The error and perturbation bounds for the absolute value equations with some applications

To our knowledge, so far, the error and perturbation bounds for the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error bounds and perturbation bounds for two types of absolute value equations (AVEs): Ax-B|x|=b and Ax-|Bx|=b. Some useful error bounds and perturbation bounds for the above two types of absolute value equations are presented. By applying the absolute value equations, we also obtain the error and perturbation bounds for the horizontal linear complementarity problem (HLCP). In addition, a new perturbation bound for the LCP without constraint conditions is given as well, which are weaker than the presented work in [SIAM J. Optim., 2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some computable estimates for the above upper bounds are given, which are sharper than some existing results under certain conditions. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds

Robust convex optimisation techniques for autonomous vehicle vision-based navigation

This thesis investigates new convex optimisation techniques for motion and pose estimation. Numerous computer vision problems can be formulated as optimisation problems. These optimisation problems are generally solved via linear techniques using the singular value decomposition or iterative methods under an L2 norm minimisation. Linear techniques have the advantage of offering a closed-form solution that is simple to implement. The quantity being minimised is, however, not geometrically or statistically meaningful. Conversely, L2 algorithms rely on iterative estimation, where a cost function is minimised using algorithms such as Levenberg-Marquardt, Gauss-Newton, gradient descent or conjugate gradient. The cost functions involved are geometrically interpretable and can statistically be optimal under an assumption of Gaussian noise. However, in addition to their sensitivity to initial conditions, these algorithms are often slow and bear a high probability of getting trapped in a local minimum or producing infeasible solutions, even for small noise levels. In light of the above, in this thesis we focus on developing new techniques for finding solutions via a convex optimisation framework that are globally optimal. Presently convex optimisation techniques in motion estimation have revealed enormous advantages. Indeed, convex optimisation ensures getting a global minimum, and the cost function is geometrically meaningful. Moreover, robust optimisation is a recent approach for optimisation under uncertain data. In recent years the need to cope with uncertain data has become especially acute, particularly where real-world applications are concerned. In such circumstances, robust optimisation aims to recover an optimal solution whose feasibility must be guaranteed for any realisation of the uncertain data. Although many researchers avoid uncertainty due to the added complexity in constructing a robust optimisation model and to lack of knowledge as to the nature of these uncertainties, and especially their propagation, in this thesis robust convex optimisation, while estimating the uncertainties at every step is investigated for the motion estimation problem. First, a solution using convex optimisation coupled to the recursive least squares (RLS) algorithm and the robust H filter is developed for motion estimation. In another solution, uncertainties and their propagation are incorporated in a robust L convex optimisation framework for monocular visual motion estimation. In this solution, robust least squares is combined with a second order cone program (SOCP). A technique to improve the accuracy and the robustness of the fundamental matrix is also investigated in this thesis. This technique uses the covariance intersection approach to fuse feature location uncertainties, which leads to more consistent motion estimates. Loop-closure detection is crucial in improving the robustness of navigation algorithms. In practice, after long navigation in an unknown environment, detecting that a vehicle is in a location it has previously visited gives the opportunity to increase the accuracy and consistency of the estimate. In this context, we have developed an efficient appearance-based method for visual loop-closure detection based on the combination of a Gaussian mixture model with the KD-tree data structure. Deploying this technique for loop-closure detection, a robust L convex posegraph optimisation solution for unmanned aerial vehicle (UAVs) monocular motion estimation is introduced as well. In the literature, most proposed solutions formulate the pose-graph optimisation as a least-squares problem by minimising a cost function using iterative methods. In this work, robust convex optimisation under the L norm is adopted, which efficiently corrects the UAV’s pose after loop-closure detection. To round out the work in this thesis, a system for cooperative monocular visual motion estimation with multiple aerial vehicles is proposed. The cooperative motion estimation employs state-of-the-art approaches for optimisation, individual motion estimation and registration. Three-view geometry algorithms in a convex optimisation framework are deployed on board the monocular vision system for each vehicle. In addition, vehicle-to-vehicle relative pose estimation is performed with a novel robust registration solution in a global optimisation framework. In parallel, and as a complementary solution for the relative pose, a robust non-linear H solution is designed as well to fuse measurements from the UAVs’ on-board inertial sensors with the visual estimates. The suggested contributions have been exhaustively evaluated over a number of real-image data experiments in the laboratory using monocular vision systems and range imaging devices. In this thesis, we propose several solutions towards the goal of robust visual motion estimation using convex optimisation. We show that the convex optimisation framework may be extended to include uncertainty information, to achieve robust and optimal solutions. We observed that convex optimisation is a practical and very appealing alternative to linear techniques and iterative methods

