Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems

LU and Cholesky factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction and implementation of these factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and vice versa. Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these round off errors altogether and to do so efficiently to ensure practicality. Firstly, this work introduces two round off-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations. Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries' bit-length--- i.e., the number of bits required for expression--- is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property. Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demonstrate the predominance of the featured framework over Q-matrices, which comprise an alternative integer-preserving approach relying on the basis adjunct matrix. Thirdly, this work develops special algorithms for updating the REF factorizations. This is necessary because applying the traditional approach to the REF factorizations is inefficient in terms of entry growth and computational effort. In fact, these inefficiencies virtually wipe out all the computational savings commonly expected of factorization updates. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. The featured REF updates are column/row addition, deletion, and replacement

Data Analytics and Performance Enhancement in Edge-Cloud Collaborative Internet of Things Systems

Based on the evolving communications, computing and embedded systems technologies, Internet of Things (IoT) systems can interconnect not only physical users and devices but also virtual services and objects, which have already been applied to many different application scenarios, such as smart home, smart healthcare, and intelligent transportation. With the rapid development, the number of involving devices increases tremendously. The huge number of devices and correspondingly generated data bring critical challenges to the IoT systems. To enhance the overall performance, this thesis aims to address the related technical issues on IoT data processing and physical topology discovery of the subnets self-organized by IoT devices. First of all, the issues on outlier detection and data aggregation are addressed through the development of recursive principal component analysis (R-PCA) based data analysis framework. The framework is developed in a cluster-based structure to fully exploit the spatial correlation of IoT data. Specifically, the sensing devices are gathered into clusters based on spatial data correlation. Edge devices are assigned to the clusters for the R-PCA based outlier detection and data aggregation. The outlier-free and aggregated data are forwarded to the remote cloud server for data reconstruction and storage. Moreover, a data reduction scheme is further proposed to relieve the burden on the trunk link for data uploading by utilizing the temporal data correlation. Kalman filters (KFs) with identical parameters are maintained at the edge and cloud for data prediction. The amount of data uploading is reduced by using the data predicted by the KF in the cloud instead of uploading all the practically measured data. Furthermore, an unmanned aerial vehicle (UAV) assisted IoT system is particularly designed for large-scale monitoring. Wireless sensor nodes are flexibly deployed for environmental sensing and self-organized into wireless sensor networks (WSNs). A physical topology discovery scheme is proposed to construct the physical topology of WSNs in the cloud server to facilitate performance optimization, where the physical topology indicates both the logical connectivity statuses of WSNs and the physical locations of WSN nodes. The physical topology discovery scheme is implemented through the newly developed parallel Metropolis-Hastings random walk based information sampling and network-wide 3D localization algorithms, where UAVs are served as the mobile edge devices and anchor nodes. Based on the physical topology constructed in the cloud, a UAV-enabled spatial data sampling scheme is further proposed to efficiently sample data from the monitoring area by using denoising autoencoder (DAE). By deploying the encoder of DAE at the UAV and decoder in the cloud, the data can be partially sampled from the sensing field and accurately reconstructed in the cloud. In the final part of the thesis, a novel autoencoder (AE) neural network based data outlier detection algorithm is proposed, where both encoder and decoder of AE are deployed at the edge devices. Data outliers can be accurately detected by the large fluctuations in the squared error generated by the data passing through the encoder and decoder of the AE