On Optimal Backward Perturbation Analysis for the Linear System with Skew Circulant Coefficient Matrix

We first give the style spectral decomposition of a special skew circulant matrix C and then get the style decomposition of arbitrary skew circulant matrix by making use of the Kronecker products between the elements of first row in skew circulant and the special skew circulant C. Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Operational modal analysis and prediction of remaining useful life for rotating machinery

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThe significance of rotating machinery spans areas from household items to vital industry sectors, such as aerospace, automotive, railway, sea transport, resource extraction, and manufacturing. Hence, our technologised society depends on efficient and reliable operation of rotating machinery. To contribute to this aim, this thesis leverages measurable quantities during its operation for structural-mechanical evaluation employing Operational Modal Analysis (OMA) and the prediction of Remaining Useful Life (RUL). Modal parameters determined by OMA are central for the design, test, and validation of rotating machinery. This thesis introduces the first open parametric simulation dataset of rotating machinery during an acceleration run. As there is a lack of similar open datasets suitable for OMA, it lays a foundation for improved reproducibility and comparability of future research. Based on this, the Averaged Order-Based Modal Analysis (AOBMA) method is developed. The novel addition of scaling and weighted averaging of individual machine orders in AOBMA alleviates the analysis effort of the existing Order-Based Modal Analysis (OBMA) method by providing a unified set of modal parameters with higher accuracy. As such, AOBMA showed a lower mean absolute relative error of 0.03 for damping ratio estimations across compared modes while OBMA provided an error value of 0.32 depending on the processed order. At excitation with high harmonic contributions, AOBMA also resulted in the highest number of accurately identified modes among the compared methods. At a harmonic ratio of 0.8, for example, AOBMA identified an average of 11.9 modes per estimation, while OBMA and baseline OMA followed with 9.5 and 9 modes, respectively. Moreover, it is the first study, which systematically evaluates the impact of excitation conditions on the compared methods and finds an advantage of OBMA and AOBMA over traditional OMA regarding mode shape estimation accuracy. While OMA can be used to evaluate significant structural changes, Machine Learning (ML) methods have seen substantially greater success in condition monitoring, including RUL prediction. However, as these methods often require large amounts of time and cost- intensive training data, a novel data-efficient RUL prediction methodology is introduced, taking advantage of distinct healthy and faulty condition data. When the number of training sequences from an open dataset is reduced to 5%, an average prediction Root Mean Square Error (RMSE) of 24.9 operation cycles is achieved, outperforming the baseline method with an RMSE of 28.1. Motivated by environmental considerations, the impact of data reduction on the training duration of several method variants is quantified. When the full training set is utilised, the most resource-saving variant of the proposed approach achieves an average training duration of 8.9% compared to the baseline method

Robust interactive simulation of deformable solids with detailed geometry using corotational FEM