Estimates of remote sensing retrieval errors by the GRASP algorithm: application to ground-based observations, concept and validation

Understanding the uncertainties in the retrieval of aerosol and surface properties is very important for an adequate characterization of the processes that occur in the atmosphere. However, the reliable characterization of the error budget of the retrieval products is a very challenging aspect that currently remains not fully resolved in most remote sensing approaches. The level of uncertainties for the majority of the remote sensing products relies mostly on post-processing validations and intercomparisons with other data, while the dynamic errors are rarely provided. Therefore, implementations of fundamental approaches for generating dynamic retrieval errors and the evaluation of their practical efficiency remains of high importance. This study describes and analyses the dynamic estimates of uncertainties in aerosol-retrieved properties by the GRASP (Generalized Retrieval of Atmosphere and Surface Properties) algorithm. The GRASP inversion algorithm, described by Dubovik et al. (2011, 2014, 2021), is designed based on the concept of statistical optimization and provides dynamic error estimates for all retrieved aerosol and surface properties. The approach takes into account the effect of both random and systematic uncertainties propagations. The algorithm provides error estimates both for directly retrieved parameters included in the retrieval state vector and for the characteristics derived from these parameters. For example, in the case of the aerosol properties, GRASP directly retrieves the size distribution and the refractive index that are used afterwards to provide phase function, scattering, extinction, single scattering albedo, etc. Moreover, the GRASP algorithm provides full covariance matrices, i.e. not only variances of the retrieval errors but also correlations coefficients of these errors. The analysis of the correlation matrix structure can be very useful for identifying less than obvious retrieval tendencies. This appears to be a useful approach for optimizing observation schemes and retrieval set-ups. In this study, we analyse the efficiency of the GRASP error estimation approach for applications to ground-based observations by a sun/sky photometer and lidar. Specifically, diverse aspects of the error generations and their evaluations are discussed and illustrated. The studies rely on a series of comprehensive sensitivity tests when simulated sun/sky photometer measurements and lidar data are perturbed by random and systematic errors and inverted. Then, the results of the retrievals and their error estimations are analysed and evaluated. The tests are conducted for different observations of diverse aerosol types, including biomass burning, urban, dust and their mixtures. The study considers observations of AErosol RObotic NETwork (AERONET) sun/sky photometer measurements at 440, 675, 870 and 1020 nm and multiwavelength elastic lidar measurements at 355, 532 and 1064 nm. The sun/sky photometer data are inverted alone or together with lidar data. The analysis shows overall successful retrievals and error estimations for different aerosol characteristics, including aerosol size distribution, complex refractive index, single scattering albedo, lidar ratios, aerosol vertical profiles, etc. Also, the main observed tendencies in the error dynamic agree with known retrieval experience. For example, the main accuracy limitations for retrievals of all aerosol types relate to the situations with low optical depth. Also, in situations with multicomponent aerosol mixtures, the reliable characterization of each component is possible only in limited situations, for example, from radiometric data obtained for low solar zenith angle observations or from a combination of radiometric and lidar data. At the same time, the total optical properties of aerosol mixtures are always retrieved satisfactorily. In addition, the study includes an analysis of the detailed structure of the correlation matrices for the retrieval errors in mono- and multicomponent aerosols. The conducted analysis of error correlation appears to be a useful approach for optimizing observation schemes and retrieval set-ups. The application of the approach to real data is provided.</p

