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Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number
Get PDFThe most popular method for computing the matrix logarithm is the
inverse scaling and squaring method, which is the basis of the recent
algorithm of [A. H. Al-Mohy and N. J. Higham, \emph{Improved inverse
scaling and squaring algorithms for the matrix logarithm}, SIAM J.
Sci.\ Comput., 34 (2012), pp.~C152--C169].
For real matrices
we develop a version of the latter algorithm
that works entirely in real arithmetic
and is twice as fast as and more accurate than the original algorithm.
We show that by differentiating the algorithms we obtain backward stable
algorithms for computing the Fr\'echet derivative.
We demonstrate experimentally that our two algorithms are more
accurate and efficient than existing algorithms for computing the
Fr\'echet derivative
and we also show how the algorithms can be used to produce reliable
estimates of the condition number of the matrix logarithm
Theoretical study of an inviscid transonic flow near a discontinuity in wall curvature (Part 2)
Get PDFThis work is Part 2 of the extensive theoretical study of an inviscid transonic flow near a discontinuity in wall
curvature, focusing on the flow regimes with the \emph{limiting characteristic}. The closed-form solutions obtained in Part 1 using the \emph{hodograph method} are extended beyond the limiting characteristic by means of analytical continuation, and the results are analysed using the \emph{phase portrait technique}. It is shown that if the flow can be extended beyond the limiting characteristic, it subsequently develops a shock wave. This leads to multiple solutions and a wider variety of flow regimes, including the ones with a concave downstream wall. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem
Shape Deformation in Two-Dimensional Electrical Impedance Tomography
Get PDFElectrical Impedance Tomography (EIT) uses mea- surements from surface electrodes to reconstruct an image of the conductivity of the contained medium. However, changes in measurements result from both changes in internal conductivity and changes in the shape of the medium relative to the electrode positions. Failure to account for shape changes results in a conductivity image with significant artifacts. Previous work to address shape changes in EIT has shown that in some cases boundary shape and electrode location can be uniquely deter- mined for isotropic conductivities; however, for geometrically conformal changes, this is not possible. This prior work has shown that the shape change problem can be partially addressed. In this paper, we explore the limits of compensation for boundary movement in EIT using three approaches: first, a theoretical model was developed to separate a deformation vector field into conformal and non-conformal components, from which the reconstruction limits may be determined; next, finite element models were used to simulate EIT measurements from a domain whose boundary has been deformed; finally, an experimental phantom was constructed from which boundary deformation measurements were acquired. Results, both in simulation and with experimental data, suggest that some electrode movement and boundary distortions can be reconstructed based on conduc- tivity changes alone while reducing image artifacts in the process
Breadth of cellular and humoral immune responses elicited in rhesus monkeys by multi-valent mosaic and consensus immunogens
No full textTo create an HIV-1 vaccine that generates sufficient breadth of immune recognition to protect against the genetically diverse forms of the circulating virus, we have been exploring vaccines based on consensus and mosaic protein designs. Increasing the valency of a mosaic immunogen cocktail increases epitope coverage but with diminishing returns, as increasingly rare epitopes are incorporated into the mosaic proteins. In this study we compared the immunogenicity of 2-valent and 3-valent HIV-1 envelope mosaic immunogens in rhesus monkeys. Immunizations with the 3-valent mosaic immunogens resulted in a modest increase in the breadth of vaccine-elicited T lymphocyte responses compared to the 2-valent mosaic immunogens. However, the 3-valent mosaic immunogens elicited significantly higher neutralizing responses to Tier 1 viruses than the 2-valent mosaic immunogens. These findings underscore the potential utility of polyvalent mosaic immunogens for eliciting both cellular and humoral immune responses to HIV-1
Yeast 5 - an Expanded Reconstruction of the Saccharomyces Cerevisiae Metabolic Network
Get PDFBackground: Efforts to improve the computational reconstruction of the Saccharomyces cerevisiae biochemical reaction network and to refine the stoichiometrically constrained metabolic models that can be derived from such a reconstruction have continued since the first stoichiometrically constrained yeast genome scale metabolic model was published in 2003. Continuing this ongoing process, we have constructed an update to the Yeast Consensus Reconstruction, Yeast 5. The Yeast Consensus Reconstruction is a product of efforts to forge a community-based reconstruction emphasizing standards compliance and biochemical accuracy via evidence-based selection of reactions. It draws upon models published by a variety of independent research groups as well as information obtained from biochemical databases and primary literature.