This thesis focuses on the interactive simulation of highly detailed deformable solids modelled with the Corotational Finite Element Method. Starting from continuum mechanics we derive the discrete equations of motion and present a simulation scheme with support for user-in-the-loop interaction, geometric constraints and contact treatment. The interplay between accuracy and computational cost is discussed in depth, and practical approximations are analyzed with an emphasis on robustness and efficiency, as required by interactive simulation. The first part of the thesis focuses on deformable material discretization using the Finite Element Method with simplex elements and a corotational linear constitutive model, and presents our contributions to the solution of widely reported robustness problems in case of large stretch deformations and finite element degeneration. First,we introduce a stress differential approximation for quasi-implicit corotational linear FEM that improves its results for large deformations and closely matches the fullyimplicit solution with minor computational overhead. Next, we address the problem ofrobustness and realism in simulations involving element degeneration, and show that existing methods have previously unreported flaws that seriously threaten robustness and physical plausibility in interactive applications. We propose a new continuous-time approach, degeneration-aware polar decomposition, that avoids such flaws and yields robust degeneration recovery. In the second part we focus on geometry representation and contact determination for deformable solids with highly detailed surfaces. Given a high resolution closed surface mesh we automatically build a coarse embedding tetrahedralization and a partitioned representation of the collision geometry in a preprocess. During simulation, our proposed contact determination algorithm finds all intersecting pairs of deformed triangles using a memory-efficient barycentric bounding volume hierarchy, connects them into potentially disjoint intersection curves and performs a topological flood process on the exact intersection surfaces to discover a minimal set of contact points. A novel contact normal definition is used to find contact point correspondences suitable for contact treatment.Aquesta tesi tracta sobre la simulació interactiva de sòlids deformables amb superfícies detallades, modelats amb el Mètode dels Elements Finits (FEM) Corotacionals. A partir de la mecànica del continuu derivem les equacions del moviment discretes i presentem un esquema de simulació amb suport per a interacció d'usuari, restriccions geomètriques i tractament de contactes. Aprofundim en la interrelació entre precisió i cost de computació, i analitzem aproximacions pràctiques fent èmfasi en la robustesa i l'eficiència necessàries per a la simulació interactiva. La primera part de la tesi es centra en la discretització del material deformable mitjançant el Mètode dels Elements Finits amb elements de tipus s'implex i un model constituent basat en elasticitat linial corotacional, i presenta les nostres contribucions a la solució de problemes de robustesa àmpliament coneguts que apareixen en cas de sobreelongament i degeneració dels elements finits. Primer introduïm una aproximació dels diferencials d'estress per a FEM linial corotacional amb integració quasi-implícita que en millora els resultats per a deformacions grans i s'apropa a la solució implícita amb un baix cost computacional. A continuació tractem el problema de la robustesa i el realisme en simulacions que inclouen degeneració d'elements finits, i mostrem que els mètodes existents presenten inconvenients que posen en perill la robustesa plausibilitat de la simulació en aplicacions interactives. Proposem un enfocament nou basat en temps continuu, la descomposició polar amb coneixement de degeneració, que evita els inconvenients esmentats i permet corregir la degeneració de forma robusta. A la segona part de la tesi ens centrem en la representació de geometria i la determinació de contactes per a sòlids deformables amb superfícies detallades. A partir d'una malla de superfície tancada construím una tetraedralització englobant de forma automàtica en un preprocés, i particionem la geometria de colisió. Proposem un algorisme de detecció de contactes que troba tots els parells de triangles deformats que intersecten mitjançant una jerarquia de volums englobants en coordenades baricèntriques, els connecta en corbes d'intersecció potencialment disjuntes i realitza un procés d'inundació topològica sobre les superfícies d'intersecció exactes per tal de descobrir un conjunt mínim de punts de contacte. Usem una definició nova de la normal de contacte per tal de calcular correspondències entre punts de contacte útils per al seu tractament.Postprint (published version

