Novel optimisation methods for numerical inverse problems

Inverse problems involve the determination of one or more unknown quantities usually appearing in the mathematical formulation of a physical problem. These unknown quantities may be boundary heat flux, various source terms, thermal and material properties, boundary shape and size. Solving inverse problems requires additional information through in-situ data measurements of the field variables of the physical problems. These problems are also ill-posed because the solution itself is sensitive to random errors in the measured input data. Regularisation techniques are usually used in order to deal with the instability of the solution. In the past decades, many methods based on the nonlinear least squares model, both deterministic (CGM) and stochastic (GA, PSO), have been investigated for numerical inverse problems. The goal of this thesis is to examine and explore new techniques for numerical inverse problems. The background theory of population-based heuristic algorithm known as quantum-behaved particle swarm optimisation (QPSO) is re-visited and examined. To enhance the global search ability of QPSO for complex multi-modal problems, several modifications to QPSO are proposed. These include perturbation operation, Gaussian mutation and ring topology model. Several parameter selection methods for these algorithms are proposed. Benchmark functions were used to test the performance of the modified algorithms. To address the high computational cost of complex engineering optimisation problems, two parallel models of the QPSO (master-slave, static subpopulation) were developed for different distributed systems. A hybrid method, which makes use of deterministic (CGM) and stochastic (QPSO) methods, is proposed to improve the estimated solution and the performance of the algorithms for solving the inverse problems. Finally, the proposed methods are used to solve typical problems as appeared in many research papers. The numerical results demonstrate the feasibility and efficiency of QPSO and the global search ability and stability of the modified versions of QPSO. Two novel methods of providing initial guess to CGM with approximated data from QPSO are also proposed for use in the hybrid method and were applied to estimate heat fluxes and boundary shapes. The simultaneous estimation of temperature dependent thermal conductivity and heat capacity was addressed by using QPSO with Gaussian mutation. This combination provides a stable algorithm even with noisy measurements. Comparison of the performance between different methods for solving inverse problems is also presented in this thesis

Sparse spectral methods for integral equations and equilibrium measures

In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems. These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost. This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels. The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces

Towards Bayesian System Identification: With Application to SHM of Offshore Structures

Within the offshore industry Structural Health Monitoring remains a growing area of interest. The oil and gas sectors are faced with ageing infrastructure and are driven by the desire for reliable lifetime extension, whereas the wind energy sector is investing heavily in a large number of structures. This leads to a number of distinct challenges for Structural Health Monitoring which are brought together by one unifying theme --- uncertainty. The offshore environment is highly uncertain, existing structures have not been monitored from construction and the loading and operational conditions they have experienced (among other factors) are not known. For the wind energy sector, high numbers of structures make traditional inspection methods costly and in some cases dangerous due to the inaccessibility of many wind farms. Structural Health Monitoring attempts to address these issues by providing tools to allow automated online assessment of the condition of structures to aid decision making. The work of this thesis presents a number of Bayesian methods which allow system identification, for Structural Health Monitoring, under uncertainty. The Bayesian approach explicitly incorporates prior knowledge that is available and combines this with evidence from observed data to allow the formation of updated beliefs. This is a natural way to approach Structural Health Monitoring, or indeed, many engineering problems. It is reasonable to assume that there is some knowledge available to the engineer before attempting to detect, locate, classify, or model damage on a structure. Having a framework where this knowledge can be exploited, and the uncertainty in that knowledge can be handled rigorously, is a powerful methodology. The problem being that the actual computation of Bayesian results can pose a significant challenge both computationally and in terms of specifying appropriate models. This thesis aims to present a number of Bayesian tools, each of which leverages the power of the Bayesian paradigm to address a different Structural Health Monitoring challenge. Within this work the use of Gaussian Process models is presented as a flexible nonparametric Bayesian approach to regression, which is extended to handle dynamic models within the Gaussian Process NARX framework. The challenge in training Gaussian Process models is seldom discussed and the work shown here aims to offer a quantitative assessment of different learning techniques including discussions on the choice of cost function for optimisation of hyperparameters and the choice of the optimisation algorithm itself. Although rarely considered, the effects of these choices are demonstrated to be important and to inform the use of a Gaussian Process NARX model for wave load identification on offshore structures. The work is not restricted to only Gaussian Process models, but Bayesian state-space models are also used. The novel use of Particle Gibbs for identification of nonlinear oscillators is shown and modifications to this algorithm are applied to handle its specific use in Structural Health Monitoring. Alongside this, the Bayesian state-space model is used to perform joint input-state-parameter inference for Operational Modal Analysis where the use of priors over the parameters and the forcing function (in the form of a Gaussian Process transformed into a state-space representation) provides a methodology for this output-only identification under parameter uncertainty. Interestingly, this method is shown to recover the parameter distributions of the model without compromising the recovery of the loading time-series signal when compared to the case where the parameters are known. Finally, a novel use of an online Bayesian clustering method is presented for performing Structural Health Monitoring in the absence of any available training data. This online method does not require a pre-collected training dataset, nor a model of the structure, and is capable of detecting and classifying a range of operational and damage conditions while in service. This leaves the reader with a toolbox of methods which can be applied, where appropriate, to identification of dynamic systems with a view to Structural Health Monitoring problems within the offshore industry and across engineering