Results: Yeast 5 refines the biochemical reactions included in the reconstruction, particularly reactions involved in sphingolipid metabolism; updates gene-reaction annotations; and emphasizes the distinction between reconstruction and stoichiometrically constrained model. Although it was not a primary goal, this update also improves the accuracy of model prediction of viability and auxotrophy phenotypes and increases the number of epistatic interactions. This update maintains an emphasis on standards compliance, unambiguous metabolite naming, and computer-readable annotations available through a structured document format. Additionally, we have developed MATLAB scripts to evaluate the model�s predictive accuracy and to demonstrate basic model applications, such as simulating aerobic and anaerobic growth. These scripts, which provide an independent tool for evaluating the performance of various
stoichiometrically constrained yeast metabolic models using flux balance analysis, are included as additional files.
Conclusions: Yeast 5 expands and refines the computational reconstruction of yeast metabolism and improves the predictive accuracy of a stoichiometrically constrained yeast metabolic model. It differs from previous reconstructions and models by emphasizing the distinction between the yeast metabolic reconstruction and the stoichiometrically constrained model, and makes both available as additional files and at http://yeast.sf.net/ as separate systems biology markup language (SBML) files. Through this separation, we intend to make the modeling process more accessible, explicit, transparent, and reproducible
Mechanistic modelling of cancer: some reflections from software engineering and philosophy of science
Get PDFThere is a growing interest in mathematical mechanistic modelling as a promising strategy for understanding tumour progression. This approach is accompanied by a methodological change of making research, in which models help to actively generate hypotheses instead of waiting for general principles to become apparent once sufficient data are accumulated. This paper applies recent research from philosophy of science to uncover three important problems of mechanistic modelling which may compromise its mainstream application, namely: the dilemma of formal and informal descriptions, the need to express degrees of confidence and the need of an argumentation framework. We report experience and research on similar problems from software engineering and provide evidence that the solutions adopted there can be transferred to the biological domain. We hope this paper can provoke new opportunities for further and profitable interdisciplinary research in the field
Unsteady turbulent plume models
No full textFour existing integral models of unsteady turbulent plumes are revisited. We demonstrate that none of these published models is ideal for general descriptions of unsteady behaviour and put forward a modified model. We show that the most recent (top-hat) plume model (Scase et al. J. Fluid Mech., vol. 563, 2006, p. 443), and the earlier (Gaussian) plume models (Delichatsios J. Fluid Mech., vol. 93, 1979, p. 241; Yu Trans. ASME, vol. 112, 1990, p.186), are all ill-posed. This ill-posedness arises from the downstream growth of short-scale waves, which have an unbounded downstream growth rate. We show that both the top-hat and the Gaussian (Yu) models can be regularized, rendering them well-posed, by the inclusion of a velocity diffusion term. The effect of including this diffusive mechanism is to include a vertical structure in the model that can be interpreted as representing the vertical extent of an eddy. The effects of this additional mechanism are small for steady applications, and cases where the plume forcing can be considered to follow a power law (both of which have been studied extensively). However, the inclusion of diffusion is shown to be crucial to the general initial-value problem for unsteady models
Purity relative to classes of finitely presented modules
Get PDFWe investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure-injectives which we make completely explicit in the case of tame hereditary algebras
Families of piecewise linear maps with constant Lyapunov exponent
Get PDFWe consider families of piecewise linear maps in which the moduli of the two slopes take different values. In some parameter regions, despite the variations in the dynamics, the Lyapunov exponent and the topological entropy remain constant. We provide numerical evidence of this fact and we prove it analytically for some special cases. The mechanism is very different from that of the logistic map and we conjecture that the Lyapunov plateaus reflect arithmetic relations between the slopes
Performance analysis of asynchronous Jacobi's method implemented in MPI, SHMEM and OpenMP
Get PDFEver-increasing core counts create the need to develop parallel algorithms that avoid closely- coupled execution across all cores. In this paper we present performance analysis of several parallel asynchronous implementations of Jacobi's method for solving systems of linear equations, using MPI, SHMEM and OpenMP. In particular we have solved systems of over 4 billion unknowns using up to 32,768 processes on a Cray XE6 supercomputer. We show that the precise implementation details of asynchronous algorithms can strongly affect the resulting performance and convergence behaviour of our solvers in unexpected ways