Optimal experimental design applied to DC resistivity problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 317-323).The systematic design of experiments to optimally query physical systems through manipulation of the data acquisition strategy is termed optimal experimental design (OED). This dissertation introduces the state-of-the-art in OED theory and presents a new design methodology, which is demonstrated by application to DC resistivity problems. The primary goal is to minimize inversion model errors and uncertainties, where the inversion is approached via nonlinear least squares with L1 smoothness constraints. An equally important goal is to find ways to expedite experimental design to make it practical for a wider variety of surveying situations than is currently possible.A fast, sequential ED strategy is introduced that designs surveys accumulatively by an efficient method that maximizes the determinant of the Jacobian matrix. An analysis of electrode geometries for multielectrode data acquisition systems reveals that experiment-space can be usefully decimated by using special subsets of observations, reducing design CPU times. Several techniques for decimating model-space are also considered that reduce design times.A law of diminishing returns is observed; compact, information-dense designed surveys produce smaller model errors than comparably sized random and standard surveys, but as the number of observations increases the utility of designing surveys diminishes. Hence, the prime advantage of OED is its ability to generate small, high-quality surveys whose data are superior for inversion.Designed experiments are examined in a Monte Carlo framework, compared with standard and random experiments on 1D, 2D and borehole DC resistivity problems in both noiseless and noisy data scenarios and for homogeneous and heterogeneous earth models. Adaptive methods are also investigated, where surveys are specifically tailored to a heterogeneous target in real time or in a two-stage process.(cont) The main contributions this thesis makes to geophysical inverse theory are: 1) a fast method of OED that minimizes a measure of total parameter uncertainty; 2) novel techniques of experiment-space and model-space decimation that expedite design times; 3) new methods of adaptive OED that tailor surveys to specific targets; and 4) though the OED method is demonstrated on geoelectrical problems, it can be applied to any inverse problem where the user controls data acquisition.by Darrell A. Coles.Ph.D

A method for the estimation of p-mode parameters from averaged solar oscillation power spectra

A new fitting methodology is presented which is equally well suited for the estimation of low-, medium-, and high-degree mode parameters from $m$-averaged solar oscillation power spectra of widely differing spectral resolution. This method, which we call the "Windowed, MuLTiple-Peak, averaged spectrum", or WMLTP Method, constructs a theoretical profile by convolving the weighted sum of the profiles of the modes appearing in the fitting box with the power spectrum of the window function of the observing run using weights from a leakage matrix that takes into account both observational and physical effects, such as the distortion of modes by solar latitudinal differential rotation. We demonstrate that the WMLTP Method makes substantial improvements in the inferences of the properties of the solar oscillations in comparison with a previous method that employed a single profile to represent each spectral peak. We also present an inversion for the internal solar structure which is based upon 6,366 modes that we have computed using the WMLTP method on the 66-day long 2010 SOHO/MDI Dynamics Run. To improve both the numerical stability and reliability of the inversion we developed a new procedure for the identification and correction of outliers in a frequency data set. We present evidence for a pronounced departure of the sound speed in the outer half of the solar convection zone and in the subsurface shear layer from the radial sound speed profile contained in Model~S of Christensen-Dalsgaard and his collaborators that existed in the rising phase of Solar Cycle~24 during mid-2010

Investigation of the structural response of masonry systems using traditional and data-driven numerical techniques.

