5,801 research outputs found

    Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

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    Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances

    Novel 129Xe Magnetic Resonance Imaging and Spectroscopy Measurements of Pulmonary Gas-Exchange

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    Gas-exchange is the primary function of the lungs and involves removing carbon dioxide from the body and exchanging it within the alveoli for inhaled oxygen. Several different pulmonary, cardiac and cardiovascular abnormalities have negative effects on pulmonary gas-exchange. Unfortunately, clinical tests do not always pinpoint the problem; sensitive and specific measurements are needed to probe the individual components participating in gas-exchange for a better understanding of pathophysiology, disease progression and response to therapy. In vivo Xenon-129 gas-exchange magnetic resonance imaging (129Xe gas-exchange MRI) has the potential to overcome these challenges. When participants inhale hyperpolarized 129Xe gas, it has different MR spectral properties as a gas, as it diffuses through the alveolar membrane and as it binds to red-blood-cells. 129Xe MR spectroscopy and imaging provides a way to tease out the different anatomic components of gas-exchange simultaneously and provides spatial information about where abnormalities may occur. In this thesis, I developed and applied 129Xe MR spectroscopy and imaging to measure gas-exchange in the lungs alongside other clinical and imaging measurements. I measured 129Xe gas-exchange in asymptomatic congenital heart disease and in prospective, controlled studies of long-COVID. I also developed mathematical tools to model 129Xe MR signals during acquisition and reconstruction. The insights gained from my work underscore the potential for 129Xe gas-exchange MRI biomarkers towards a better understanding of cardiopulmonary disease. My work also provides a way to generate a deeper imaging and physiologic understanding of gas-exchange in vivo in healthy participants and patients with chronic lung and heart disease

    Modelling, Monitoring, Control and Optimization for Complex Industrial Processes

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    This reprint includes 22 research papers and an editorial, collected from the Special Issue "Modelling, Monitoring, Control and Optimization for Complex Industrial Processes", highlighting recent research advances and emerging research directions in complex industrial processes. This reprint aims to promote the research field and benefit the readers from both academic communities and industrial sectors

    Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

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    We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any n×nn \times n matrix pencil (A,B)(A,B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability as long as the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling (A,B)(A,B) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (A,B)(A,B) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible S,TS,T and diagonal DD such that ASDT12ε||A - SDT^{-1}||_2 \leq \varepsilon and BSIT12ε||B - SIT^{-1}||_2 \leq \varepsilon in at most O(log(n)log2(nε)TMM(n))O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right) operations, where TMM(n)T_{\text{MM}}(n) is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of exact arithmetic matrix pencil diagonalization.Comment: 58 pages, 8 figures, 2 table

    GMMap: Memory-Efficient Continuous Occupancy Map Using Gaussian Mixture Model

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    Energy consumption of memory accesses dominates the compute energy in energy-constrained robots which require a compact 3D map of the environment to achieve autonomy. Recent mapping frameworks only focused on reducing the map size while incurring significant memory usage during map construction due to multi-pass processing of each depth image. In this work, we present a memory-efficient continuous occupancy map, named GMMap, that accurately models the 3D environment using a Gaussian Mixture Model (GMM). Memory-efficient GMMap construction is enabled by the single-pass compression of depth images into local GMMs which are directly fused together into a globally-consistent map. By extending Gaussian Mixture Regression to model unexplored regions, occupancy probability is directly computed from Gaussians. Using a low-power ARM Cortex A57 CPU, GMMap can be constructed in real-time at up to 60 images per second. Compared with prior works, GMMap maintains high accuracy while reducing the map size by at least 56%, memory overhead by at least 88%, DRAM access by at least 78%, and energy consumption by at least 69%. Thus, GMMap enables real-time 3D mapping on energy-constrained robots.Comment: 15 pages, 9 figure

    Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients

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    Solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical results by providing a quasi-error reduction results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement

    The influence of vision on the perceptual compensation for reverberation in simulated environments

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    In typical listening environments, auditory signals arrive at the ear as a fusion of the direct energy from sound sources and the indirect reflections via reverberation. The listener thus cannot directly access the source and reverberation components individually, highlighting that the perceptual separation of these components can be subject to ambiguity. Accurate expectations of reverberation have been shown to reduce such ambiguity. The visible features of the physical environment (e.g., spatial and surface properties) can reveal aspects of reverberation that inform such expectations, suggesting an inferential role of vision in disambiguating the source and reverberation components. The aim of this thesis was to evaluate the degree to which visual information from simulated environments can affect the expectations of reverberation to consequently improve judgements of sound sources. To investigate this aim, we conducted three behavioural studies that assessed perception in audiovisual environments via online simulations created from a database of real-world locations. Chapter 3 assessed whether visual cues to the environment could inform of the reverberant properties of physical locations in an audiovisual congruence task. The results indicated a greater impression of congruence when reverberant cues were identical or similar to those represented by the depicted environment, demonstrating a capacity for vision to inform meaningful expectations of reverberation. Chapter 4 evaluated the degree to which vision contributed to the identification of speech sources within reverberation by prior exposure to visual environments. We found that exposure to the visual environment had hardly any effect on improving the identification of reverberant speech sources in this context. Chapter 5 investigated if a concurrent visual depiction of the environment would affect the tendency for estimates of sound source duration to be consistent despite varying reverberation. The results showed that source duration estimates were influenced by the degree of reverberation present, and were seemingly unaffected by any visual exposure. Taken together, the findings of this thesis suggest that scene understanding from vision contributes to the overall spatial understanding of environments and their reverberant properties, but appears to contribute little to enhancing the perceptual separation of source and reverberation components used to improve judgements of auditory sources

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum
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