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Nonlinear state and parameter estimation of spatially distributed systems

In this thesis two probabilistic model-based estimators are introduced that allow the reconstruction and identification of space-time continuous physical systems. The Sliced Gaussian Mixture Filter (SGMF) exploits linear substructures in mixed linear/nonlinear systems, and thus is well-suited for identifying various model parameters. The Covariance Bounds Filter (CBF) allows the efficient estimation of widely distributed systems in a decentralized fashion

Detailed Numerical Simulation of Multi-Dimensional Plasma Assisted Combustion

Interaction between flames and plasmas are the guiding thread of this work. Nanosecond Repetitively Pulsed (NRP) discharges are non-thermal plasmas which have shown interesting features for combustion control. They can interact with flames not only through heat, but also chemically by producing active species. In this work, fully-coupled plasma assited combustion simulations are targeted. To achieve this goal, plasma discharge capabilities are built in the low temperature plasma code, AVIP. The corresponding numerical methods, as well as validation cases regarding each set of equations, are first presented. To simulate plasma discharges, the coupled drift-diffusion equations and the Poisson equation are considered. AVIP is coupled to the AVBP code which solves the reactive Navier-Stokes equations to describe combustion phenomena. In a second part, we start by constructing and validating a fully-detailed chemistry for methane-air mixtures in zero-dimensional reactors before reducing it for multi dimensional simulations. The multi-dimensional streamer simulation capabilities of the code are then assessed using simple chemistries. All the validated parts of the code come together in a fully detailed simulation of ignition using NRP discharges. We finish by discussing phenomenological models built upon the knowledge that we gained from fully-detailed simulations. In a last part, finally, attempt to solve the Poisson and generalized Poisson equations using neural networks, which have a potential for speedup compared to classical linear solvers, is carried out

Molecular Dynamics Simulation

Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...

Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen : 20. bis 22.7. 2015, Bauhaus-Universität Weimar

The 20th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering will be held at the Bauhaus University Weimar from 20th till 22nd July 2015. Architects, computer scientists, mathematicians, and engineers from all over the world will meet in Weimar for an interdisciplinary exchange of experiences, to report on their results in research, development and practice and to discuss. The conference covers a broad range of research areas: numerical analysis, function theoretic methods, partial differential equations, continuum mechanics, engineering applications, coupled problems, computer sciences, and related topics. Several plenary lectures in aforementioned areas will take place during the conference. We invite architects, engineers, designers, computer scientists, mathematicians, planners, project managers, and software developers from business, science and research to participate in the conference