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Ontologies for Models and Algorithms in Applied Mathematics and Related Disciplines
No full textIn applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies
Barrier Algorithms for constrained Non-Convex Optimization
No full textIn this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely O(ε−2) (first-order) and O(ε−3/2) (second-order), respectively
Quantification of Biophysical Parameters in Medical Imaging
No full textThe second edition of this book offers six new chapters covering the latest developments in quantitative medical imaging, including artificial intelligence, MRI mapping, sonography, elastography and cardiac CT. All the other existing chapters have been updated and expanded, many with new text and figures, to reflect the rapid translation and advancement of technology in this exciting area of biomedical research. This updated edition presents fundamental knowledge on the imaging quantification of biophysical parameters for clinical diagnostic purposes. Clinical imaging scanners are considered by the authors as physical measurement systems capable of quantifying intrinsic parameters for the representation of the constitution and biophysical properties of tissues in vivo. In one respect, this approach fosters the development of new imaging methods for highly reproducible, system-independent, and quantitative biomarkers. These methods are greatly detailed in the book. Alternatively, this new edition equips the reader with a better understanding of how the physical properties of tissues interact with signal generation in medical imaging, opening up new insights into the complex and fascinating relationship between structure and function in living tissues. This updated edition is of interest to all those who recognize the limitations of clinical diagnosis based primarily on visual inspection of images, and who wish to learn more about the diagnostic potential of quantitative, biophysically-based medical imaging markers, as well as the challenges posed by the scarcity of such markers for next-generation imaging technologies
A novel approach to hydrodynamics for long-range generalized exclusion
Get PDFWe consider a class of generalized long-range exclusion processes evolving either on or on a finite lattice with an open boundary. The jump rates are given in terms of a general kernel depending on both the departure and destination sites, and it is such that the particle displacement has an infinite expectation, but some tail bounds are satisfied. We study the superballisitic scaling limit of the particle density and prove that its space-time evolution is concentrated on the set of weak solutions to a non-local transport equation. Since the stationary states of the dynamics are unknown, we develop a new approach to such a limit relying only on the algebraic structure of the Markovian generator
Barrier algorithms for constrained non-convex optimization
Get PDFIn this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely (first-order) and (second-order), respectively
Investigating Apparent Differences between Standard DKI and Axisymmetric DKI and its Consequences for Biophysical Parameter Estimates.
No full textPurpose The purpose of the study is to identify differences between axisymmetric diffusion kurtosis imaging (DKI) and standard DKI, their consequences for biophysical parameter estimates, and the protocol choice influence on parameter estimation. Methods Noise-free and noisy, synthetic diffusion MRI human brain data is simulated using standard DKI for a standard and the fast “199” acquisition protocol. First the noise-free “baseline” difference between both DKI models is estimated and the influence of fiber complexity is investigated. Noisy data is used to establish the signal-to-noise ratio at which the baseline difference exceeds noise variability. The influence of protocol choices and denoising is investigated. The five axisymmetric DKI tensor metrics (AxTM), the parallel and perpendicular diffusivity and kurtosis and mean of the kurtosis tensor are used to compare both DKI models. Additionally, the baseline difference is also estimated for the five parameters of the WMTI-Watson model. Results The parallel and perpendicular kurtosis and all of the WMTI–Watson parameters had large baseline differences. Using a Westin or FA mask reduced the number of voxels with large baseline difference, that is, by selecting voxels with less complex fibers. For the noisy data, precision was worsened by the fast “199” protocol but adaptive denoising can help counteract these effects. Conclusion For the diffusivities and mean of the kurtosis tensor, axisymmetric DKI with a standard protocol delivers similar results as standard DKI. Fiber complexity is one main driver of the baseline differences. Using the “199” protocol worsens precision in noisy data but adaptive denoising mitigates these effects
Inf–Sup Stabilized Scott–Vogelius Pairs on General Shape-Regular Simplicial Grids by Raviart–Thomas Enrichment
No full textThis paper considers the discretization of the Stokes equations with Scott–Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf–sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order k of the Scott–Vogelius velocity space with appropriately chosen and explicitly given Raviart–Thomas bubbles. This approach is inspired by [X. Li and H. Rui, A low-order divergence-free H(div)-conforming finite element method for Stokes flows, IMA J. Numer. Anal.42 (2022) 3711–3734], where the case k=1 was studied. The proposed method is pressure-robust, with optimally converging H1-conforming velocity and a small H(div)-conforming correction rendering the full velocity divergence-free. For k≥d, with d being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart–Thomas enrichment and also all non-constant pressure degrees of freedom can be eliminated, effectively leading to a pressure-robust, inf–sup stable, optimally convergent Pk×P0 scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results
Gibbs Measures for Hardcore-Solid-on-Solid Models on Cayley Trees
No full textWe investigate the finite-state p-solid-on-solid (p-SOS) model for on Cayley trees of order and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as . This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p > 0 and, in the case of k = 2, we demonstrate that, on the level of the functional equations, the transition is continuous
Optimizing the Economic Dispatch of Weakly-Connected Mini-Grids under Uncertainty using Joint Chance Constraints
No full textIn this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classical power system’s operation tasks. Three alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The results show the dispatch modifications for battery and Diesel reserve planning, with the JCC model providing the most robust results, albeit with a small increase in costs
Rectifiable Paths with Polynomial Log‐Signature are Straight Lines
No full textThe signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log-signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log-signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log-signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization