Randomized Rank-Revealing QLP for Low-Rank Matrix Decomposition

The pivoted QLP decomposition is computed through two consecutive pivoted QR decompositions, and provides an approximation to the singular value decomposition. This work is concerned with a partial QLP decomposition of low-rank matrices computed through randomization, termed Randomized Unpivoted QLP (RU-QLP). Like pivoted QLP, RU-QLP is rank-revealing and yet it utilizes random column sampling and the unpivoted QR decomposition. The latter modifications allow RU-QLP to be highly parallelizable on modern computational platforms. We provide an analysis for RU-QLP, deriving bounds in spectral and Frobenius norms on: i) the rank-revealing property; ii) principal angles between approximate subspaces and exact singular subspaces and vectors; and iii) low-rank approximation errors. Effectiveness of the bounds is illustrated through numerical tests. We further use a modern, multicore machine equipped with a GPU to demonstrate the efficiency of RU-QLP. Our results show that compared to the randomized SVD, RU-QLP achieves a speedup of up to 7.1 times on the CPU and up to 2.3 times with the GPU

Likelihood-informed dimension reduction for nonlinear inverse problems

The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively low-dimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the "likelihood-informed subspace" (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao-Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations

A randomized algorithm for the QR decomposition-based approximate SVD

Matrix decomposition is a very important mathematical tool in numerical linear algebra for data processing. In this paper, we introduce a new randomized matrix decomposition algorithm, which is called randomized approximate SVD based on Qatar Riyal decomposition (RCSVD-QR). Our method utilize random sampling and the OR decomposition to address a serious bottlenck associated with classical SVD. RCSVD-QR gives satisfactory convergence speed as well as accuracy as compared to those state-of-the-art algorithms. In addition, we provides an estimate for the expected approximation error in Frobenius norm. Numerical experiments verify these claims.Comment: 6 pages,6 figure

Advanced control systems for fast orbit feedback of synchrotron electron beams

Diamond Light Source is the UK’s national synchrotron facility that produces synchrotron radiation for research. At source points of synchrotron radiation, the electron beam stability relative to the beam size is critical for the optimal performance of synchrotrons. The current requirement at Diamond is that variations in the beam position should not exceed 10% of the beam size for frequencies up to 140Hz. This is guaranteed by the fast orbit feedback that actuates hundreds of corrector magnets at a sampling rate of 10kHz to reduce beam vibrations down to sub-micron levels. For the next-generation upgrade, Diamond-II, the beam stability requirements will be raised to 3% up to 1kHz. Consequently, the sampling rate will be increased to 100kHz and an additional array of fast correctors will be introduced, which precludes the use of the existing controller. This thesis develops two different control approaches to accommodate the additional array of fast correctors at Diamond-II: internal model control based on the generalised singular value decomposition (GSVD) and model predictive control (MPC). In contrast to existing controllers, the proposed approaches treat the control problem as a whole and consider both arrays simultaneously. To achieve the sampling rate of 100kHz, this thesis proposes to reduce the computational complexity of the controllers in several ways, such as by exploiting symmetries of the magnetic lattice. To validate the controllers for Diamond-II, a real-time control system is implemented on high-performance hardware and integrated in the existing synchrotron. As a first-of-its-kind application to electron beam stabilisation in synchrotrons, this thesis presents real-world results from both MPC and GSVD-based controllers, demonstrating that the proposed approaches meet theoretical expectations with respect to performance and robustness in practice. The results from this thesis, and in particular the novel GSVD-based method, were successfully adopted for the Diamond-II upgrade. This may enable the use of more advanced control systems in similar large-scale and high-speed applications in the future