Doctoral Degree. University of KwaZUlu-Natal, Durban.The understanding of the structural behaviour of masonry structures is of great importance for the preservation of their structural integrity and restoration. Masonry arches are among the oldest structural systems in the world. The failure of these structures can lead to loss of the architectural inheritance and therefore, a full understanding of their structural behaviour is of paramount importance. Over the years, several approaches have been developed for the investigation of failure of masonry structures. Emphasis is given in the heterogeneous nature of masonry (masonry blocks and mortar joints), which imposes a difficulty in simulating the response of this structural type. Continuum damage and discrete models can be adopted to simulate damage in masonry structures. Finite element analysis is one of the numerical tools, which are widely used for this task. In this thesis, a methodology is proposed for the structural evaluation of masonry systems, such as buildings and arches, using nonlinear finite element analysis. Traditional constitutive descriptions, including non-smooth contact mechanics, as well as damage mechanics, are adopted for the investigation of the ultimate, failure response of masonry structures. Within this framework, the existing interfaces between masonry blocks, standing for potential damage surfaces, are simulated using unilateral contact and friction. To capture the compressive damage mode on the blocks, damage plasticity laws are introduced. Compressive and tensile damage plasticity laws can also be used to simulate the failure response of complex masonry systems. A new approach is also provided in the thesis, relying on data-driven structural engineering using machine learning principles. According to this approach, artificial neural networks are adopted to replace time-consuming numerical simulations, providing a fast and computationally efficient evaluation of the failure response for masonry arches. Datasets are built for this purpose, using finite element analysis simulations. For the implementation of the parametric simulations, which are needed for the development of the datasets, programming codes in Python and Matlab are developed, in collaboration with commercial finite element models. The proposed concept can be adopted to predict the mechanical response, failure load and collapse mechanism of masonry arches and thus, it can be used for the structural health monitoring of these structures. To provide a holistic investigation of the structural response, the thesis focuses on the evaluation of both the static structural and the dynamic response of masonry buildings. Case studies in real structural systems are included, highlighting the applicability and efficiency of the proposed methodologies. In particular, the structural response of a three-span masonry arch bridge in Turkey, as well as the response of a seven-span shipyard building in Greece, has been investigated. Among the outcomes of this thesis, is the evaluation of the collapse mechanisms of multi-span masonry arches, as these compare to the collapse mechanisms of single-span arches. It is proved that a four-hinge failure mechanism arises when a vertical load is applied at the middle arch of a three-span masonry arch bridge, which is a typical response observed on single span masonry arches. It is also noted that a hinge-mechanism is the critical failure pattern for discrete models of multi-span masonry arches, under in-plane and out-of-plane loads. For the structural assessment of masonry buildings, it is proved in this thesis that finite element analysis can be used to explain real and possibly undocumented structural damages experienced by the buildings, due to static and/or dynamic actions. An effort is also made in the thesis, to propose an innovative data-driven methodology, aiming to capture the structural response and collapse mechanism of masonry arches. Thus, it is shown how machine learning can be integrated within structural analysis and used to solve the complex problem of the structural evaluation of circular masonry arches. The computational cost of this methodology is significantly reduced, comparing to conventional finite element simulations. The extension of this concept can be adopted for the structural health monitoring of masonry structures

Applied Mathematics to Mechanisms and Machines

This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics

Numerical Optimisation Problems in Finance

This thesis consists of four projects regarding numerical optimisation and financial derivative pricing. The first project deals with the calibration of the Heston stochastic volatility model. A method using the Levenberg-Marquardt algorithm with the analytical gradient is developed. It is so far the fastest Heston model calibrator and meets the speed requirement of practical trading. In the second project, a triply-nested iterative method for the implementation of interior-point methods for linear programs is proposed. It is the first time that an interior-point method entirely based on iterative solvers succeeds in solving a fairly large number of linear programming instances from benchmark libraries under the standard stopping criteria. The third project extends the Black-Scholes valuation to a complex volatility parameter and presents its singularities at zero and infinity. Fractals that describe the chaotic nature of the Newton-Raphson calculation of the implied volatility are shown for different moneyness values. Among other things, these fractals visualise dramatically the effect of an existing modification for improving the stability and convergence of the search. The project studies scientifically an interesting problem widespread in the financial industry, while revealing artistic values stemming from mathematics. The fourth project investigates the consistency of a class of stochastic volatility models under spot rate inversion, and hence their suitability in the foreign exchange market. The general formula of the model parameters for the inversion rate is given, which provides basis for further investigation. The result is further extended to the affine stochastic volatility model. The Heston model, among the other members in the stochastic volatility family, is the only one that we found to be consistent under the spot inversion. The conclusion on the Heston model verifies the arbitrage opportunity in the variance swap

