrocarbon reservoirs by using electromagnetic field (EM) measurements to obtain information about the conductivity profile of the sub-seabed. Traditionally, algorithms for deterministic models have been employed to solve the inverse problem in CSEM by optimisation and regularisation methods, which aside from the image reconstruction provide no quantitative information on the credibility of its features. This work employs instead stochastic models where the conductivity is represented as a lognormal random field, with the objective of providing a more informative characterisation of the model observables and the unknown parameters. The variational formulation of these stochastic models is analysed and proved to be well-posed under suitable assumptions. For computational purposes the stochastic formulation is recast as a deterministic, parametric problem with distributed uncertainty, which leads to an infinite-dimensional integration problem with respect to the prior and posterior measure. One of the main challenges is thus the approximation of these integrals, with the standard choice being some variant of the Monte-Carlo (MC) method. However, such methods typically fail to take advantage of the intrinsic properties of the model and suffer from unsatisfactory convergence rates. Based on recently developed theory on high-dimensional approximation, this thesis advocates the use of Sparse Quadrature (SQ) to tackle the integration problem. For the models considered here and under certain assumptions, we prove that for forward UQ, Sparse Quadrature can attain dimension-independent convergence rates that out-perform MC. Typical CSEM models are large-scale and thus additional effort is made in this work to reduce the cost of obtaining forward solutions for each sampling parameter by utilising the weighted Reduced Basis method (RB) and the Empirical Interpolation Method (EIM). The proposed variant of a combined SQ-EIM-RB algorithm is based on an adaptive selection of training sets and a primal-dual, goal-oriented formulation for the EIM-RB approximation. Numerical examples show that the suggested computational framework can alleviate the computational costs associated with forward UQ for the pertinent large-scale models, thus providing a viable methodology for practical applications

Best practice guide for the assessment of EMF exposure from vehicle Wireless Power Transfer systems

open11sì(Editors: Roberta Guilizzoni, Stuart Harmon, Mauro Zucca)This document is based on the experience gained by the partners involved in the EMPIR Project 16ENG08 "Metrology for inductive charging of electric vehicles (MICEV)" (www.micev.eu). The project addressed the electromagnetic metrology and human exposure problems related to inductive charging of electric vehicles, both from a modelling and a measurement point of view. The guidelines reported here are designed for people who approach the assessment of human exposure in vehicles and around inductive charging stations. These guidelines are intended to complement the published standards in use and those currently being developed by international technical organisations and bodies. This document concerns the charging of electric vehicles, for transmitted power up to 200 kW. The frequency range of interest is related to resonant coils that produce significant electromagnetic field (EMF) emissions from the charging station. Resonant coils operate in the frequency range between 20 kHz and 85 kHz. Their electric current, and thus the magnetic field and harmonic distortion, is very low and not significant in relation to human exposure guidelines. Consequently, the frequency range of interest for human exposure does not exceed 100 kHz. This guide seeks to assemble the experience gained in the field of human exposure assessment and to provide information for the assessment of exposure through experimental measurements and validated calculations. The calculation of the induced quantities, in particular the induced electric field and electric currents in the tissues, is of fundamental importance for the determination of human exposure. From the point of view of dosimetry, for obvious reasons of feasibility, the calculation replaces the measurement. Therefore, a whole chapter of this guide covers the choice of instruments and the description of the correct settings for both the magnetic field calculations and the dosimetric calculations. The document particularly focuses on the following challenges: • the testing framework, including the common layout of charging stations, with reference to the normative and EU Directive on magnetic field exposure (Sections 4 to 6); • means and methods to perform: o measurements of the magnetic flux density in and around a vehicle; o measurements of limb currents (Section 7); • means and methods to perform: o analytical calculation of magnetic flux density levels for EMF exposure assessment; o computation of the induced electric fields in human beings (Section 8). The guidelines contain some appendices, which include the following: a real example of a charging station; some tables with the exposure limits referred to in this guide; a brief comparison between two existing standards; a test case of a numerical code to calculate the sources; some results on the sensitivity of simulated exposure metrics to the variations in tissue properties and, finally, the measurement capabilities of European national metrological institutes concerning AC magnetic fields at the frequency range of interest for Wireless Power Transfer systems (WPTs). These guidelines do not intend to discuss the implementation of wireless charging systems, the design of their components or the optimisation of their performance, as they do not discuss the interoperability or the techniques for building the systems, or their classification. Risk analysis and mitigation measures are beyond the scope of this guideopenAnkarson, Peter; Bottauscio, Oriano; Clarke, Bob; Freschi, Fabio; Guilizzoni, Roberta; Harmon, Stuart; Laporta, Erika; Pichon, Lionel; Bruna Romero, Jorge; Zilberti, Luca; Zucca, MauroAnkarson, Peter; Bottauscio, Oriano; Clarke, Bob; Freschi, Fabio; Guilizzoni, Roberta; Harmon, Stuart; Laporta, Erika; Pichon, Lionel; Bruna Romero, Jorge; Zilberti, Luca; Zucca, Maur

