Quantum and Classical Multilevel Algorithms for (Hyper)Graphs

Combinatorial optimization problems on (hyper)graphs are ubiquitous in science and industry. Because many of these problems are NP-hard, development of sophisticated heuristics is of utmost importance for practical problems. In recent years, the emergence of Noisy Intermediate-Scale Quantum (NISQ) computers has opened up the opportunity to dramaticaly speedup combinatorial optimization. However, the adoption of NISQ devices is impeded by their severe limitations, both in terms of the number of qubits, as well as in their quality. NISQ devices are widely expected to have no more than hundreds to thousands of qubits with very limited error-correction, imposing a strict limit on the size and the structure of the problems that can be tackled directly. A natural solution to this issue is hybrid quantum-classical algorithms that combine a NISQ device with a classical machine with the goal of capturing “the best of both worlds”. Being motivated by lack of high quality optimization solvers for hypergraph partitioning, in this thesis, we begin by discussing classical multilevel approaches for this problem. We present a novel relaxation-based vertex similarity measure termed algebraic distance for hypergraphs and the coarsening schemes based on it. Extending the multilevel method to include quantum optimization routines, we present Quantum Local Search (QLS) – a hybrid iterative improvement approach that is inspired by the classical local search approaches. Next, we introduce the Multilevel Quantum Local Search (ML-QLS) that incorporates the quantum-enhanced iterative improvement scheme introduced in QLS within the multilevel framework, as well as several techniques to further understand and improve the effectiveness of Quantum Approximate Optimization Algorithm used throughout our work

MULTIVARIATE CALIBRATION FOR ICP-AES

The analysis of metals is now a major application area for ICP-AES, however, the technique suffers from both spectral and non-spectral interferences. This thesis details the application of univariate and multivariate calibration methods for the prediction of Pt, Pd, and Rh in acid-digested and of Au, Ag and Pd in fusion-digested autocatalyst samples. Of all the univariate calibration methods investigated matrix matching proved the most accurate method with relative root mean square errors (RRMSEs) for Pt, Pd and Rh of 2.4, 3.7, and 2.4 % for a series of synihelic lest solutions, and 12.0, 2.4, and 8.0 % for autocatalyst samples. In comparison, the multivariate calibration method (PLSl) yielded average relative errors for Pt, Pd, and RJi of 5.8, 3.0, and 3.5 % in the test solutions, and 32.0, 7.5, and 75.0 % in the autocatalyst samples. A variable selection procedure has been developed enabling multivariate models to be built using large parts of the atomic emission spectrum. The first stage identified and removed wavelengths whose PLS regression coefficients were equal to zero. The second stage ranked the remaining wavelengths according to their PLS regression coefficient and estimated standard error ratio. The algorithms were applied to the emission spectra for the determination of Pt, Pd and Rh in a synthetic matrix. For independent test samples variable selection gave RRMSEs of 5.3, 2.5 and 1.7 % for Pt, Pd and Rh respectively compared with 8.3, 7.0 and 3.1 % when using integrated atomic emission lines. Variable selection was then applied for the prediction of Au, Ag and Pd in independent test fusion digests. This resulted in RRMSEs of 74.2, 8.8 and 12.2 % for Au, Ag and Pd respectively which were comparable to those obtained using a more traditional univariate calibration approach. A preliminary study has shown that calibration drift can be corrected using Piecewise Direct Standardisation (PDS). The application of PDS to synthetic test samples analysed 10 days apart resulted in RRMSEs of 4.14, 3.03 and 1.88%, compared to 73.04, 44.39 and 28.06 % without correction, for Pt, Pd, and Rh respectively.The Analytical Innovation Programme, Johnson Matthey Ltd. and Department of Trade and Industr

Multivariate Analysis in Management, Engineering and the Sciences

Recently statistical knowledge has become an important requirement and occupies a prominent position in the exercise of various professions. In the real world, the processes have a large volume of data and are naturally multivariate and as such, require a proper treatment. For these conditions it is difficult or practically impossible to use methods of univariate statistics. The wide application of multivariate techniques and the need to spread them more fully in the academic and the business justify the creation of this book. The objective is to demonstrate interdisciplinary applications to identify patterns, trends, association sand dependencies, in the areas of Management, Engineering and Sciences. The book is addressed to both practicing professionals and researchers in the field