Optimal nonlinear control and estimation using global domain linearization

Alan Turing teaches that cognition is symbol processing. Norbert Wiener teaches that intelligence rests on feedback control. Thus, there are discrete symbols and continuous sensory-motor signals. Sensorimotor dynamics are well-represented by nonlinear differential equations. A possible construction of symbols could be based on equilibria. Language is a symbol system and is one of the highest expressions of cognition. Much of this comes from spatial reasoning, which requires embodied cognition. Spatial reasoning derives from motor function. This thesis introduces a novel generalized non-heuristic method of linearizing nonlinear differential equations over a finite domain. It is used to engineer optimal convergence to target sets, a general form of spatial reasoning. Ordinary differential equations are ubiquitous models in physics and engineering that describe a wide range of phenomena including electromechanical systems. This thesis considers ordinary differential equations expressed in state-space form. For a given initial state, these equations generate signals that are continuous in both time and state. The control engineering objective is to find input functions that steer these states to desired target sets using only the measured output of the system. The state-space domain containing the target set, along with its cost, can be thought of as a symbol for high-level planning. Consider a basis state equal to a vector of nonlinear basis functions, computed from the state, where the state is generated from a multivariate nonlinear dynamic system. The basis state derivative can be expressed as a linear dynamic system with an additional error term. This thesis describes radial basis functions that minimize the error over the entire state-space domain, where the basis state equals zero if and only if the state is in a desired target set. This gives an approximate linear-dynamic system, and if the basis state goes to zero, then the state goes to the target set. This form of linear approximation is global over the domain. Careful selection of the basis gives a fully generalizable relationship between linear stability and nonlinear stability. This form of linearization can be applied to optimal state feedback and state estimation problems. This thesis carefully introduces optimal state feedback control with an emphasis on optimal infinite horizon solutions to linear-dynamic systems that have quadratic cost. A thorough introduction is also given to the optimal output feedback of linear-dynamics systems. The detectable and stabilizable subspaces of a linear-dynamic system are expressed in a generalized closed form. After introducing optimal control for linear systems, this thesis explores adaptive control from several different perspectives including: tuning, system identification, and reinforcement learning. Each of these approaches can be characterized as an optimal nonlinear output feedback problem. In each case, generalized representations can be found using a single layer of appropriately chosen nonlinear basis functions with linear parameterization. The primary focus of this thesis is to select these basis functions, in a fully generalized way, so that they have linear-dynamics. When this can be achieved, infinite horizon state feedback and state estimation can be computed using well-known closed-form solutions. This thesis demonstrates how multivariate nonlinear dynamic systems defined on a finite domain can be approximated by computationally equivalent high-dimensional linear-dynamic systems using a generalized basis state. This basis state is computed with a single layer of biologically inspired radial basis functions. The method of linearization is described as "global domain linearization" because it holds over a specified domain, and therefore provides a global linear approximation with respect to that domain. Any optimal state estimation or state feedback is globally optimal over the domain of linearization. The tools of optimal linear control theory can be applied. In particular, control and estimation problems involving under-actuated under-measured nonlinear systems with generalized nonlinear reward can be solved with closed-form infinite horizon linear-quadratic control and estimation. The controllable, uncontrollable, stabilizable, observable, unobservable, and detectable subspaces can all be described in a meaningful generalized way. State estimation and state feedback can then be implemented in computationally efficient low-dimensional highly nonlinear form. Generalized optimal state estimation and state feedback for continuous-time continuous-state systems is necessary machinery for any high-level symbolic planning that might involve unstable electromechanical systems. Symbols naturally form in the presence of more than one target state. This could provide a natural method of language acquisition. Given a state, all symbolic domains that intersect the state would have equilibrium and cost. These intersections define the legal grammar of symbol transition. An engineer or agent can design these symbols for high-level planning. Generalized infinite horizon state feedback and state estimation can then be computed for the continuous system that each symbol represents using traditional linear tools with domain linearization

Bayesian matrix factorisation: inference, priors, and data integration

In recent years the amount of biological data has increased exponentially. Most of these data can be represented as matrices relating two different entity types, such as drug-target interactions (relating drugs to protein targets), gene expression profiles (relating drugs or cell lines to genes), and drug sensitivity values (relating drugs to cell lines). Not only the size of these datasets is increasing, but also the number of different entity types that they relate. Furthermore, not all values in these datasets are typically observed, and some are very sparse. Matrix factorisation is a popular group of methods that can be used to analyse these matrices. The idea is that each matrix can be decomposed into two or more smaller matrices, such that their product approximates the original one. This factorisation of the data reveals patterns in the matrix, and gives us a lower-dimensional representation. Not only can we use this technique to identify clusters and other biological signals, we can also predict the unobserved entries, allowing us to prune biological experiments. In this thesis we introduce and explore several Bayesian matrix factorisation models, focusing on how to best use them for predicting these missing values in biological datasets. Our main hypothesis is that matrix factorisation methods, and in particular Bayesian variants, are an extremely powerful paradigm for predicting values in biological datasets, as well as other applications, and especially for sparse and noisy data. We demonstrate the competitiveness of these approaches compared to other state-of-the-art methods, and explore the conditions under which they perform the best. We consider several aspects of the Bayesian approach to matrix factorisation. Firstly, the effect of inference approaches that are used to find the factorisation on predictive performance. Secondly, we identify different likelihood and Bayesian prior choices that we can use for these models, and explore when they are most appropriate. Finally, we introduce a Bayesian matrix factorisation model that can be used to integrate multiple biological datasets, and hence improve predictions. This model hybridly combines different matrix factorisation models and Bayesian priors. Through these models and experiments we support our hypothesis and provide novel insights into the best ways to use Bayesian matrix factorisation methods for predictive purposes.UK Engineering and Physical Sciences Research Council (EPSRC), grant reference EP/M506485/1