Hydroinformatics and diversity in hydrological ensemble prediction systems

Nous abordons la prévision probabiliste des débits à partir de deux perspectives basées sur la complémentarité de multiples modèles hydrologiques (diversité). La première exploite une méthodologie hybride basée sur l’évaluation de plusieurs modèles hydrologiques globaux et d’outils d’apprentissage automatique pour la sélection optimale des prédicteurs, alors que la seconde fait recourt à la construction d’ensembles de réseaux de neurones en forçant la diversité. Cette thèse repose sur le concept de la diversité pour développer des méthodologies différentes autour de deux problèmes pouvant être considérés comme complémentaires. La première approche a pour objet la simplification d’un système complexe de prévisions hydrologiques d’ensemble (dont l’acronyme anglais est HEPS) qui dispose de 800 scénarios quotidiens, correspondant à la combinaison d’un modèle de 50 prédictions météorologiques probabilistes et de 16 modèles hydrologiques globaux. Pour la simplification, nous avons exploré quatre techniques: la Linear Correlation Elimination, la Mutual Information, la Backward Greedy Selection et le Nondominated Sorting Genetic Algorithm II (NSGA-II). Nous avons plus particulièrement développé la notion de participation optimale des modèles hydrologiques qui nous renseigne sur le nombre de membres météorologiques représentatifs à utiliser pour chacun des modèles hydrologiques. La seconde approche consiste principalement en la sélection stratifiée des données qui sont à la base de l’élaboration d’un ensemble de réseaux de neurones qui agissent comme autant de prédicteurs. Ainsi, chacun d’entre eux est entraîné avec des entrées tirées de l’application d’une sélection de variables pour différents échantillons stratifiés. Pour cela, nous utilisons la base de données du deuxième et troisième ateliers du projet international MOdel Parameter Estimation eXperiment (MOPEX). En résumé, nous démontrons par ces deux approches que la diversité implicite est efficace dans la configuration d’un HEPS de haute performance.In this thesis, we tackle the problem of streamflow probabilistic forecasting from two different perspectives based on multiple hydrological models collaboration (diversity). The first one favours a hybrid approach for the evaluation of multiple global hydrological models and tools of machine learning for predictors selection, while the second one constructs Artificial Neural Network (ANN) ensembles, forcing diversity within. This thesis is based on the concept of diversity for developing different methodologies around two complementary problems. The first one focused on simplifying, via members selection, a complex Hydrological Ensemble Prediction System (HEPS) that has 800 daily forecast scenarios originating from the combination of 50 meteorological precipitation members and 16 global hydrological models. We explore in depth four techniques: Linear Correlation Elimination, Mutual Information, Backward Greedy Selection, and Nondominated Sorting Genetic Algorithm II (NSGA-II). We propose the optimal hydrological model participation concept that identifies the number of meteorological representative members to propagate into each hydrological model in the simplified HEPS scheme. The second problem consists in the stratified selection of data patterns that are used for training an ANN ensemble or stack. For instance, taken from the database of the second and third MOdel Parameter Estimation eXperiment (MOPEX) workshops, we promoted an ANN prediction stack in which each predictor is trained on input spaces defined by the Input Variable Selection application on different stratified sub-samples. In summary, we demonstrated that implicit diversity in the configuration of a HEPS is efficient in the search for a HEPS of high performance