A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography

In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute the voltage-to-current map in an embarrassingly parallel manner, which is an important issue with regard to the corresponding inverse problem. Our method uses the well-known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions. We analyze the global bias and the variance of the new estimator both theoretically and experimentally. In a second step, the variance is considerably reduced via a novel control variate conditional sampling technique

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges

Developing an electron multipacting-free cathode unit of the superconducting radio frequency photoinjector

Future light sources such as synchrotron radiation sources have in common that they require injectors, which provide high-brilliance, high-current electron beams in almost continuous operation. Superconducting radio frequency photoinjector (SRF gun) provided a promising approach. However, some limitations occur caused by electron multipacting in the cathode vicinity, which prevent the superconducting radio frequency photoinjector (SRF gun) from maximum productivity. The aim of this thesis is to develop a new design of the photocathode channel

Kinetic modelling of rarefied gas flows with radiation

Two kinetic models are proposed for high-temperature rarefied (or non-equilibrium) gas flows with radiation. One of the models uses the Boltzmann collision operator to model the translational motion of gas molecules, which has the ability to capture the influence of intermolecular potentials, while the other adopts the relaxation time approximations, which has higher computational efficiency. In the kinetic modelling, not only the transport coefficients such as the shear/bulk viscosity and thermal conductivity but also their fundamental relaxation processes are recovered. Also, the non-equilibrium dynamics of gas flow and radiation are coupled in a self-consistent manner. The two proposed kinetic models are first validated by the direct simulation Monte Carlo method in several non-radiative rarefied gas flows, including the normal shock wave, Fourier flow, Couette flow, and the creep flow driven by the Maxwell demon. Then, the rarefied gas flows with strong radiation are investigated, not only in the above one-dimensional gas flows, but also in the two-dimensional radiative hypersonic flow passing cylinder. In addition to the Knudsen number of gas flow, the influence of the photon Knudsen number and relative radiation strength is scrutinised. It is found that the radiation can make a profound contribution to the total heat transfer on obstacle surface.Comment: 34 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:2201.0685

Simulation and Robust Optimization for Electric Devices with Uncertainties

This dissertation deals with modeling, simulation and optimization of low-frequency electromagnetic devices and quantification of the impact of uncertainties on these devices. The emphasis of these methods is on their application for electric machines. A Permanent Magnet Synchronous Machine (PMSM) is simulated using Iso-Geometric Analysis (IGA). An efficient modeling procedure has been established by incorporating a harmonic stator-rotor coupling. The procedure is found to be stable. Furthermore, it is found that there is strong reduction in computational time with respect to a classical monolithic finite element method. The properties of the ingredients of IGA, i.e. B-splines and Non-Uniform B-Splines, are exploited to conduct a shape optimization for the example of a Stern-Gerlach magnet. It is shown that the IGA framework is a reliable and promising tool for simulating and optimizing electric devices. Different formulations for robust optimization are recalled. The formulations are tested for the optimization of the size of the permanent magnet in a PMSM. It is shown that under the application of linearization the deterministic and the stochastic formulation are equivalent. An efficient deterministic optimization algorithm is constructed by the implementation of an affine decomposition. It is shown that the deterministic algorithm outperforms the widely used stochastic algorithms for this application. Finally, different models to incorporate uncertainties in the simulation of PMSMs are developed. They incorporate different types of rotor eccentricity, uncertainties in the permanent magnets (geometric and material related) and uncertainties that are introduced by the welding processes during the manufacturing. Their influences are studied using stochastic collocation and using the classical Monte Carlo method. Furthermore, the Multilevel Monte Carlo approach is combined with error estimation and applied to determine high dimensional uncertainties in a PMSM